0907/0907.3054.md

HARDY INEQUALITIES FOR FRACTIONAL INTEGRALS ON GENERAL DOMAINS

MICHAEL LOSS ¹ AND CRAIG SLOANE ¹

Abstract

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.

1. INTRODUCTION

In this note we prove a conjecture by Bogdan and Dyda [3] concerning Hardy inequalities for fractional integrals. It was shown in [3] that for any function $f$ supported in the half-space $\mathbb{H}^n = {x \in \mathbb{R}^n : x = (x_1, \dots, x_n), x_n > 0}$

$$\frac{1}{2}{\int }_{{\mathbb{H}}^n\times {\mathbb{H}}^n}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\kappa }_{n,\alpha }{\int }_{{\mathbb{H}}^n}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^2}{x_n^{\alpha }}dx\mathrm{.}(1)\backslash frac\{1\}\{2\}\; \backslash int\_\{\backslash mathbb\{H\}^n\; \backslash times\; \backslash mathbb\{H\}^n\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{n,\backslash alpha\}\; \backslash int\_\{\backslash mathbb\{H\}^n\}\; \backslash frac\{|f(x)|^2\}\{x\_n^\backslash alpha\}\; dx.\; \backslash quad\; (1)$$

Here $0 < \alpha < 2$ and

$${\kappa }_{n,\alpha }={\pi }^{\frac{n-1}{2}}\frac{\mathrm{\Gamma }(\frac{1+\alpha }{2})}{\mathrm{\Gamma }(\frac{n+\alpha }{2})}\frac{1}{\alpha }[\frac{2^{1-\alpha }}{\sqrt{\pi }}\mathrm{\Gamma }\left(\frac{2-\alpha }{2}\right)\mathrm{\Gamma }\left(\frac{1+\alpha }{2}\right)-1](2)\backslash kappa\_\{n,\backslash alpha\}\; =\; \backslash pi^\{\backslash frac\{n-1\}\{2\}\}\; \backslash frac\{\backslash Gamma(\backslash frac\{1+\backslash alpha\}\{2\})\}\{\backslash Gamma(\backslash frac\{n+\backslash alpha\}\{2\})\}\; \backslash frac\{1\}\{\backslash alpha\}\; \backslash left[\; \backslash frac\{2^\{1-\backslash alpha\}\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash Gamma\backslash left(\backslash frac\{2-\backslash alpha\}\{2\}\backslash right)\; \backslash Gamma\backslash left(\backslash frac\{1+\backslash alpha\}\{2\}\backslash right)\; -\; 1\; \backslash right]\; \backslash quad\; (2)$$

is the sharp constant. Note that $\kappa_{n,1} = 0$ and $\kappa_{n,\alpha} > 0$ otherwise.

It was conjectured in [3] that for $1 < \alpha < 2$ this inequality continues to hold with the same constant for any convex set $\Omega$ , i.e., for functions $f$ supported in $\Omega$

$$\frac{1}{2}{\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\kappa }_{n,\alpha }{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^2}{d_{\mathrm{\Omega }}(x{)}^{\alpha }}dx,(3)\backslash frac\{1\}\{2\}\; \backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{n,\backslash alpha\}\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{|f(x)|^2\}\{d\_\{\backslash Omega\}(x)^\backslash alpha\}\; dx,\; \backslash quad\; (3)$$

where $d_{\Omega}(x)$ denotes the distance from the point $x \in \Omega$ to the boundary of $\Omega$ . This is a precise analogue of the Hardy inequality due to Davies [8]. For $0 < \alpha < 1$ the inequality cannot hold for compact sets. A counterexample is given in [5].

Sharp Hardy inequalities analogous to (3) but for the $L^p$ -norms of gradients of functions are well known. The first result is due to Davies [8] for the case $p = 2$ . The case for arbitrary $p$ is derived in [14] and [13]. For a review the reader may consult [9]. Let us add that these results have been considerably generalized in [1].

Hardy inequalities for fractional integrals are of a more recent provenience, in particular the higher dimensional versions were investigated by Dyda (see [5]) in great generality following previous work in [12] and [6]. While Hardy inequalities for fractional integrals are of interest in their own right, they deliver also spectral information on the generators of censored stable processes. The generator of a censored stable process is defined by the closure of the quadratic

---

Date: November 3, 2018.

¹Work partially supported by NSF Grant DMS 0901304.

©2009 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.form on the left side of (3). Loosely speaking it is a stable process with the jumps between $\Omega$ and its complement suppressed. The reference [2] contains the construction of censored stable processes and a wealth of information about these. For the connection between Hardy inequalities and censored stable processes the reader may consult [6].

Since we prove a stronger result than (3) we need a few concepts before we can state the result. Let $\Omega$ be any domain in $\mathbb{R}^n$ with non-empty boundary. The following notion is taken from Davies [7]. Fix a direction $w \in \mathbb{S}^{n-1}$ and define

$$d_{w,\mathrm{\Omega }}(x)=\min \{\mathrm{\mid }t\mathrm{\mid }:x+tw\mathrm{\notin }\mathrm{\Omega }\}\mathrm{.}(4)d\_\{w,\backslash Omega\}(x)\; =\; \backslash min\backslash \{|t|\; :\; x\; +\; tw\; \backslash notin\; \backslash Omega\backslash \}\; .\; \backslash quad\; (4)$$

Further, define the function

$${\delta }_{w,\mathrm{\Omega }}(x)=\sup \{\mathrm{\mid }t\mathrm{\mid }:x+tw\in \mathrm{\Omega }\},(5)\backslash delta\_\{w,\backslash Omega\}(x)\; =\; \backslash sup\backslash \{|t|\; :\; x\; +\; tw\; \backslash in\; \backslash Omega\backslash \}\; ,\; \backslash quad\; (5)$$

i.e., $\delta_{w,\Omega}(x)$ is the point in the intersection of the line $x + tw$ and $\Omega$ that is farthest away from $x$ and set

$$\frac{1}{M_{\alpha }(x{)}^{\alpha }}:=\frac{{\int }_{{\mathbb{S}}^{n-1}}dw{[\frac{1}{d_{w,\mathrm{\Omega }}(x)}+\frac{1}{{\delta }_{w,\mathrm{\Omega }}(x)}]}^{\alpha }}{{\int }_{{\mathbb{S}}^{n-1}}dw\mathrm{\mid }{w}_n{\mathrm{\mid }}^{\alpha }}\mathrm{.}(6)\backslash frac\{1\}\{M\_\backslash alpha(x)^\backslash alpha\}\; :=\; \backslash frac\{\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; \backslash left[\; \backslash frac\{1\}\{d\_\{w,\backslash Omega\}(x)\}\; +\; \backslash frac\{1\}\{\backslash delta\_\{w,\backslash Omega\}(x)\}\; \backslash right]^\backslash alpha\}\{\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; |w\_n|^\backslash alpha\}\; .\; \backslash quad\; (6)$$

The integral in the denominator can be easily computed to be

$${\int }_{{\mathbb{S}}^{n-1}}dw\mathrm{\mid }{w}_n{\mathrm{\mid }}^{\alpha }=2{\pi }^{\frac{n-1}{2}}\frac{\mathrm{\Gamma }(\frac{1+\alpha }{2})}{\mathrm{\Gamma }(\frac{n+\alpha }{2})}(7)\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; |w\_n|^\backslash alpha\; =\; 2\backslash pi^\{\backslash frac\{n-1\}\{2\}\}\; \backslash frac\{\backslash Gamma(\backslash frac\{1+\backslash alpha\}\{2\})\}\{\backslash Gamma(\backslash frac\{n+\backslash alpha\}\{2\})\}\; \backslash quad\; (7)$$

These definitions are analogous to the one in [7] where all estimates are expressed in terms of

$$\frac{1}{m_2(x{)}^2}=\frac{{\int }_{{\mathbb{S}}^{n-1}}dw\frac{1}{d_{w,\mathrm{\Omega }}(x{)}^2}}{\mathrm{\mid }{\mathbb{S}}^{n-1}\mathrm{\mid }\mathrm{/}n}\mathrm{.}\backslash frac\{1\}\{m\_2(x)^2\}\; =\; \backslash frac\{\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; \backslash frac\{1\}\{d\_\{w,\backslash Omega\}(x)^2\}\}\{|\backslash mathbb\{S\}^\{n-1\}|/n\}\; .$$

In case the domain $\Omega$ is convex, the quantity $M_\alpha(x)$ can be bounded in terms of $d_\Omega(x)$ and $D_\Omega(x)$ , the ‘width of $\Omega$ with respect to $x$ ’. For convex domains with smooth boundary, this quantity is given by the width of the smallest slab that contains $\Omega$ and consists of two parallel hyper-planes one of which is tangent to $\partial\Omega$ at the point closest to $x$ . For general convex sets we define it as follows. Fix $x \in \Omega$ arbitrary and pick a point $z$ on the boundary of $\Omega$ that is closest to $x$ , so that $d_\Omega(x) = |x - z|$ . In general, there may be more than one such point. Denote by $P_z$ the set of supporting hyper-planes of $\Omega$ that pass through the point $z$ and set

$${\mathcal{P}}_x={\cup }_{z\in \mathrm{\partial }\mathrm{\Omega },\mathrm{\mid }z-x\mathrm{\mid }=d_{\mathrm{\Omega }}(x)}{P}_z\mathrm{.}\backslash mathcal\{P\}\_x\; =\; \backslash cup\_\{z\; \backslash in\; \backslash partial\backslash Omega,\; |z-x|=d\_\backslash Omega(x)\}\; P\_z\; .$$

For $P \in \mathcal{P}x$ , we denote by $S(P)$ the smallest slab that contains $\Omega$ and is bounded by $P$ on one side and a hyper-plane parallel to it on the other. Such a slab might be a half space if $\Omega$ is unbounded. The width $D{S(P)}$ of the slab $S(P)$ is, naturally, the distance between the two bounding hyper-planes. We set $D_{S(P)} = \infty$ if $S(P)$ is a half space. Now we define

$$D_{\mathrm{\Omega }}(x)=\underset{P\in {\mathcal{P}}_x}{\inf }{D}_{S(P)}\mathrm{.}(8)D\_\backslash Omega(x)\; =\; \backslash inf\_\{P\; \backslash in\; \backslash mathcal\{P\}\_x\}\; D\_\{S(P)\}\; .\; \backslash quad\; (8)$$

The inequality

$$\frac{1}{M_{\alpha }(x{)}^{\alpha }}\ge {[\frac{1}{d_{\mathrm{\Omega }}(x)}+\frac{1}{D_{\mathrm{\Omega }}(x)-d_{\mathrm{\Omega }}(x)}]}^{\alpha },(9)\backslash frac\{1\}\{M\_\backslash alpha(x)^\backslash alpha\}\; \backslash geq\; \backslash left[\; \backslash frac\{1\}\{d\_\backslash Omega(x)\}\; +\; \backslash frac\{1\}\{D\_\backslash Omega(x)\; -\; d\_\backslash Omega(x)\}\; \backslash right]^\backslash alpha\; ,\; \backslash quad\; (9)$$

follows from

$${\int }_{{\mathbb{S}}^{n-1}}dw{[\frac{1}{d_{w,\mathrm{\Omega }}(x)}+\frac{1}{{\delta }_{w,\mathrm{\Omega }}(x)}]}^{\alpha }\ge {\int }_{{\mathbb{S}}^{n-1}}dw\mathrm{\mid }{w}_n{\mathrm{\mid }}^{\alpha }{[\frac{1}{d_{\mathrm{\Omega }}(x)}+\frac{1}{D_{\mathrm{\Omega }}(x)-d_{\mathrm{\Omega }}(x)}]}^{\alpha }\mathrm{.}(10)\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; \backslash left[\; \backslash frac\{1\}\{d\_\{w,\backslash Omega\}(x)\}\; +\; \backslash frac\{1\}\{\backslash delta\_\{w,\backslash Omega\}(x)\}\; \backslash right]^\backslash alpha\; \backslash geq\; \backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; |w\_n|^\backslash alpha\; \backslash left[\; \backslash frac\{1\}\{d\_\backslash Omega(x)\}\; +\; \backslash frac\{1\}\{D\_\backslash Omega(x)\; -\; d\_\backslash Omega(x)\}\; \backslash right]^\backslash alpha\; .\; \backslash quad\; (10)$$Indeed, for given $P$ pick coordinates such that the standard vector $e_n$ is normal to the plane $P$ . Clearly $d_{w,\Omega}(x) \leq d_{w,S(P)}(x)$ and $\delta_{w,\Omega}(x) \leq \delta_{w,S(P)}(x)$ . Further, note that $d_{w,S(P)}(x) + \delta_{w,S(P)}(x)$ is the length of the segment given by intersecting the slab $S(P)$ with the line $x + tw$ . Projecting this segment onto the line normal to the slab yields

$$d_{w,S(P)}(x)\mathrm{\mid }{w}_n\mathrm{\mid }=d_{\mathrm{\Omega }}(x),{\delta }_{w,S(P)}(x)\mathrm{\mid }{w}_n\mathrm{\mid }=D_{S(P)}-d_{\mathrm{\Omega }}(x)\mathrm{.}d\_\{w,S(P)\}(x)|w\_n|\; =\; d\_\backslash Omega(x),\; \backslash quad\; \backslash delta\_\{w,S(P)\}(x)|w\_n|\; =\; D\_\{S(P)\}\; -\; d\_\backslash Omega(x).$$

Note that there may exist directions $w$ where the length of this segment is not finite in which case we set $D_{S(P)} = \infty$ . Thus,

$${[\frac{1}{d_{w,\mathrm{\Omega }}(x)}+\frac{1}{{\delta }_{w,\mathrm{\Omega }}(x)}]}^{\alpha }\ge \mathrm{\mid }{w}_n{\mathrm{\mid }}^{\alpha }{[\frac{1}{d_{\mathrm{\Omega }}(x)}+\frac{1}{D_{S(P)}-d_{\mathrm{\Omega }}(x)}]}^{\alpha }\backslash left[\; \backslash frac\{1\}\{d\_\{w,\backslash Omega\}(x)\}\; +\; \backslash frac\{1\}\{\backslash delta\_\{w,\backslash Omega\}(x)\}\; \backslash right]^\backslash alpha\; \backslash geq\; |w\_n|^\backslash alpha\; \backslash left[\; \backslash frac\{1\}\{d\_\backslash Omega(x)\}\; +\; \backslash frac\{1\}\{D\_\{S(P)\}\; -\; d\_\backslash Omega(x)\}\; \backslash right]^\backslash alpha$$

holds for all $P \in \mathcal{P}_x$ . Taking the supremum over $\mathcal{P}_x$ and integrating with respect to $w$ over the unit sphere yields (10). With these preparations we can state our main theorem.

Theorem 1.1. Let $\Omega$ be a domain with non-empty boundary and $1 < \alpha < 2$ . For any $f \in C_c^\infty(\Omega)$

$$\frac{1}{2}{\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\kappa }_{n,\alpha }{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^2}{M_{\alpha }(x{)}^{\alpha }}dx\mathrm{.}(11)\backslash frac\{1\}\{2\}\; \backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{n,\backslash alpha\}\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{|f(x)|^2\}\{M\_\backslash alpha(x)^\backslash alpha\}\; dx.\; \backslash quad\; (11)$$

In particular, if $\Omega$ is a convex region then for any $f \in C_c^\infty(\Omega)$

$$\frac{1}{2}{\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\kappa }_{n,\alpha }{\int }_{\mathrm{\Omega }}\mathrm{\mid }f(x){\mathrm{\mid }}^2{[\frac{1}{d_{\mathrm{\Omega }}(x)}+\frac{1}{D_{\mathrm{\Omega }}(x)-d_{\mathrm{\Omega }}(x)}]}^{\alpha }dx(12)\backslash frac\{1\}\{2\}\; \backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{n,\backslash alpha\}\; \backslash int\_\{\backslash Omega\}\; |f(x)|^2\; \backslash left[\; \backslash frac\{1\}\{d\_\backslash Omega(x)\}\; +\; \backslash frac\{1\}\{D\_\backslash Omega(x)\; -\; d\_\backslash Omega(x)\}\; \backslash right]^\backslash alpha\; dx\; \backslash quad\; (12)$$

where $d_\Omega(x)$ is the distance of $x \in \Omega$ to the boundary of $\Omega$ and $D_\Omega(x)$ is defined in (8). The constant $\kappa_{n,\alpha}$ is best possible.

It was pointed out to us by Rupert Frank and Robert Seiringer that this Theorem 1.1 can be generalized, albeit in a weaker form, by replacing the powers 2 by $p > 1$ . More precisely we have,

Theorem 1.2. Let $1 < p < \infty$ and $1 < \alpha < p$ . Then for any domain $\Omega \subset \mathbb{R}^n$ and any $f \in C_c^\infty(\Omega)$

$${\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\mathcal{D}}_{n,p,\alpha }{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{m_{\alpha }(x{)}^{\alpha }}dx(13)\backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash mathcal\{D\}\_\{n,p,\backslash alpha\}\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{|f(x)|^p\}\{m\_\backslash alpha(x)^\backslash alpha\}\; dx\; \backslash quad\; (13)$$

where

$$\frac{1}{m_{\alpha }(x{)}^{\alpha }}:=\frac{{\int }_{{\mathbb{S}}^{n-1}}dw\frac{1}{d_{w,\mathrm{\Omega }}(x{)}^{\alpha }}}{{\int }_{{\mathbb{S}}^{n-1}}dw\mathrm{\mid }{w}_n{\mathrm{\mid }}^{\alpha }}\mathrm{.}(14)\backslash frac\{1\}\{m\_\backslash alpha(x)^\backslash alpha\}\; :=\; \backslash frac\{\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; \backslash frac\{1\}\{d\_\{w,\backslash Omega\}(x)^\backslash alpha\}\}\{\backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; |w\_n|^\backslash alpha\}.\; \backslash quad\; (14)$$

and

$${\mathcal{D}}_{n,p,\alpha }=2{\pi }^{\frac{n-1}{2}}\frac{\mathrm{\Gamma }(\frac{1+\alpha }{2})}{\mathrm{\Gamma }(\frac{n+\alpha }{2})}{\int }_0^1\frac{\mathrm{\mid }1-r^{\frac{\alpha -1}{p}}{\mathrm{\mid }}^p}{(1-r{)}^{1+\alpha }}dr(15)\backslash mathcal\{D\}\_\{n,p,\backslash alpha\}\; =\; 2\backslash pi^\{\backslash frac\{n-1\}\{2\}\}\; \backslash frac\{\backslash Gamma(\backslash frac\{1+\backslash alpha\}\{2\})\}\{\backslash Gamma(\backslash frac\{n+\backslash alpha\}\{2\})\}\; \backslash int\_0^1\; \backslash frac\{|1\; -\; r^\{\backslash frac\{\backslash alpha-1\}\{p\}\}|^p\}\{(1-r)^\{1+\backslash alpha\}\}\; dr\; \backslash quad\; (15)$$

is the sharp constant. In particular, for $\Omega$ convex

$${\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdy\ge {\mathcal{D}}_{n,p,\alpha }{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{d_{\mathrm{\Omega }}(x{)}^{\alpha }}dx\mathrm{.}(16)\backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash mathcal\{D\}\_\{n,p,\backslash alpha\}\; \backslash int\_\{\backslash Omega\}\; \backslash frac\{|f(x)|^p\}\{d\_\backslash Omega(x)^\backslash alpha\}\; dx.\; \backslash quad\; (16)$$

The constant $\mathcal{D}_{n,p,s}$ has been computed before in [11] as the sharp constant for the Hardy inequality for the half-space. For $0 < p \leq 1$ the inequality continues to hold (see [5]), however, the sharp constant is not known.

In the next section we establish the analogous one dimensional inequalities and then show how an averaging argument leads to the general result. At the end of Section 2 we indicatehow to obtain the result for general values of $p$ . We are grateful to Rupert Frank and Robert Seiringer to allow us to include arguments in our work. We present them at the end of our paper.

Acknowledgment: M.L. would like to thank the Erwin Schrödinger Institut for its kind hospitality and useful discussions with Thomas Hoffmann-Ostenhof. The authors thank an anonymous referee for suggesting various improvements of the manuscript.

2. THE ONE DIMENSIONAL PROBLEM

The proof of Theorem 1.1 will rely heavily on the following one dimensional inequality.

Theorem 2.1. Let $f \in C_c^\infty((a, b))$ . Then for all $1 < \alpha < 2$ we have

$$\frac{1}{2}{\int }_{(a,b)\times (a,b)}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\kappa }_{1,\alpha }{\int }_a^b\mathrm{\mid }f(x){\mathrm{\mid }}^2{(\frac{1}{x-a}+\frac{1}{b-x})}^{\alpha }dx\mathrm{.}(17)\backslash frac\{1\}\{2\}\; \backslash int\_\{(a,b)\; \backslash times\; (a,b)\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{1,\backslash alpha\}\; \backslash int\_a^b\; |f(x)|^2\; \backslash left(\; \backslash frac\{1\}\{x-a\}\; +\; \backslash frac\{1\}\{b-x\}\; \backslash right)^\backslash alpha\; dx.\; \backslash quad\; (17)$$

The idea of proving Theorem 2.1 is to reduce the problem on the interval to a problem on the half-line via a fractional linear mapping. The reader may consult [4] for further examples where inversion symmetry is used to obtain sharp functional inequalities.

Lemma 2.2 (Invariance under fractional linear transformations). Let $f$ be any function in $C_c^\infty(\mathbb{R} \setminus {0})$ . Consider the inversion $x \rightarrow 1/x$ and set

$$g(x)=I(f)(x):=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha -1}f\left(\frac{1}{x}\right)\mathrm{.}g(x)\; =\; I(f)(x)\; :=\; |x|^\{\backslash alpha-1\}\; f\backslash left(\backslash frac\{1\}\{x\}\backslash right).$$

Then $g \in C_c^\infty(\mathbb{R})$ and

$${\int }_{\mathbb{R}\times \mathbb{R}}\frac{\mathrm{\mid }g(x)-g(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy={\int }_{\mathbb{R}\times \mathbb{R}}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\mathrm{.}(18)\backslash int\_\{\backslash mathbb\{R\}\; \backslash times\; \backslash mathbb\{R\}\}\; \backslash frac\{|g(x)\; -\; g(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; =\; \backslash int\_\{\backslash mathbb\{R\}\; \backslash times\; \backslash mathbb\{R\}\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy.\; \backslash quad\; (18)$$

Proof. For fixed $\varepsilon$ consider the regions

$$R_1:=\{(x,y)\in {\mathbb{R}}^2:\mid \frac{x}{y}\mid >1+\epsilon \},R\_1\; :=\; \backslash \{(x,\; y)\; \backslash in\; \backslash mathbb\{R\}^2\; :\; \backslash left|\; \backslash frac\{x\}\{y\}\; \backslash right|\; >\; 1\; +\; \backslash varepsilon\backslash \},$$

and likewise,

$$R_2:=\{(x,y)\in {\mathbb{R}}^2:\mid \frac{y}{x}\mid >1+\epsilon \}\mathrm{.}R\_2\; :=\; \backslash \{(x,\; y)\; \backslash in\; \backslash mathbb\{R\}^2\; :\; \backslash left|\; \backslash frac\{y\}\{x\}\; \backslash right|\; >\; 1\; +\; \backslash varepsilon\backslash \}.$$

Now by changing variables $x \rightarrow 1/x$ and $y \rightarrow 1/y$ we find that

which, by symmetry under exchange of $x$ and $y$ ,

$$={\int }_{R_1\cup R_2}\frac{\mathrm{\mid }g(x)-g(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy+2{\int }_{R_1\cup R_2}\frac{\mathrm{\mid }f(1\mathrm{/}x){\mathrm{\mid }}^2(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha -1}\mathrm{\mid }y{\mathrm{\mid }}^{\alpha -1}-\mathrm{\mid }x{\mathrm{\mid }}^{2(\alpha -1)})}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\mathrm{.}=\; \backslash int\_\{R\_1\; \backslash cup\; R\_2\}\; \backslash frac\{|g(x)\; -\; g(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; +\; 2\; \backslash int\_\{R\_1\; \backslash cup\; R\_2\}\; \backslash frac\{|f(1/x)|^2\; (|x|^\{\backslash alpha-1\}\; |y|^\{\backslash alpha-1\}\; -\; |x|^\{2(\backslash alpha-1)\})\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy.$$We can write the second term as

$${\int }_{\mathbb{R}}\mathrm{\mid }f(1\mathrm{/}x){\mathrm{\mid }}^2\mathrm{\mid }x{\mathrm{\mid }}^{\alpha -2}{\int }_{\{\mathrm{\mid }s\mathrm{\mid }>1+\epsilon \}\cup \{\frac{1}{\mathrm{\mid }s\mathrm{\mid }}>1+\epsilon \}}\frac{\mathrm{\mid }s{\mathrm{\mid }}^{\alpha -1}-1}{\mathrm{\mid }1-s{\mathrm{\mid }}^{1+\alpha }}ds\mathrm{.}\backslash int\_\{\backslash mathbb\{R\}\}\; |f(1/x)|^2\; |x|^\{\backslash alpha-2\}\; \backslash int\_\{\backslash \{|s|>1+\backslash varepsilon\backslash \}\; \backslash cup\; \backslash \{\backslash frac\{1\}\{|s|\}>1+\backslash varepsilon\backslash \}\}\; \backslash frac\{|s|^\{\backslash alpha-1\}\; -\; 1\}\{|1-s|^\{1+\backslash alpha\}\}\; ds\; .$$

The integral

and by changing the variable $s \rightarrow 1/s$ in the last integral we find that this sum vanishes. Letting $\varepsilon \rightarrow 0$ yields (18). $\square$

Proof of Theorem 2.1. By translation and scaling it suffices to prove the result for the interval $(0, 1)$ . Let $f \in C_c^\infty((0, 1))$ . We have to show that

$$\frac{1}{2}{\int }_{(0,1)\times (0,1)}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\kappa }_{1,\alpha }{\int }_0^1\mathrm{\mid }f(x){\mathrm{\mid }}^2{(\frac{1}{x}+\frac{1}{1-x})}^{\alpha }dx\mathrm{.}(19)\backslash frac\{1\}\{2\}\; \backslash int\_\{(0,1)\; \backslash times\; (0,1)\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{1,\backslash alpha\}\; \backslash int\_0^1\; |f(x)|^2\; \backslash left(\; \backslash frac\{1\}\{x\}\; +\; \backslash frac\{1\}\{1-x\}\; \backslash right)^\backslash alpha\; dx\; .\; \backslash quad\; (19)$$

Set

$$g(x)=\mathrm{\mid }x+1{\mathrm{\mid }}^{\alpha -1}f\left(\frac{1}{1+x}\right)\mathrm{.}g(x)\; =\; |x+1|^\{\backslash alpha-1\}\; f\backslash left(\backslash frac\{1\}\{1+x\}\backslash right)\; .$$

Clearly, $g \in C_c^\infty((0, \infty))$ . Note that

$$g(x)=I(f)(x+1)g(x)\; =\; I(f)(x+1)$$

and hence we may use Lemma 2.2 and find that

Some of the integrals are easily evaluated and yield

Using the sharp Hardy inequality of Bogdan and Dyda [3] on the half-line yields

Changing variables, i.e., expressing everything in terms of the function $f$ , we arrive at the inequality

Finally, we note that for $1 < \alpha < 2$

$$1-x^{\alpha }-(1-x{)}^{\alpha }\ge 01\; -\; x^\backslash alpha\; -\; (1-x)^\backslash alpha\; \backslash geq\; 0$$

which proves the inequality (19). $\square$

Theorem 2.1 generalizes easily to open sets on the real line.

Corollary 2.3. Let $J \subset \mathbb{R}$ be an open set and $1 < \alpha < 2$ . Then for any $f \in C_c^\infty(J)$

$$\frac{1}{2}{\int }_{J\times J}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\kappa }_{1,\alpha }{\int }_J\mathrm{\mid }f(x){\mathrm{\mid }}^2{(\frac{1}{d_J(x)}+\frac{1}{{\delta }_J(x)})}^{\alpha }dx,(24)\backslash frac\{1\}\{2\}\; \backslash int\_\{J\; \backslash times\; J\}\; \backslash frac\{|f(x)\; -\; f(y)|^2\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash kappa\_\{1,\backslash alpha\}\; \backslash int\_J\; |f(x)|^2\; \backslash left(\; \backslash frac\{1\}\{d\_J(x)\}\; +\; \backslash frac\{1\}\{\backslash delta\_J(x)\}\; \backslash right)^\backslash alpha\; dx,\; \backslash quad\; (24)$$

where $\delta_J(x)$ is defined in (5).

Proof. Since any open set $J \subset \mathbb{R}$ is a countable union of disjoint intervals $I_k$ we find, using Theorem 2.1, that

$\square$

Lemma 2.4 (Reduction to one dimension). Let $\Omega$ be any region in $\mathbb{R}^n$ and assume that $f \in C_c^\infty(\Omega)$ . Then

where $\mathcal{L}_w$ denotes the $(n-1)$ dimensional Lebesgue measure on the plane $x \cdot w = 0$ .

Proof. We write the expression

$$I_{\mathrm{\Omega }}(f):={\int }_{\mathrm{\Omega }\times \mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+\alpha }}dxdyI\_\backslash Omega(f)\; :=\; \backslash int\_\{\backslash Omega\; \backslash times\; \backslash Omega\}\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{n+\backslash alpha\}\}\; dx\; dy$$

in the form

$${\int }_{\mathrm{\Omega }}dx{\int }_{\{x+z\in \mathrm{\Omega }\}}dz\frac{\mathrm{\mid }f(x)-f(x+z){\mathrm{\mid }}^p}{\mathrm{\mid }z{\mathrm{\mid }}^{n+\alpha }}\backslash int\_\backslash Omega\; dx\; \backslash int\_\{\backslash \{x+z\; \backslash in\; \backslash Omega\backslash \}\}\; dz\; \backslash frac\{|f(x)\; -\; f(x+z)|^p\}\{|z|^\{n+\backslash alpha\}\}$$and using polar coordinates $z = rw$ we arrive at the expression

Thus, the domain of integration in the innermost integral is the line $x + hw$ intersected with the domain $\Omega$ . Splitting the variable $x$ into a component perpendicular to $w$ and parallel to $w$ , i.e., replacing $x$ by $x + sw$ , where $x \cdot w = 0$ , we arrive at

$$\frac{1}{2}{\int }_{{\mathbb{S}}^{n-1}}dw{\int }_{\{x:x\cdot w=0\}}d{\mathcal{L}}_w(x){\int }_{\{x+sw\in \mathrm{\Omega }\}}ds{\int }_{\{x+(s+h)w\in \mathrm{\Omega }\}}dh\frac{\mathrm{\mid }f(x+sw)-f(x+(s+h)w){\mathrm{\mid }}^p}{\mathrm{\mid }h{\mathrm{\mid }}^{1+\alpha }}\backslash frac\{1\}\{2\}\; \backslash int\_\{\backslash mathbb\{S\}^\{n-1\}\}\; dw\; \backslash int\_\{\backslash \{x:\; x\; \backslash cdot\; w\; =\; 0\backslash \}\}\; d\backslash mathcal\{L\}\_w(x)\; \backslash int\_\{\backslash \{x+sw\; \backslash in\; \backslash Omega\backslash \}\}\; ds\; \backslash int\_\{\backslash \{x+(s+h)w\; \backslash in\; \backslash Omega\backslash \}\}\; dh\; \backslash frac\{|f(x+sw)\; -\; f(x+(s+h)w)|^p\}\{|h|^\{1+\backslash alpha\}\}$$

The variable change $t = s + h$ yields (26). $\square$

Proof of Theorem 1.1. By Lemma 2.4 and Corollary 2.3 we find that

where we have used (7) in the last equation. It remains to show that the constant $\kappa_{n,\alpha}$ in the inequality (12) is best possible. Pick a hyper-plane $H$ that is tangent to $\Omega$ at a point $P$ . Such hyper-planes exist since $\Omega$ is convex. It was shown in [3] that the constant for the half-space problem, $\kappa_{n,\alpha}$ , is best possible by constructing a sequence of trial functions. Transplanting these trial functions to $\Omega$ near the point $P$ one can show that $\kappa_{n,\alpha}$ is also optimal for (12). The actual proof is a straightforward imitation of the proof of Theorem 5 in [13] and we omit the details. $\square$

We finally come to the proof of Theorem 1.2. We thank Rupert Frank and Robert Seiringer for allowing us to present their argument.

Theorem 2.5. Let $1 < p < \infty$ and $1 < \alpha < p$ . Then for all smooth functions $f$ with $f(0) = 0$ ,

$${\int }_0^1{\int }_0^1\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\mathcal{D}}_{1,p,\alpha }{\int }_0^1\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{x^{\alpha }}dx\mathrm{.}\backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash mathcal\{D\}\_\{1,p,\backslash alpha\}\; \backslash int\_0^1\; \backslash frac\{|f(x)|^p\}\{x^\{\backslash alpha\}\}\; dx.$$

Proof. Let $\omega(x) = x^{(\alpha-1)/p}$ . Then by [11, Lemma 2.4]

$$2{\int }_0^{\mathrm{\infty }}(\omega (x)-\omega (y))\mathrm{\mid }\omega (x)-\omega (y){\mathrm{\mid }}^{p-2}\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}=\frac{{\mathcal{D}}_{1,p,\alpha }}{x^{\alpha }}\omega (x{)}^{p-1}2\; \backslash int\_0^\{\backslash infty\}\; (\backslash omega(x)\; -\; \backslash omega(y))\; |\backslash omega(x)\; -\; \backslash omega(y)|^\{p-2\}\; \backslash frac\{dy\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; =\; \backslash frac\{\backslash mathcal\{D\}\_\{1,p,\backslash alpha\}\}\{x^\{\backslash alpha\}\}\; \backslash omega(x)^\{p-1\}$$

where the integral is understood in principal value sense. Since

$${\int }_1^{\mathrm{\infty }}(\omega (x)-\omega (y))\mathrm{\mid }\omega (x)-\omega (y){\mathrm{\mid }}^{p-2}\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}\le 0\text{for }x\in [0,1],\backslash int\_1^\{\backslash infty\}\; (\backslash omega(x)\; -\; \backslash omega(y))\; |\backslash omega(x)\; -\; \backslash omega(y)|^\{p-2\}\; \backslash frac\{dy\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; \backslash leq\; 0\; \backslash quad\; \backslash text\{for\; \}\; x\; \backslash in\; [0,\; 1],$$we conclude that

$$V(x):=\frac{2}{\omega (x{)}^{p-1}}{\int }_0^1(\omega (x)-\omega (y))\mathrm{\mid }\omega (x)-\omega (y){\mathrm{\mid }}^{p-2}\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}\ge \frac{{\mathcal{D}}_{1,p,\alpha }}{x^{\alpha }}\text{for }x\in [0,1]\mathrm{.}V(x)\; :=\; \backslash frac\{2\}\{\backslash omega(x)^\{p-1\}\}\; \backslash int\_0^1\; (\backslash omega(x)\; -\; \backslash omega(y))\; |\backslash omega(x)\; -\; \backslash omega(y)|^\{p-2\}\; \backslash frac\{dy\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; \backslash geq\; \backslash frac\{\backslash mathcal\{D\}\_\{1,p,\backslash alpha\}\}\{x^\backslash alpha\}\; \backslash quad\; \backslash text\{for\; \}\; x\; \backslash in\; [0,\; 1].$$

Now [10, Prop. 2.2] implies that

$${\int }_0^1{\int }_0^1\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\int }_0^{1}V(x)\mathrm{\mid }f(x){\mathrm{\mid }}^{p}dx,\backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash int\_0^1\; V(x)\; |f(x)|^p\; dx,$$

which proves the claim. $\square$

An easy consequence is

Theorem 2.6. Let $f \in C_c^\infty((a, b))$ . Then for all $1 < p < \infty$ and $1 < \alpha < p$ we have

$$\frac{1}{2}{\int }_{(a,b)\times (a,b)}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\mathcal{D}}_{1,p,\alpha }{\int }_a^b\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{\min \{(x-a),(b-x){\}}^{\alpha }}dx\mathrm{.}(29)\backslash frac\{1\}\{2\}\; \backslash int\_\{(a,b)\; \backslash times\; (a,b)\}\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash mathcal\{D\}\_\{1,p,\backslash alpha\}\; \backslash int\_a^b\; \backslash frac\{|f(x)|^p\}\{\backslash min\backslash \{(x\; -\; a),\; (b\; -\; x)\backslash \}^\backslash alpha\}\; dx.\; \backslash quad\; (29)$$

Exactly the same proof as the one of Corollary 2.3 yields

Corollary 2.7. Let $1 < p < \infty$ and $1 < \alpha < p$ . Let $J \subset \mathbb{R}$ be open and $f$ a function on $J$ with $f \in C_c^\infty(J)$ , then

$${\int }_J{\int }_J\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{1+\alpha }}dxdy\ge {\mathcal{D}}_{1,p,\alpha }{\int }_J\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{d_J(x{)}^{\alpha }}dx\mathrm{.}\backslash int\_J\; \backslash int\_J\; \backslash frac\{|f(x)\; -\; f(y)|^p\}\{|x\; -\; y|^\{1+\backslash alpha\}\}\; dx\; dy\; \backslash geq\; \backslash mathcal\{D\}\_\{1,p,\backslash alpha\}\; \backslash int\_J\; \backslash frac\{|f(x)|^p\}\{d\_J(x)^\backslash alpha\}\; dx.$$

Proof of Theorem 1.2. The proof is a repetition of the arguments in the proof of Theorem 1.1. $\square$

REFERENCES

• [1] Barbatis, G., Filippas, S., Tertikas, A. A, unified approach to improved $L^p$ Hardy inequalities with best constants. Trans. Amer. Math. Soc. 356 (2004), no. 6, 2169–2196
• [2] Bogdan, K., Burdzy, K. and Chen, Z-Q., Censored stable processes, Probab. Theory Relat. Fields, 127, 89–152, 2003.
• [3] Bogdan, K. and Dyda, B., The best constant in a fractional Hardy inequality, arXiv:0807.1825.
• [4] Carlen, E. A. and Loss, M., On the minimization of symmetric functionals, Reviews in Mathematical Physics, Special Issue, (1994) 1011–1032.
• [5] Dyda, B., A fractional order Hardy inequality, Illinois J. of Math., 48, 575–588, 2004.
• [6] Chen, Z-Q. and Song, R., Hardy inequality for censored stable processes, Tohoku Math. J. 55, 439–450, 2003.
• [7] Davies, E.B., Heat kernels and spectral theory, Cambridge Tracts in Mathematics, Cambridge University Press, 1989.
• [8] Davies, E.B., Some norm bounds and quadratic form inequalities for Schrödinger operators, II, J. Oper. Theory 12, 177–196, 1984.
• [9] Davies, E. B. A review of Hardy inequalities. The Maz’ya anniversary collection, Vol. 2 (Rostock, 1998), 55–67, Oper. Theory Adv. Appl., 110, Birkhuser, Basel, 1999.
• [10] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), 3407 – 3430.
• [11] Frank, R. and Seiringer, R., Sharp fractional Hardy inequalities in half-spaces, arXiv:0906.1561.
• [12] H. P. Heinig, A. Kufner, and L.-E. Persson, On some fractional order Hardy inequalities, J. Inequal. Appl. 1 (1997), 2546.
• [13] Marcus M., Mizel V.J. and Pinchover Y. , On the best constant for Hardys inequality in $\mathbb{R}^n$ . Trans. Amer. Math. Soc. 350 (1998), 3237–3255.
• [14] Matskewich T. and Sobolevskii P.E. , The best possible constant in generalized Hardys inequality for convex domain in $\mathbb{R}^n$ . Nonlinear Anal., Theory, Methods and Appl., 28 (1997), 1601–1610.GEORGIA INSTITUTE OF TECHNOLOGY, SCHOOL OF MATHEMATICS, ATLANTA GA 30332-0160

loss@math.gatech.edu , csloane@math.gatech.edu