Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multidimensional Frank-Laptev-Weidl improvement of the Hardy Inequality

Prasun Roychowdhury, Michael Ruzhansky, Durvudkhan Suragan

TL;DR

This work extends the classical $L^p$ Hardy inequality on $\mathbb{R}^N$ to the supercritical range $p>N$ by proving a sharp multidimensional improvement formulated in polar coordinates as the maximum of two radial supremum functionals. The authors first establish a one-dimensional rearrangement-based Hardy-type estimate and then lift it to higher dimensions via symmetric decreasing rearrangements and radialisation, yielding a radial version and a non-radial extension with the sharp constant $|\frac{p}{N-p}|^p$. The resulting inequality strengthens Hardy’s bound and has implications for PDE and spectral theory, plus a derived HPW-type uncertainty principle. Overall, the paper provides a novel multidimensional framework that extends the Frank–Laptev–Weidl improvement from 1D to $1\le N<p$, highlighting the power of rearrangement techniques in combination with polar geometry.

Abstract

We establish a new improvement of the classical $L^p$-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.

Multidimensional Frank-Laptev-Weidl improvement of the Hardy Inequality

TL;DR

This work extends the classical Hardy inequality on to the supercritical range by proving a sharp multidimensional improvement formulated in polar coordinates as the maximum of two radial supremum functionals. The authors first establish a one-dimensional rearrangement-based Hardy-type estimate and then lift it to higher dimensions via symmetric decreasing rearrangements and radialisation, yielding a radial version and a non-radial extension with the sharp constant . The resulting inequality strengthens Hardy’s bound and has implications for PDE and spectral theory, plus a derived HPW-type uncertainty principle. Overall, the paper provides a novel multidimensional framework that extends the Frank–Laptev–Weidl improvement from 1D to , highlighting the power of rearrangement techniques in combination with polar geometry.

Abstract

We establish a new improvement of the classical -Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.
Paper Structure (9 sections, 9 theorems, 46 equations)

This paper contains 9 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric decreasing rearrangements
  4. Polar coordinates and radial gradient
  5. Supporting lemmas
  6. Improvement of the classical Hardy inequality
  7. Radial version of the results
  8. Non-radial setting of the results
  9. Uncertainty principle

Key Result

Lemma 3.1

Let $1<p<\infty$. Let $g$ be any nonnegative function on $(0,\infty)$. Assume $h$ is a strictly positive non-decreasing function on $(0,\infty)$ such that $s h(r)\leq r h(s)$ for any $r,s\in(0,\infty)$ with $r\leq s$. Let $f$ be a locally absolutely continuous function on $(0,\infty)$. Then we have where $f^*$ is the non-increasing rearrangement of $f$.

Theorems & Definitions (18)

  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • Lemma 3.2
  • Theorem 4.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 8 more