diff --git "a/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt" "b/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9NE4T4oBgHgl3EQf3Q1N/content/tmp_files/load_file.txt" @@ -0,0 +1,842 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf,len=841 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='05304v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='RT] 12 Jan 2023 A characterization of the L2-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces Abdelhamid Boussejra ∗Achraf Ouald Chaib† Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Faculty of Sciences University Ibn Tofail,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Kénitra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Morocco Abstract We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles Sp(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1)×Sp(n)×Sp(1) Vτ over the quaternionic hyperbolic spaces Sp(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1)/Sp(n)× Sp(1) associated with irreducible representations τ of Sp(n)×Sp(1) which are trivial on Sp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As a consequence, we describe the image of the section space L2(Sp(n, 1)×Sp(n)×Sp(1) Vτ) under the generalized spectral projections associated to a family of eigensections of the Casimir operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Keywords: Vector Poisson transform, Fourier restriction estimate, Strichartz conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1 Introduction Let G be a connected real semisimple noncompact Lie group with finite center, and K a maximal compact subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then X = G/K is a Riemannian symmetric space of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let G = KAN be an Iwasawa decomposition of G, and let M be the centralizer of A in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We write g = κ(g)eH(g)n(g), for each g ∈ G according to G = KAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' A central result in harmonic analysis (see [17]) asserts that all joint eigenfunctions F of the algebra D(X) of invariant differential operators, are Poisson integrals F(g) = Pλf(g) := � K e(iλ+ρ)H(g−1k)f(k) dk, of a hyperfunction f on K/M, for a generic λ ∈ a∗ c (the complexification of a∗ the real dual of a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since then a characterization of the Lp-range of the Poisson transform was developed in several articles such as [3], [5], [6], [7], [15], [20], [21], [22], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The problem of characterizing the image of the Poisson transform Pλ of L2(K/M) with real and regular spectral parameter λ is intimately related to Strichartz conjecture [[25], Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5] on the uniform L2-boundedness of the generalized spectral projections associated with D(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To be more specific, consider the generalized spectral projections Qλ defined initially for F ∈ C∞ c (X) by QλF(x) =| c(λ) |−2 Pλ(FF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' )(x), λ ∈ a∗, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) where FF is the Helgason Fourier transform of F and c(λ) is the Harish-Chandra c-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Conjecture (Strichartz [[25], Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' There exists a positive constant C such that for any Fλ = QλF with ∗e-mail: boussejra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='abdelhamid@uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='ma †e-mail:achraf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='oualdchaib@uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='ma 1 F ∈ L2(X) we have C−1 ∥ F ∥2 L2(X)≤ sup R>0,y∈X � a∗ + 1 Rr � B(y,R) | Fλ(x) |2 dx dλ ≤ C ∥ F ∥2 L2(X), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) and ∥ F ∥2 L2(X)= γr lim R→∞ � a∗ + 1 Rr � B(y,R) | Fλ(x) |2 dx dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) Conversely, if Fλ is any family of joint eigenfunctions for which the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) is finite, then there exists F ∈ L2(X) such that Fλ = QλF for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' λ ∈ a∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Here r = rank X, and B(y, R) denotes the open ball in X of radius R about y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The constant γr depends on the normalizations of the measures dx and dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The strichartz conjecture has been recently settled by Kaizuka, see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Most of the proof consists in proving a uniform estimate for the Poisson transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' More precisely, the following was proved by Kaizuka [[16], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3]: Let F be a joint eigenfunction with eigenvalue corresponding to a real and regular spectral parameter λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then F is the Poisson transform by Pλ of some f ∈ L2(K/M) if and only if sup R>1 1 Rr � B(0,R) | F(x) |2 dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Moreover there exists a positive constant C independent of such λ, C−1 | c(λ) |2∥ f ∥2 L2(K/M)≤ sup R>1 1 Rr � B(0,R) | Pλf(x) |2 dx ≤ C | c(λ) |2∥ f ∥2 L2(K/M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The generalization of these results to vector bundles setting has only just begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In [8] we extend Kaizuka result to homogeneous line bundles over non-compact complex Grassmann manifolds (See also [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Our aim in this paper is to generalize theses results to a class of homogeneous vector bundles over the quaternionic hyperbolic space G/K, where G is the symplectic group Sp(n, 1) with maximal compact subgroup K = Sp(n)×Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To state our results in rough form, let us first introduce the class of the homogenous vector bundles that we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let τν be a unitary irreducible representation of Sp(1) realized on a (ν + 1)-dimensional Hilbert space (V, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' )ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We extend τν to a representation of K by setting τν ≡ 1 on Sp(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As usual the space of sections of the homogeneous vector bundle G ×K V associated with τν will be identified with the space Γ(G, τν) of vector valued functions F : G → Vν which are right K-covariant of type τν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=', F(gk) = τν(k)−1F(g), ∀g ∈ G, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) We denote by C∞(G, τν) and C∞ c (G, τν) the elements of Γ(G, τν) that are respectively smooth, smooth with compact support in G, and by L2(G, τν) the elements of Γ(G, τν) such that ∥ F ∥L2(G,τν)= �� G/K ∥ F(g) ∥2 ν dgK � 1 2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In above ∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ∥ν is the norm in Vν and ∥ F(gK) ∥ν=∥ F(g) ∥ν is well defined for F satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let σν denote the restriction of τν to the group M ≃ Sp(n−1)×Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Over K/M we have the associated homogeneous vector bundle K ×M Vν with L2-sections identified with L2(K, σν) the space of all functions f : K → Vν which are M-covariant of type σν and satisfy ∥ f ∥2 L2(K,σν)= � K ∥ f(k) ∥2 ν dk < ∞, 2 where dk is the normalized Haar measure of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For λ ∈ C and f ∈ L2(K, σν), the Poisson transform Pν λf is defined by Pν λf(g) = � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k))f(k) dk Let Ω denote the Casimir element of the Lie algebra g of G, viewed as a differential operator acting on C∞(G, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then the image Pν λ(L2(K, σν)) is a proper closed subspace of Eλ(G, τν) the space of all F ∈ C∞(G, τν) satisfying Ω F = −(λ2 + ρ2 − ν(ν + 2))F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For more details see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For λ ∈ R \\ {0}, we define a weighted L2-space E2 λ(G, τν) consisting of all F in Eλ(G, τν) that satisfy ∥ F ∥∗= sup R>1 � 1 R � B(R) ∥F(g)∥2 ν dgK � 1 2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Our first main result is an image characterization of the Poisson transform Pν λ of L2(K, σν) for λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let λ ∈ R\\{0} and ν a nonnegative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) There exists a positive constant Cν independent of λ such that for f ∈ L2(K, σν) we have C−1 ν |cν(λ)| ∥f∥L2(K,σν) ≤ ∥Pν λf∥∗ ≤ Cν| | cν(λ) | ∥f∥L2(K,σν), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) with cν(λ) = 2ρ−iλ Γ(ρ − 1)Γ(iλ) Γ( iλ+ν+ρ 2 )Γ( iλ+ρ−ν−2 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Furthermore we have the following Plancherel type formula for the Poisson transform lim R→+∞ 1 R � B(R) ∥Pν λf(g)∥2 ν dgK = 2 | cν(λ) |2 ∥f∥2 L2(K,σν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) ii) Pν λ is a topological isomorphism from L2(K, σν) onto E2 λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This generalizes the result of Kaizuka [[16], (i) and (ii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3] which corresponds to τν trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Consequence For λ ∈ R we define the space E∗ λ(G, τν) = {F ∈ Eλ(G, τν) : M(F) < ∞}, where M(F) = lim sup R→∞ � 1 R � B(R) | F(g) |2 dgK � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then as an immediate consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 we obtain the following result which generalizes a conjecture of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Bray [10] which corresponds to τν trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' If λ ∈ R \\ {0} then E∗ λ(G, τν), M) is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In the case of the trivial bundle (the scalar case) the conjecture of Bray was proved by Ionescu [15] for all rank one symmetric spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It was generalized to Riemannian symmetric spaces of higher rank by Kaizuka, see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 3 Next, let us introduce our second main result on the L2-range of the generalized spectral projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For F ∈ C∞ c (G, τν) the vector valued Helgason-Fourier transform FνF is given by (see [11]) Fν F(λ, k) = � G e(iλ−ρ)H(g−1k)τν(κ(g−1k)−1)F(g) dg λ ∈ C, Then the following inversion formula holds (see section 4) F(g) = 1 2π � ∞ 0 � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) | cν(λ) |−2 dλ dk + � λj∈Dν dν(λj) � K e−(iλj+ρ)H(g−1k)τν(κ(g−1k))FνF(λj, k) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) In above dν(λ) = −iResµ=λ(cν(µ)cν(−µ))−1, λ ∈ Dν and Dν is a finite set in {λ ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ℑ(λ) > 0} which parametrizes the τν-spherical functions arising from the discrete series of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It is empty if ν ≤ ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) gives rise to the decomposition of L2(G, τν) into a continuous part and a discrete part: L2(G, τν) = L2 cont(G, τν) ⊕ L2 disc(G, τν) Our aim here is to study the operator Qν λ, λ ∈ R, defined for F ∈ L2 cont(G, τν) ∩ C∞ c (C, τν) by Qν λF(g) =| cν(λ) |−2 Pν λ[Fν F(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' )](g), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='8) More precisely, following Strichartz idea, we are interested in the following question: Characterize those Fλ ∈ Eλ(G, τν) (λ ∈ (0, ∞)) for which there exists F ∈ L2 cont(G, τν) such that Fλ = Qν λF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To do so, we introduce the space E2 +(G, τν) consisting of all Vτν-valued measurable functions ψ on (0, ∞) × G such that (i) Ω ψ(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=') = −(λ2 + ρ2 − ν(ν + 2)) ψ(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=') a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' λ ∈ (0, ∞) (ii) ∥ ψ ∥+< ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' where ∥ ψ ∥2 += sup R>1 � ∞ 0 1 R � B(R) ∥ ψ(λ, g) ∥2 ν dgK dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The second main result we prove in this paper can be stated as follows Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) There exists a positive constant C such that for F ∈ L2(G, τν) we have C−1 ∥ F ∥L2(G,τν)≤∥ Qν λF ∥+≤ C ∥ F ∥L2(G,τν) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='9) Furthermore we have lim R→∞ � ∞ 0 1 R � B(R) ∥ Qν λF ∥2 ν dgK dλ = 2 ∥ F ∥2 L2(G,τν) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='10) (ii) The linear map Qν λ is a topological isomorphism from L2 cont(G, τν) onto E2 +(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This extends Kaizuka result [ [16], (i) and (ii) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6] on the Strichartz conjecture (see [25] Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5] to the class of vector bundles considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Before giving the outline of the paper, let us mention that a number of authors have obtained an image characterization for the Poisson transform Pλ (λ ∈ a∗ \\ {0}) of L2-functions on K/M in the rank one case, see [[3], [5], [7], [15]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Nevertheless, the obtained characterization is weaker than the one conjectured by Strichartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The approach taken in 4 the quoted papers is based on the theory of Calderon-Zygmund singular integrals (see also [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Using a different approach based on the techniques used in the scattering theory, Kaizuka [16] settled the Strichartz conjecture on Riemannian symmetric spaces of noncompact type, of arbitrary rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now describe the contents of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proofs of our results are a generalisation of Kaizuka’s method [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In section 2 we recall some basic facts on the quaternionc hyperbolic spaces and introduce the vector Poisson transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In section 3, we define the Helgason-Fourier transform on the vector bundles G ×K Vν and give the inversion and Plancherel Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2 follows from the Plancherel formula and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The main ingredients in proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 are a Fourier restriction estimate for the vector valued Helgason-Fourier transform (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 in section 4) and an asymptotic formula for the vector Poisson transform in the framework of Agmon- Hörmander spaces [2] (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 will be derived from the Key lemma of this paper giving the asymptotic behaviour of the translate of the τν-spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Section 6 is devoted to the proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In section 7 we prove the Key Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 The quaternionic hyperbolic space Let G = Sp(n, 1) be the group of all linear transformations of the right H-vector space Hn+1 which preserve the quadratic form n � j=1 | uj |2 − | un+1 |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let K = Sp(n) × Sp(1) be the subgroup of G consisting of pairs (a, d) of unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then K is a maximal compact subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The quaternionic hyperbolic space is the rank one symmetric space G/K of the noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It can be realized as the unit ball B(Hn) = {x ∈ Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' | x |< 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The group G acts on B(Hn) by the fractional linear mappings x �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='x = (ax + b)(cx + d)−1, if g = � a b c d � , with a ∈ Hn×n, b ∈ Hn×1, c ∈ H1×n and d ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Denote by g the Lie algebra of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' g = k ⊕ p the Cartan decomposition of g, where p is a vector space of matrices of the form �� 0 x x∗ 0 � , x ∈ Hn � , and k = �� X 0 0 q � , X∗ + X = 0, q + q = 0 � , where X∗ is the conjugate transpose of the matrix X and q ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let H = � 0n e1 te1 0 � ∈ p with te1 = (1, 0, · · · , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then a = R H is a Cartan subspace in p, and the corresponding analytic subgroup A = {at = exp t H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' t ∈ R}, where at = \uf8eb \uf8ec \uf8ed cht 0 sht 0 0n−1 0 sht 0 cht \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' With A determined we then have that M = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 g = \uf8eb \uf8ec \uf8ed q 0 0 0 m 0 0 0 q \uf8f6 \uf8f7 \uf8f8 , m ∈ Sp(n − 1), | q |= 1 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe ≃ Sp(n − 1) × Sp(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let α ∈ a∗ be defined by α(H) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then a system Σ of restricted roots of the pair (g, a) is Σ = {±α, ±2α} if n ≥ 2 and Σ = {±2α} if n = 1, with Weyl group W ≃ {±Id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' A positive subsystem of roots corresponding to the positive Weyl chamber a+ ≃ (0, ∞) in a is Σ+ = {α, 2α} if n ≥ 2 and Σ+ = {2α} if n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let n = gα + g2α be the direct sum of the positive root subspaces, with dim gα = 4(n − 1) and dim g2α = 3 and N the corresponding analytic subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then the half sum of the positive restricted roots with multiplicities counted ρ equals to (2n + 1)α, and shall be viewed as a real number ρ = 2n + 1 by the identification a∗ c ≃ C via λα ↔ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let A+ = {at ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have the Cartan decomposition G = KA+K, that is any g ∈ G can be written g = k1(g) eA+(g) k2(g), k1(g), k2(g) ∈ K and A+(g) ∈ a+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 5 If we write g ∈ G in (n + 1) × (n + 1) block notation as g = � a b c d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then a straightforward computation gives cosh A+(g) =| d | and H(g) = log | ce1 + d | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) We normalize the invariant measure dgK on G/K so that the following integral formula holds: for all h ∈ L1(G/K), � G/K h(gK)dgK = � G h(g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0)dg = � K � ∞ 0 h(k at)∆(t) dk dt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) where dt is the Lebesgue measure, ∆(t) = (2 sinh t)4n−1(2 cosh t)3, and dk is the Haar measure of K with � K dk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2 The vector Poisson transform In this subsection we define the Poisson transform associated to the vector bundles G×KVν over Sp(n, 1)/Sp(n)×Sp(1) and derive some results referring to [23], [27], and [28] for more informations on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let σν denote the restriction of τν to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For λ ∈ C we consider the representation σν,λ of P = MAN on Vν defined by σν,λ(man) = aρ−iλσν(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then σν,λ defines a principal series representations of G on the Hilbert space Hν,λ := {f : G → Vν | f(gman) = σ−1 ν,λ(man)f(g) ∀man ∈ MAN, f|K ∈ L2}, where G acts by the left regular representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We shall denote by C−ω(G, σν,λ) the space of its hyperfunctions vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By the Iwasawa decomposition, the restriction map from G to K gives an isomorphism from Hν,λ onto the space L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This yields, the so-called compact picture of Hν,λ, with the group action given by πσν,λ(g)f(k) = e(iλ−ρ)H(g−1k)f(κ(g−1k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By C−ω(K, σν) we denote the space of its hyperfunctions vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' A Poisson transform is the continuous, linear, G-equivariant map Pν λ from C−ω(G, σν,λ) to C∞(G, τν) defined by Pν λ f(g) = � K τν(k)f(gk) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In the compact picture the Poisson transform is given by Pν λ f(g) = � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k)) f(k) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let D(G, τν) denote the algebra of left invariant differential operators on C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let Eν,λ(G) be the space of all F ∈ C∞(G, τν) such that Ω F = −(λ2 + ρ2 − ν(ν + 2)) F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) D(G, τν) is the algebra generated by the Casimir operator Ω of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) For λ ∈ C, ν ∈ N, the Poisson transform Pν λ maps C−ω(G, σν,λ) to Eν,λ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) Let U(a) be the universal enveloping algebra of the complexification of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since the restriction of τν to M is irreducible, then D(G, τν) ≃ U(a)W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As a is one dimensional, then D(G, τν) ≃ C[s2], symmetric functions of one variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus D(G, τν) is generated by the Casimir element Ω of the Lie algebra g of G, viewed as a differential operator acting on C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) Since σν is irreducible, the image of Pν λ consists of joint eigenfunctions with respect to the action of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Moreover Ω acts by the infinitesimal character of the the principal series representations πσν,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='22 and Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='28 in [18], that πσν,λ(Ω) = −(λ2 + ρ2 − c(σν))Id on C−ω(G, σν,λ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) where c(σν) is the Casimir value of σν given by c(σν) = ν(ν + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 6 Let Φν,λ be the τν-spherical function associated to σν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then Φν,λ admits the following Eisenstein integral repre- sentation (see [[11], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2]): Φν,λ(g) = � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k)k−1) dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Note that Φν,λ lies in C∞(G, τν, τν) the space of smooth functions F : G → End(Vτν) satisfying F(k1gk2) = τν(k−1 2 )F(g)τν(k−1 1 ), the so called τν-radial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Being τν-radial, Φν,λ is completely determined by its restriction to A, by the Cartan decomposition G = KAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Moreover, since σν is irreducible, it follows that Φν,λ(at) ∈ EndM(Vν) ≃ CIdVν, ∀at ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore there exists ϕν : R → C such that Φν,λ(at) = ϕν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='IdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We have ϕν,λ(t) = 1 ν + 1 � K e−(iλ+ρ)H(g−1k)χν(κ(g−1k)k−1) dk, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) where χν is the character of τν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This so-called trace τν-spherical function has been computed explicitly in [12] using the radial part of the Casimir operator Ω (see also [26] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We have ϕν,λ(t) = (cosh t)νφ(ρ−2,ν+1) λ (t), where φ(ρ−2,ν+1) λ (t) is the Jacobi function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' [19]) φ(ρ−2,ν+1) λ (t) = 2F1(iλ + ρ + ν 2 , −iλ + ρ + ν 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ρ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' − sinh2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We deduce from (A4) the asymptotic behaviour of ϕν,λ ϕλ,ν(at) = e(iλ−ρ)t[cν(λ) + ◦(1)], as t → ∞ if ℑ(λ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) where cν(λ) = 2ρ−iλΓ(ρ − 1)Γ(iλ) Γ( iλ+ρ+ν 2 )Γ( iλ+ρ−ν−2 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) For λ ∈ C the c-function of Harish-Chandra associated to τν is defined by c(τν, λ) = � N e−(iλ+ρ)H(n)τν(κ(n)) dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The integral converges for λ such that ℜ(iλ) > 0 and it has a meromorphic continuation to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In above dn is the Haar measure of N = θ(N), θ being the Cartan involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We may use formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) to give explicitly c(τν, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Indeed, one easily check that c(τν, λ) ∈ EndM(Vν) = CIdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then using the following result on the behaviour of Φν,λ(at) ([28], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) Φν,λ(at) = e(iλ−ρ)t(c(τν, λ) + ◦(1))as t → ∞, together with Φν,λ(at) = ϕν,λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='Id, we find then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) that c(τν, λ) = cν(λ)IdVν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We end this section by recalling a result of Olbrich [23] on the range of the Poisson transform on vector bundles which reads in our case as follows Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' [23] Let ν ∈ N and λ ∈ C such that (i) −2iλ /∈ N (ii) iλ + ρ /∈ −2N − ν ∪ −2N + ν + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then the Poisson transform Pν λ is a K-isomorphism from C−ω(K, σν) onto Eν,λ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 7 3 The vector-valued Helgason-Fourier transfrorm In this section we give the inversion and the Plancherel formulas for the Helgason-Fourier transform on the vector bundle G ×K Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' According to [11] the vector-valued Helgason-Fourier transform of f ∈ C∞ c (G, τν) is the Vν-valued function on C × K defined by: Fνf(λ, k) = � G eλ,ν(k−1g) f(g)dg, where eλ,ν is the vector valued function eλ,ν : G → End(Vν) given by eλ,ν(g) = e(iλ−ρ)H(g−1)τ −1 ν (κ(g−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Notice that our sign on "λ" is the opposite of the one in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In order to state the next theorem, we introduce the finite set in {λ, ℑ(λ) ≥ 0} Dν = {λj = i(ν − ρ + 2 − 2j), j = 0, 1, · · · , ν − ρ + 2 − 2j > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Note that Dν is empty if ν ≤ ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It parametrizes the discrete series representation of G containing τν, see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let dν(λj) = 2−2(ρ−ν−1)(ν − ρ − 2j + 2)(ρ − 2 + j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ν − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Γ2(ρ − 1)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ν − ρ − j + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' , λj ∈ Dν For λj ∈ Dν, we define the operators Qν j L2(G, τν) → Eν,λj(G, τν) F �→ dν(λj) Φν,λj ∗ F We denote the image by A2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We set L2 disc(G, τν) = � j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ν−ρ+2−2j>0 A2 j, and denote by L2 cont(G, τν) its orthocomplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let L2 σν(R+ × K, | cν(λ) |−2 dλ dk) be the space of vector functions φ : R+ × K → Vν satisfying (i) For each fixed λ, φ(λ, km) = σν(m)−1φ(λ, k), ∀m ∈ M (ii) � R+×K ∥ Fνφ(λ, k) ∥2 | cν(λ) |−2 dλ dk < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) For F ∈ C∞ c (G, τν) we have the following inversion and Plancherel formulas F(g) = 1 2π � ∞ 0 � K e∗ λ,ν(k−1g)FνF(λ, k) | cν(λ) |−2 dλ dk + � λj∈Dν dν(λj) � K e∗ λj,ν(k−1g)FνF(λj, k) dk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) � G ∥ F(g) ∥2 ν dgK = 1 2π � ∞ 0 � K ∥ FνF((λ, k) ∥2 ν| cν(λ) |−2 dλ dk+ � λj∈Dν dν(λj) � K < FνF(λj, k), FνF(−λj, k) >ν dk (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) (ii) The Fourier transform Fν extends to an isometry from L2 cont(G, τν) onto the space L2 σν(R+ ×K, | cν(λ) |−2 dλ dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The first part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 can be easily deduced from the inversion and Plancherel formulas for the spherical transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 8 Let C∞ c (G, τν, τν) denote the space of smooth compactly supported τν-radial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The spherical transform of F ∈ C∞ c (G, τν, τν) is the C-valued function HνF defined by: HνF(λ) = 1 ν + 1 � G T r[Φν,λ(g−1)F(g))]dg, λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The inversion and the Plancherel formulas for the τ-spherical transform have been given explicitly in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For the convenience of the reader we give an elementary proof by using the Jacobi transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For F ∈ C∞ c (G, τν, τν) we have the following inversion and Plancherel formulas F(g) = 1 2π � +∞ 0 Φν,λ(g)HνF(λ) | cν(λ) |−2 dλ + � λj∈Dν Φν,λj(g)Hνf(λj) dν(λj), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) � G ∥ F(g) ∥2 HS dg = ν + 1 2π � +∞ 0 | HνF((λ) |2| cν(λ) |−2 dλ + (ν + 1) � λj∈Dν dν(λj) | HνF((λj) |2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) In above ∥ ∥HS stands for the Hilbert-Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ C∞ c (G, τν, τν) and let fν be its scalar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Using the integral formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2), the identity Φν,λ(at) = Φν,λ(a−t) = (cosh t)νφ(ρ−2,ν+1) λ (t) and the fact that ∆(t) = (2 cosh t)−2ν∆ρ−2,ν+1, we have HνF(λ) = � ∞ 0 fν(t)(cosh t)νφ(ρ−2,ν+1) λ (t) ∆(t) dt = � ∞ 0 fν(t)(22 cosh t)−νφ(ρ−2,ν+1) λ (t) ∆ρ−2,ν+1(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) Thus the τν-spherical transform HνF may be written in terms of the Jacobi transform J α,β, with α = ρ − 2 and β = ν + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Namely, we have HνF(λ) = J ρ−2,ν+1[(22 cosh t)−νfν](λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We refer to (A5) in the Appendix for the definition of the Jacobi transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Now the theorem follows from the inversion and the Plancherel formulas for the Jacobi transform (A6), (A6’) and (A7) in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For the proof of the surjectivity statement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 we shall need the following result Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ C∞ c (G, τν) and Φ ∈ C∞(G, τν, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have Fν(F ∗ Φ)(λ, k) = HνΦ(λ)FνF(λ, k), λ ∈ C, k ∈ K, where the convolution is defined by (Φ ∗ F)(g) = � G Φν,λ(x−1g)F(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let Φ ∈ C∞(G, τν, τν), v ∈ Vν, and set Fv = Φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' )v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have the following relation between the Fourier transform and the spherical transform FνFv(λ, k) = HνΦ(λ)τ(k−1)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) By definition Fν(F ∗ Φ)(λ, k) = � G � G eν λ(k−1g)Φ(x−1g)F(x)dxdg = � G dx � G eν λ(k−1xy)Φ(y)F(x)dy 9 Using the following cocycle relations for the Iwasawa function H(x) H(xy) = H(xκ(y)) + H(y), and κ(xy) = κ(xκ(y)), for all x, y ∈ G, we get the following identity eν λ(k−1xy) = e(iλ−ρ)H(x−1k)eν λ(κ−1(x−1k)y), from which we obtain Fν(Φ ∗ F)(λ, k) = � G e(iλ−ρ)H(x−1k) �� G eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Next, put hv(y) = Φ(y)v, v ∈ Vτν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) implies � G eλ,ν(κ−1(x−1k)y)Φ(y)F(x) dy = Fν(hF (x))(λ, κ−1(x−1k)) = H(Φ)(λ)τν(κ−1(x−1k))F(x), from which we deduce Fν(Φ ∗ F)(λ, k) = H(Φ)(λ) � G e(iλ−ρ)H(x−1k)τν(κ−1(x−1k))F(x)dx, and the proposition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now come to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) We may follow the same method as in [11] to prove the inversion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) and the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We give an outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ C∞ c (G, τν) and consider the τν-radial function defined for any g ∈ G by Fg,v(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='w = � K < τν(k)w, v >ν F(gkx) dk, v being a fixed vector in Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then a straightforward calculation shows that HνFg,v(λ) = 1 ν + 1 < (Φν,λ ∗ F)(g), v >ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The inversion formula for the spherical transform together with T rFg,v(e) =< F(g), v >ν imply F(g) = 1 2π � ∞ 0 (Φν,λ ∗ F)(g) | cν(λ) |−2 dλ + � λj∈Dν (Φν,λj ∗ F)(g)dν(λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To conclude use the following result for the translated spherical function ( see [11] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) Φν,λ(x−1y) = � K e−(iλ+ρ)H(y−1k)e(iλ−rho)H(x−1k)τν(κ(y−1k))τν(κ−1(x−1k)) dk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) to get (Φν,λ ∗ F)(g) = � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k))FνF(λ, k) dk, 10 and the inversion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proof of the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) is essentially the same as in the scalar case, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Note that as a consequence of the Plancherel formula not involving the discrete series, we have � G ∥ F(g) ∥2 dgK = 1 π � ∞ 0 � K ∥ FνF(λ, k) ∥2 | cν(λ) |−2 dλ dk, for every F ∈ L2 cont(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) We prove the surjectivity statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Suppose that there exists a function f in L2 σν(R+ × K, | cν(λ) |−2 dλ dk) such that � ∞ 0 � K < f(λ, k), FνF(λ, k) >| cν(λ) |−2 dλ dk = 0 for all F ∈ C∞ c (G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Changing F into F ∗ Φ where Φ ∈ C∞(G, τν, τν) and using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1, we have � ∞ 0 � K < f(λ, k), FνF(λ, k) > Hνφ(λ) | cν(λ) |−2 dλ dk = 0 By the Stone-Weierstrass theorem, the algebra {HνΦ, Φ ∈ C∞(G, τν, τν)} is dense in C∞ e (R) the space of even continuous functions on R vanishing at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore for every F ∈ C∞ c (G, τν) there is a set EF of measure zero in R such that � K < f(λ, k), FνF(λ, k) > dk = 0 for all λ not in EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The rest of the proof is based on an adaptation of the arguments given in [14] Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5, for the scalar case, and the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 4 Fourier restriction estimate The main result of this section is the following uniform continuity estimate for the Fourier-Helgason restriction operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' There exists a positive constant Cν such that for λ ∈ R\\{0} and R > 1, we have � � K ∥FνF(λ, k)∥2 νdk �1/2 ≤ Cν|cν(λ)|R1/2 � � G/K ∥F(g)∥2 ν dgK �1/2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) for every F ∈ L2(G, τν) with suppF ⊂ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To prove this result we shall need estimates of the Harish-Chandra c-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To this end we introduce the function bν(λ) defined on R by bν(λ) = \uf8f1 \uf8f2 \uf8f3 cν(λ) if ν−ρ+2 2 ∈ Z+ λ cν(λ) if ν−ρ+2 2 /∈ Z+ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Assume ν > ρ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) The function bν(λ) has no zero in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) There exists a positive constant C such that for λ ∈ R, we have C−1(1 + λ2) 2ρ−4−ε(ν) 4 ≤| bν(λ) |−1≤ C(1 + λ2) 2ρ−4−ε(ν) 4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) 11 with ε(ν) = ±1 according to ν−ρ+2 2 /∈ Z+ or ν−ρ+2 2 ∈ Z+ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) If ν−ρ+2 2 /∈ Z+, then bν(λ) = 2ρ+ν−iλΓ(ρ−1)Γ(iλ+1) Γ( iλ+ρ+ν 2 )Γ( iλ+ρ−ν−2 2 ), and clearly bν(λ) has no zero on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' If ν−ρ+2 2 ∈ Z+ then bν(λ) a priori can have zero and pole at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This is not the case, since lim λ→0 bν(λ) = (−1) ν−ρ+2 2 2ρ+νΓ(ρ − 1)( ν−ρ+2 2 )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Γ( ρ+ν 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) To prove the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) we shall use the following property of the Γ-function lim |z|→∞ Γ(z + a) Γ(z) z−a = 1, | arg(z) |< π − δ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) where a is any complex number, and log is the principal value of the logarithm and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Assume first that ν−ρ+2 2 /∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Using the duplicata formula for the function gamma Γ(2z) = 22z−2 √π Γ(z)Γ(z + 1 2), we rewrite bν(λ) as bν(λ) = 2ρ+ν−1 √π Γ( iλ+1 2 )Γ( iλ+2 2 ) Γ( iλ+ρ+ν 2 )Γ( iλ+ρ−ν−2 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) that for every λ ∈ R, we have | bν(λ) |≤ C(1 + λ2)− 2ρ−5 4 and | bν(λ) |−1≤ C(1 + λ2) 2ρ−5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proof for the case ν−ρ+2 2 ∈ Z+ follows the same line as in the case ν−ρ+2 2 /∈ Z+, so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This finishes the proof of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let us recall from [1] an auxiliary lemma which will be useful for the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let η be a positive Schwartz function on R whose Fourier transform has a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For m ∈ R, set ηm(x) = � R η(t)(1 + |t − x|)m/2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' i) ηm is a positive C∞-function with C−1(1 + t2) m 2 ≤ ηm(t) ≤ C(1 + t2) m 2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ii) The Fourier transform of ηm has a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In order to prove the Fourier restriction Theorem, we need to introduce the bundle valued Radon transform, see [9] for more informations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The Radon transform for F ∈ C∞ c (G, τν) is defined by RF(g) = eρH(g) � N F(gn)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 12 We set RF(t, k) = RF(kat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then, using the Iwaswa decomposition G = NAK, we may rewrite the Helgason-Fourier transform as FνF(λ, k) = FR(RF(·, k))(λ), where FRφ(λ) = � R e−iλtφ(t) dt, is the Euclidean Fourier transform of φ a Vν-valued smooth function with compact support in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We define on p the scalar product < X, Y >= 1 2T r(XY ) and denote by | | the corresponding norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It induces a distance function d on G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By the Cartan decomposition G = K exp p, any g ∈ G may be written uniquely as g = k exp X, so that d(0, gK) =| X |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Define the open ball centred at 0 and of radius R by B(R) = {gK ∈ G/K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' d(0, gK) < R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ C∞ 0 (G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' If supp F ⊂ B(R), then supp RF ⊂ [−R, R] × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As (see [[13], page 476] d(0, ketHnK) ≥| t |, k ∈ K, n ∈ N, t ∈ R it follows that supp RF ⊂ [−R, R] × K if supp F ⊂ B(R) Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It suffices to prove the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) for functions F ∈ C∞ c (G, τν) supported in B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) that � B(R) ∥ F(g) ∥2 ν dgK ≥ � K � R ∥ FνF(λ, k) ∥2 ν | cν(λ) |−2 dλ dk Therefore it is sufficient to show � K � R ∥ FνF(λ, k) ∥2 ν | cν(λ) |−2 dλ dk ≥ C | cν(λ) |−2 R � R ∥ FνF(λ, k) ∥2 ν dk, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) fir some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) we have | cν(λ) |−1≍ η 2ρ−3 2 (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) is equivalent to η 2ρ−3 2 (λ) R � K ∥ FνF(λ, k) ∥2 ν dk ≤ � K � R ∥ FνF(λ, k) ∥2 ν η 2ρ−3 2 (λ)dλ dk (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) Let T be the tempered distribution on R defined by T := F−1 R η 2ρ−3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2, T is compactly supported .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let R0 > 1 such that supp T ⊂ [−R0, R0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) is equivalent to � K ∥ FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' , k))(λ) ∥2 ν dk ≤ CR � K � R FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' , k))(λ) ∥2 ν dλ dk, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) where ∗ denotes the convolution on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' From suppT ⊂ [−R0, R0] and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3, it follows that for any k ∈ K, supp (T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' , k)) ⊂ [−(R + R0), R + R0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus � K ∥ FR(T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' , k)(λ) ∥2 ν dk ≤ 2(R + R0) � K � R ∥ (T ∗ RF(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' k))(t) ∥2 ν dt dk Next use the Euclidean Plancherel formula to get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7), and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As a consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1, we obtain the uniform continuity estimate for the Poisson transform Pν λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' There exists a positive constant Cν such that for λ ∈ R\\{0}, we have sup R>1 � 1 R � B(R) ∥ Pν λf(g) ∥2 ν dgK �1/2 ≤ Cν |cν(λ)| ∥ f ∥L2(K,σν) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='8) for every f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ L2(G, τν) with supp F ⊂ B(R), and let f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since λ is real and τν is unitary, the Poisson transform and the restriction Fourier transform are related by the following formula � B(R) < Pν λf(g), F(g) >ν dg = � K < f(k), FνF(λ, k) >ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus | � B(R) < Pν λf(g), F(g) >ν dg | ≤ ∥f∥L2(K,σν)( � K ∥ FνF(λ, k) ∥2 ν dk) 1 2 ≤ Cν|cν(λ)|R1/2 ∥ f ∥L2(K,τν)∥ F ∥L2(G,τν), by the restriction Fourier theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Taking the supermum over all F with ∥ F ∥L2(G,τν)= 1, the corollary follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 5 Asymptotic expansion for the Poisson transform In this section we give an asymptotic expansion for the Poisson transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We first start by establishing some intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let L2 λ(K, σν) denote the finite linear span of the functions f g λ,v : k �−→ f g λ,v(k) = e(iλ−ρ)H(g−1k)τ −1 ν (κ(g−1k))v, g ∈ G, v ∈ Vν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For λ ∈ R \\ {0}, ν ∈ N the space L2 λ(K, σν) is a dense subspace of L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As λ ∈ R \\ {0}, the density is just a reformulation of the injectivity of the Poisson transform Pν,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let λ ∈ R \\ {0}, ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then there exists a unique unitary isomorphism U ν λ on L2(K, σν) such that : U ν λ f g λ,v = f g −λ,v, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Moreover, for f1, f2 ∈ L2(K, σν), we have Pν λF1 = Pν −λF2 if and only if U ν λF1 = F2 ( i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' U ν λ = (Pν −λ)−1 ◦ Pν λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The proof is the same as in the scalar case so we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now introduce the function space B∗(G, τν) on G, consisting of functions F in L2 loc(G, τν) satisfying ∥ F ∥B∗(G,τν)= sup j∈N [2− j 2 � Aj ∥ F(g) ∥2 ν dgK] < ∞, where A0 = {g ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' d(0, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0) < 1} and Aj = {g ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 2j−1 ≤ d(0, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0) < 2j}, for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' One could easily show that ∥ F ∥B∗(G,τν)≤∥ F ∥∗≤ 2 ∥ F ∥B∗(G,τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We define an equivalent relation on B∗(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For F1, F2 ∈ B∗(G, τν) we write F1 ≃ F2 if lim R→+∞ 1 R � B(R) ∥ F1(g) − F2(g) ∥2 ν dg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Note that by using the polar decomposition we see that F1 ≃ F2 if lim R→+∞ 1 R � K×[0,R] ∥ F1(ketH)) − F2(ketH)) ∥2 ν ∆(t) dt dk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now state the main result of this section 14 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let ν ∈ N, λ ∈ R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For f ∈ L2(K, σν) we have the following asymptotic expansions for the Poisson transform in B∗(G, τν) Pλ,νf(x) ≃ τ −1 ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν λf(k1(x))], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) where x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Most of the proof of the above theorem consists in proving the following Key Lemma, giving the asymptotic ex- pansion for the translates of the τν-spherical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' KEY LEMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For λ ∈ R \\ {0}, g ∈ G and v ∈ Vν, we have the following asymptotic expansion in B∗(G, τν) Φν,λ(g−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v ≃ τ −1 ν (k2(x)) � s∈{±1} cν(sλ)e(isλ−ρ)A+(x)f g sλ,v(k1(x)), x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We first note that both side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) depend continuously on f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This can be proved in the same manner as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore we only have to prove that the asymptotic expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) holds for f ∈ L2 λ(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let f = f g λ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then according to [[11], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3], we have Pν λf(x) = Φν,λ(g−1x)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The theorem follows from the Key lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' As a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 we obtain the following result giving the behaviour of the Poisson integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For any f ∈ L2(K, σν) we have the Plancherel-Poisson formula lim R→+∞ 1 R � B(R) ∥ Pν λf(g) ∥2 ν dgK = 2 | cν(λ) |2 ∥ f ∥2 L2(K,σν) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' There exists a positive constant Cν such that for any λ ∈ R \\ {0}, we have C−1 ν | cν(λ) | ∥ f ∥L2(K,σν)≤∥ Pλ ν f ∥∗≤ Cν | cν(λ) | ∥ f ∥L2(K,σν), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) for every f ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We define for f ∈ L2(K, σν) Sν λf(x) := τ −1 ν (k2(x))[cν(λ)e(iλ−ρ)(A+(x)f(k1(x)) + cν(−λ)e(−iλ−ρ)(A+(x))U ν λf(k1(x))], x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By the unitarity of Uλ, we have 1 R � B(R) ∥Sν λf(g)∥2dgK = 2|cν(λ)|2∥f∥2 L2(K,τν) � 1 R � R 0 e−2ρt∆(t)dt � + 2|cν(λ)|2ℜ � < f, Uλf >L2(K,σν) 1 R � R 0 e2(iλ−ρ)t∆(t)dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' From lim R→+∞ 1 R � R 0 e−2ρt∆(t)dt = 1, and lim R→+∞ 1 R � R 0 e2(iλ−ρ)t∆(t)dt = 0, we deduce that lim R→+∞ 1 R � B(R) ∥ Sν λf(g) ∥2 ν dgK = 2 | cν(λ) |2∥ f ∥2 L2(K,σν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) 15 Next write 1 R � B(R) ∥ Pν λf(g) ∥2 ν dgK = 1 R � B(R) (∥ Sν λf(g) ∥2 ν + ∥ Pν λf(g) − Sν λf(g) ∥2 ν + 2Re[< Pν λf(g) − Sν λf(g), Sν λf(g) >])dgK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) then follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 and the Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The right hand side of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) has already been proved, see corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The left hand side of the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) obviously follows from the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This finishes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let f1, f2 ∈ L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then using the polarization identity as well as the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2), we get lim R→+∞ 1 R � B(R) < Pν λf1(g), Pν λf2(g) >ν dgK = 2 | cν(λ) |2< f1, f2 >L2(K,σν) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) 6 Proof of the main results In this section we shall prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 on the L2-range of the vector Poisson transform and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2 charac- terizing the image Qν λ(L2(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 The L2-range of the Poisson transform We first recall some results of harmonic analysis on the homogeneous vector bundle K ×M Vν associated to the representation σν of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let �K be the unitary dual of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For δ ∈ �K let Vδ denote a representation space of δ with dδ = dim Vδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We denote by �K(σν) the set of δ ∈ �K such that σν occurs in δ |M with multiplicity mδ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The decomposition of L2(K, σν) under K (the group K acts by left translations on this space) is given by the Frobenius reciprocity law L2(K, σν) = � δ∈� K(σν) Vδ ⊗ HomM(Vν, Vδ), where v ⊗L, for v ∈ Vδ, L ∈ HomM(Vν, Vδ) is identified with the function (v ⊗L)(k) = L∗(δ(k−1)v), where L∗ denotes the adjoint of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For each δ ∈ �K(σν) let (Lj)mδ j=1 be an orthonormal basis of HomM(Vν, Vδ) with respect to the inner product < L1, L2 >= 1 ν + 1T r(L1L∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let {v1, · · · , vdδ} be an orhonormal basis of Vδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then f δ ij : k → � dδ ν + 1L∗ i δ(k−1)vj, 1 ≤ i ≤ mδ, 1 ≤ j ≤ dδ, δ ∈ �K(σ) form an orthonormal basis of L2(K, σν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For f ∈ L2(K, σν) we have the Fourier series expansion f(k) = � δ∈� K(σ) mδ � i=1 dδ � j=1 aδ ijf δ ij(k) with ∥ f ∥2 L2(K,σ)= � δ∈� K(σ) mδ � i=1 dδ � j=1 | aδ ij |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 16 We define for δ ∈ �K(σ) and λ ∈ C, the generalized Eisenstein integral ΦL λ,δ(g) = � K e−(iλ+ρ)H(g−1k)τν(κ(g−1k))L∗δ(k−1)dk, L ∈ HomM(Vν, Vδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It is easy to see that ΦL λ,δ satisfies the following identity ΦL λ,δ(k1gk2) = τν(k−1 2 )ΦL λ,δ(g)δ(k−1 1 ), k1, k2 ∈ K, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now prove an asymptotic estimate for the generalized Eisenstein integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let ν ∈ N, λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then for δ ∈ �K(σν), T, S ∈ HomM(Vν, Vδ) we have lim R→+∞ 1 R � B(R) Tr � ΦT λ,δ(g)∗ΦS λ,δ(g) � dgK = 2 | cν(λ) |2 Tr(T S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By definition we have lim R→+∞ 1 R � B(R) Tr � ΦT λ,δ(g)∗ΦS λ,δ(g) � dgK = dδ � j=1 lim R→+∞ 1 R � B(R) < ΦS λ,δ(g)vj, ΦT λ,δ(g)vj >ν dgK Noting that ΦT λ,δ(g)vj is the Poisson transform of the function k �→ L∗δ(k−1)vj and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5), we get lim R→+∞ 1 R � B(R) Tr � ΦT λ,δ(g)∗ΦS λ,δ(g) � dgK = 2 | cν(λ) |2 dδ � j=1 � K < S∗δ(k−1)vj, T ∗δ(k−1)vj >ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Hence Schur Lemma lead us to conclude that lim R→+∞ 1 R � B(R) Tr � ΦT λ,δ(g)∗ΦS λ,δ(g) � dgK = 2 | cν(λ) |2 Tr(T S∗), and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Noting that T r( � ΦT λ,δ(g)∗ΦS λ,δ(g) � = T r( � ΦT λ,δ(a)∗ΦS λ,δ(a) � , g = k1 a k2, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) that lim R→+∞ 1 R � R 0 T r � ΦT λ,δ(at)∗ΦS λ,δ(at) � ∆(t)dt =| cν(λ) |2 Tr(T S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (i) The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) implies that the Poisson transform Pλ,ν maps L2(K, σν) into Eλ(G, τν) and that the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (ii) We now prove that the Poisson transform maps L2(K, σν) onto E2 λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ E2 λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since λ ∈ R \\ {0}, we know by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 that there exists a hyperfunction f ∈ C−ω(K, σν) such that F = Pλ,νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let f = � δ∈� K(σ) dδ � j=1 mδ � i=1 aδ ijf δ ij, be the Fourier series expansion of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have F(g) = � δ∈� K(σ) � dδ ν + 1 dδ � j=1 mδ � i=1 aδ ijΦLi λ,δ(g)vj in C∞(G, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By the Schur relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' we have � K < ΦLi λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ(kat)vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ΦLm λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ′(kat)vn >ν dk = � 0 if δ ≁ δ′ 1 dδ T r(ΦLm λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ′ (at))∗ΦLi λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ(at) < vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' vn > if δ′ = δ 17 Therefore � K ∥ F(kat) ∥2 dk = 1 ν + 1 � δ∈� K(σ) dδ � j=1 � 1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='j≤mδ aδ ijaδ mjT r[(ΦLm λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ (at))∗ΦLi λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ(at)] = 1 ν + 1 � δ∈� K(σ) dδ � j=1 T r \uf8ee \uf8f0 � 1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='m≤mδ (aδ mjΦLm λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ (at))∗(aδ ijΦLi λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ(at) \uf8f9 \uf8fb = 1 ν + 1 � δ∈� K(σ) dδ � j=1 ∥ mδ � i=1 aδ ijΦLi λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='δ(at) ∥2 HS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let Λ be a finite subset in �K(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since ∥ F ∥∗< ∞, it follows that, for any R > 1 we have ∞ >∥ F ∥2 ∗≥ 1 ν + 1 � δ∈Λ dδ � j=1 1 R � R 0 ∥ mδ � i=1 aδ ijΦLi λ,δ(at) ∥2 HS ∆(t) dt By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) we have lim R→∞ 1 R � R 0 ∥ mδ � i=1 aδ ijΦLi λ,δ(at) ∥2 HS ∆(t) dt = lim R→∞ � 1≤i,m≤mδ aδ ijaδ mj 1 R � R 0 T r[(ΦLm λ,δ (at))∗ΦLi λ,δ(at)] ∆(t)dt = 2 | cν(λ) |2 � 1≤i,m≤mδ aδ ijaδ mjT r(LiL∗ m) = 2(ν + 1) | cν(λ) |2 mδ � i=1 | aδ ij |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus ∞ >∥ F ∥2 ∗≥| cν(λ) |2 � δ∈Λ dδ � j=1 mδ � i=1 | aδ ij |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since Λ is arbitrary, it follows that | cν(λ) |2 � δ∈� K(σ) dδ � j=1 mδ � i=1 | aδ ij |2≤∥ F ∥2 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This shows that f ∈ L2(K, σν) with | cν(λ) |∥ f ∥L2(K,σν)≤∥ Pν λf ∥∗ and the proof of the theorem is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2 The L2-range of the generalized spectral projections We now proceed to the poof of the second main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let F ∈ L2 c(G, τν) ∩ C∞(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from the definition ( see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='8)) that the operator Qν λ may be written as Qν λF(g) =| cν(λ) |−2 Pν λ(FνF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' ))(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 we deduce that sup R>1 1 R � B(R) ∥ Qν λF(g) ∥2 ν dgK ≤ Cν | cν(λ) |−2 � K ∥ FνF(λ, k) ∥2 ν dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The above inequality and the Plancherel formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) imply � ∞ 0 (sup R>1 1 R � B(R) ∥ Qν λF(g) ∥2 ν dgK) dλ ≤ Cν � ∞ 0 � K ∥ FνF(λ, k) ∥2 ν| cν(λ) |−2 dk dλ ≤ Cν ∥ F ∥2 L2(G,τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 18 This prove the right hand side of the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) we have lim R→∞ 1 R � B(R) ∥ Qν λF(g) ∥2 ν dgK = 2 | cν(λ) |−2 � K ∥ FνF(λ, k) ∥2 ν dk, and since for all R > 1 1 R � B(R) ∥ Qν λF(g) ∥2 dgK ≤ Cν | cν(λ) |−2 � K ∥ FνF(λ, k) ∥2 dk, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' λ ∈ (0, ∞), we may apply the Lebesgue’s dominated convergence theorem to get lim R→∞ � ∞ 0 � 1 R � B(R) ∥ Qν λF(g) ∥2 ν dgK � dλ = 2 ∥ F ∥2 L2(G,τν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from the above equality that C ∥ F ∥2 L2(G,τν)≤ � ∞ 0 (sup R>1 � B(R) ∥ Qν λF(x) ∥2 dx) dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This complete the proof of the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We now prove that Qν λ maps L2 c(G, τν) onto E2 λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let Fλ ∈ E2 λ(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have sup R>1 1 R � B(R) ∥ Fλ(g) ∥2 ν dgK < ∞, for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' λ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1, there exists fλ ∈ L2(K, σν) such that Fλ(g) =| cν(λ) |−2 Pν λfλ(g) with sup R>1 1 R � B(R) ∥ Fλ(g) ∥2 ν dgK ≥ C−1 ν | cν(λ) |−2 � K ∥ fλ(k) ∥2 dk Integrating the both side of the above inequality over (0, ∞), we get ∞ >∥ Fλ ∥2 ∗≥ C−1 ν � ∞ O � K ∥ fλ(k) ∥2 ν | cν(λ) |−2 dk dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1, that there exists F ∈ L2 c(G, τν) such that FνF(λ, k) = fλ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Henceforth Fλ(g) =| cν(λ) |−2 Pλ,ν(FνF(λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' )(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This finishes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 7 Proof of the Key Lemma In this section we prove the Key Lemma of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' To this end we need to establish some auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We first prove an asymptotic formula for the τν-spherical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For any v ∈ Vν we have Φν,λ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v ≃ � s∈{±1} cν(sλ)e(isλ−ρ)A+(g)τ −1 ν (κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) g = κ1(g)eA+(g)κ2(g) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since ∆(t) ≤ e2ρ t, we get 1 R � B(R) ∥ e(iλ−ρ)A+(g)τ −1 ν (κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v ∥2 dg = 1 R ∥ v ∥2 � R 0 e−2ρ t∆(t)dt ≤∥ v ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 19 This shows that the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) belongs to B∗(G, τν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since λ ∈ R \\ {0}, we may use the identity (A3) to write ϕν,λ(t) − � s∈{±1} cν(sλ)e(isλ−ρ)t = � s∈{±1} cν(sλ) � (2 cosh t)νΨρ−2,ν+1 sλ (t) − e(isλ−ρ)t� = � s∈{±1} cν(sλ)e(isλ−ρ)t � (1 + e−2t)νe(ρ+ν−isλ)tΨρ−2,ν+1 sλ (t) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It follows from (A2’) that ϕν,λ(t) − � s∈{±1} cν(sλ)e(isλ−ρ)t = � s∈{±1} cν(sλ)e(isλ−ρ)t � (1 + e−2t)ν − 1) + e−2tEsλ(t) � , where | Esλ(t) |≤ 2νC if t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore | ϕν,λ(t) − � s∈{±1} cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρe−2t, if t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This together with | ϕν,λ(t) − � s∈{±1} cν(sλ)e(isλ−ρ)t |≤ Cν,λe−ρt, for t ∈ [0, 1], imply that lim R→∞ 1 R � B(R) ∥ Φν,λ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v − � s∈{±1} cν(sλ)e(isλ−ρ)A+(g)τ −1(κ1(g)κ2(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' v ∥2 ν dgK = =∥ v ∥2 lim R→∞ 1 R � R 0 | ϕν,λ(t) − � s∈{±1} cν(sλ)e(isλ−ρ)t |2 ∆(t) dt = 0, and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let g ∈ G, k ∈ K and t a non negative real number .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have 0 ≤ A+(g−1k exp(tH)) − H(g−1k exp(tH)) ≤ 1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 | 1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 |e−2t, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let g−1 = � a b c d, � and k == � u 0 O v, � , where a, b, c and d are n×n, n×1, 1×n and 1×1 matrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' A direct computation yields g−1k exp(tH) = � ∗ ∗ ∗ c1 d1 � , where c1 = c u � cosh t 0 0 In−1 � and d1 = sinh t cue1 + cosh t dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1) we have eH(g−1k exp(tH)) = et | cue1 + dv |, and eA+(g−1k exp(tH)) =| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 20 From eA+(g−1k exp(tH))−H(g−1k exp(tH)) = e−t | cue1 + dv |[| sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1) 1 2 ], together with | sinh t cue1 + cosh t dv | +(| sinh t cue1 + cosh t dv |2 −1) 1 2 ≤ 2 | sinh t cue1v−1 + cosh t d | ≤| cue1v−1 + d | et+ | d − cue1v−1 | e−t we deduce that e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | d − cue1v−1 | | cue1v−1 + d |e−2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Noting that (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0)∗ = −(d−1c), and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e1 = ue1v−1, we get e(A+(g−1k exp(tH))−H(g−1k exp(tH)) ≤ 1 + | 1+ < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e1 >| | 1− < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e1 >|e−2t ≤ 1 + 1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 | 1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 |e−2t, from which we deduce (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2), and the proof of the lemma is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof of the Key Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Since B∗(G, τν) is G-invariant, we may apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='1 to get Φν,λ(g−1x)v ≃ τ −1 ν (κ1(g−1x)κ2(g−1x) � s∈{±} cν(sλ)e(isλ−ρ)A+(g−1x)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus it suffices to show that τ −1 ν (κ1(g−1x)κ2(g−1x) � s∈{±} cν(sλ)e(isλ−ρ)A+(g−1x)v ≃ τ −1 ν (k2(x)) � s∈{±1} cν(sλ)e(isλ−ρ)A+(x)f g sλ,v(k1(x)), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) Note that τ −1 ν [k1(g−1k1(x)eA+(x)k2(x))k2(g−1k1(x)eA+(x)k2(x))] = τ −1 ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))k2(x))], x = k1(x)eA+(x)k2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Henceforth (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='3) is equivalent to τ −1 ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))] � s∈{±1} cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x)) v ≃ � s∈{±1} cν(λ)e(isλ−ρ)A+(x)f g sλ,v(k1(x)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) We write the left hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='4) as � s∈{±1} cν(λ)e(isλ−ρ)A+(x)f g sλ,v(k1(x)) + rg(x)v, where rg(x) =τ −1 ν [k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))] � s∈{±1} cν(λ)e(isλ−ρ)A+(g−1k1(x)eA+(x)) − � s∈{±1} cν(λ)e(isλ−ρ)[A+(x)+H(g−1k1(x))]τ −1 ν (κ(g−1k1(x)), x ∈ G (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='5) 21 To finish the proof we show that for each g ∈ G, rg ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Noting that H(g−1k1(x)eA+(x)) = H(g−1k1(x)) + A+(x), we rewrite rg as rg(x) = [τ −1 ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))) − τ−1 ν (κ(g−1k1(x))] � s∈{±1} cν(λ)e(isλ−ρ)H(g−1k1(x)eA+(x)) + τ −1 ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x))) \uf8eb \uf8ed � s∈{±1} cν(λ)[e(isλ−ρ)A+(g−1k1(x)eA+(x))) − e(isλ−ρ)H((g−1k1(x)eA+(x))] \uf8f6 \uf8f8 =: Ig(x) + Jg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Using the following Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Let g = � a b c d � ∈ Sp(n, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we have τν(κ1(g)κ2(g)) = τν( d | d |) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) τν(κ(g)) = τν( ce1 + d | ce1 + d |) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) lim R→∞ τν(κ1(g exp(RH))κ2(g exp(RH))) = τν(κ(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='8) we easily see that Igv ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We have Jg(x) =τ −1 ν (k1(g−1k1(x)eA+(x))k2(g−1k1(x)eA+(x)))e(isλ−ρ)H(g−1k1(x)eA+(x)) � s∈{±1} cν(λ) � e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 � As τν is unitary we have 1 R � K×[0,R] ∥ Jg(ketH)v ∥2 ν ∆(t)dt dk ≤∥ v ∥2 2 | cν(λ) |2 R � K×[0,R] e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2 From | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |≤ C(| λ | +ρ) | A+(g−1k1(x)eA+(x)) − H(g−1k1(x)eA+(x) | together with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2 we get � K×[0,R] e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2 ≤ � C(| λ | +ρ)1+ | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 | 1− | g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='0 | �2 1 R � K×[0,R] e−2ρH(g−1k)e−2(ρ+2t)∆(t) dk dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 22 As � K e−2ρH(g−1k) dk = 1 and ∆(t) ≤ 2ρe2ρt we obtain lim R→∞ 1 R � K×[0,R] e−2ρH(g−1ketH) | e(isλ−ρ)(A+(g−1k1(x)eA+(x))−H(g−1k1(x)eA+(x))] − 1 |2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This shows that Jg ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Therefore we have proved that for each g ∈ G, rg ≃ 0 as to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' It remain to prove Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' If g = � a b c d � = � u1 0 0 v1 � at � u2 0 0 v2 � with respect to the Cartan decomposition G = KAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then we easily see that d = cosh t v1v2 and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Analogously if g = � a b c d � = � u 0 0 v � at n with respect to the Iwasawa decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then from g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='e1 = � ae1 + b ce1 + d � = et � u v � we get et v = ce1 + d and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='7) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' We have g exp(RH) = � ∗ ∗∗ ∗ ∗ ∗ sinh Re1 + cosh Rd � Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='6) imply that τν(κ1(g)κ2(g)) = τν( tanh Rce1+d |tanh Rce1+d|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Thus limR→∞ τν(κ1(g)κ2(g)) = τν( ce1+d |ce1+d|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' This finishes the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content='2, and the proof of the Key Lemma is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 8 Appendix In this section we collect some results on the Jacobi functions, referring to [19] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For α, β, λ ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' α ̸= −1, −2, · · · and t ∈ R, the Jacobi function is defined by φ(α,β) λ (t) = 2F1(iλ + ρα,β 2 , −iλ + ρα,β 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' α + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' − sinh2 t), where 2F1 is the Gauss hypergeometric function and ρα,β = α + β + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The Jacobi function φ(α,β) λ is the unique even smooth function on R which satisfy φ(α,β) λ (0) = 1 and the differential equation { d2 dt2 + [(2α + 1) coth t + (2β + 1) tanh t] d dt + λ2 + ρ2 α,β}φ(α,β) λ (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (A1) For λ /∈ −iN another solution Ψα,β λ of (A1) such that Ψα,β λ (t) = e(iλ−ρα,β)t(1 + ◦(1)), as t → ∞ (A2) is given by Ψα,β λ (t) = (2 sinh t)iλ−ρα,β 2F1(ρα,β − iλ 2 , β − α + 1 − iλ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' 1 − iλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' − 1 sinh2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Moreover there exists a constant C > 0 such that for all λ ∈ R and all t ≥ 1 we have Ψα,β λ (t) = e(iλ−ρα,β)t(1 + e−2tΘλ(t)), with | Θλ(t) |≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (A2’) For λ /∈ iZ, we have φ(α,β) λ (t) = � s=±1 cα,β(sλ)Ψα,β sλ (t) (A3) 23 where cα,β(λ) = 2ρα,β−iλ Γ(α + 1)Γ(iλ) Γ( iλ+ρα,β 2 )Γ( iλ+α−β+1 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For ℜ(iλ) > 0, the asymptotic behaviour of φ(α,β) λ as t → ∞ is then given by lim t→∞ e(ρα,β−iλ)tφ(α,β) λ (t) = cα,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (A4) Let De(R) denote the space of even smooth function with compact support on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' For f ∈ De(R), the Fourier-Jacobi transform J α,βf (λ ∈ C) is defined by J α,βf(λ) = � ∞ 0 f(t)φ(α,β) λ (t)∆α,β(t) dt, (A5) where ∆α,β(t) = (2 sinh t)2α+1(2 cosh t)2β+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' In the sequel, we assume that α > −1, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' Then the meromorphic function cα,β(−λ)−1 has only simple poles for ℑλ ≥ 0 which occur in the set Dα,β = {λk = i(| β | −α − 1 − 2k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' k = 0, 1, · · · , | β | −α − 1 − 2k > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (If | β |≤ α + 1, then Dα,β is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' The following inversion and Plancherel formulas for the Jacobi transform hold for every f ∈ De(R): f(t) = 1 2π � ∞ 0 (J α,βf)(λ) φ(α,β) λ (t) | cα,β(λ) |−2 dλ + � λk∈Dα,β dk(J α,βf)(λk) φ(α,β) λk (t), (A6) � ∞ 0 | f(t) |2 ∆(t) dt = 1 2π � ∞ 0 | (J α,βf)(λ) |2 | cα,β(λ) |−2 dλ + � λk∈Dα,β dk | (J α,βf)(λk) |2 (A6’) where dk = −i Resλ=λk(cα,β(λ)cα,β(−λ))−1, is given explicitly by dk = (β − α − 2k − 1)2−2(α+β)Γ(α + k + 1)Γ(β − k) Γ2(α + 1)Γ(β − α − k)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' (A7) References [1] Anker,J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE4T4oBgHgl3EQf3Q1N/content/2301.05304v1.pdf'} +page_content=': A basis inequality for Scattering Theory for Riemannian Symmetric 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