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$L^p$ Hardy inequalities with homogeneous weights

Subhajit Roy

Abstract

For $p\in (1,\infty)$ and $α\in\mathbb{R}$, we consider measurable functions $g$ on $\mathbb{S}^{N-1}$ that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant $C>0$. Depending on $N$, $p$, and $α$, we identify suitable function spaces for $g$ so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when $g \equiv 1$. Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights.

$L^p$ Hardy inequalities with homogeneous weights

Abstract

For and , we consider measurable functions on that satisfy the following weighted Hardy inequality: \begin{equation}\label{abs} \int_{\mathbb{R}^N}\frac{ g (x/|x|)}{|x|^{p+α}}|u(x)|^p dx \leq C\int_{\mathbb{R}^N}\frac{|\nabla u(x)|^p}{|x|^α} dx, \quad\forall\,u\in \mathcal{C}_c^\infty(\mathbb{R}^N), \end{equation} for some constant . Depending on , , and , we identify suitable function spaces for so that \eqref{abs} holds. The constant obtained is sharp, in the sense that it is sharp when . Furthermore, we establish the sharp fractional Hardy inequality with homogeneous weights.

Paper Structure

This paper contains 6 sections, 11 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetrisation
  4. Sobolev embedding
  5. Weighted Hardy inequality:
  6. Weighted Hardy inequality

Key Result

Theorem 1.1

ari2015 Let $N\ge 3$ and $0\le g\in L^q(\mathbb{S}^{N-1}),$ where $q={\frac{(N-2)^2}{2(N-1)}+1}.$ Then the following inequality holds where Moreover, the inequality sharp-Hardy is sharp and reduces to the classical sharp form when $g\equiv 1.$

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 9 more