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One-dimensional integral Rellich type inequalities

Tohru Ozawa, Prasun Roychowdhury, Durvudkhan Suragan

TL;DR

The paper addresses sharp one-dimensional Hardy–Rellich and Rellich-type inequalities in integral form, motivated by a recent sharp Hardy-type inequality. It leverages symmetric decreasing rearrangement and a key supremum identity to reduce the problem to rearranged functions, yielding a sharp Hardy–Rellich integral inequality in the $p=2$ case with constant $4$ and a sharp Rellich-type integral inequality for general $p>1$ with constant $\frac{p^{2p}}{(p-1)^p(2p-1)^p}$. Sharpness is established via explicitly constructed minimizing sequences, filling a gap in one-dimensional integral-inequality theory. These results contribute precise integral-forms and sharp constants with potential applications in spectral theory and PDE analysis.

Abstract

The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy-Rellich type inequality improves the previously known result. Meanwhile, the established sharp Rellich type integral inequality seems new.

One-dimensional integral Rellich type inequalities

TL;DR

The paper addresses sharp one-dimensional Hardy–Rellich and Rellich-type inequalities in integral form, motivated by a recent sharp Hardy-type inequality. It leverages symmetric decreasing rearrangement and a key supremum identity to reduce the problem to rearranged functions, yielding a sharp Hardy–Rellich integral inequality in the case with constant and a sharp Rellich-type integral inequality for general with constant . Sharpness is established via explicitly constructed minimizing sequences, filling a gap in one-dimensional integral-inequality theory. These results contribute precise integral-forms and sharp constants with potential applications in spectral theory and PDE analysis.

Abstract

The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy-Rellich type inequality improves the previously known result. Meanwhile, the established sharp Rellich type integral inequality seems new.
Paper Structure (3 sections, 4 theorems, 31 equations)

This paper contains 3 sections, 4 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Proofs of Theorems \ref{['hardy-rel-th']} and \ref{['rel-th']}

Key Result

Lemma 2.1

Let $f$ be a locally absolutely continuous function on $(0,\infty)$. Then for a fixed $r>0$, the following identity holds: where $f^*$ is the non-increasing rearrangement of $f$.

Theorems & Definitions (5)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1