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Hardy-Sobolev interpolation inequalities

Charlotte Dietze, Phan Thành Nam

TL;DR

The paper develops Hardy–Sobolev interpolation inequalities that extend Hardy's inequality to the Sobolev critical exponent by replacing the $L^2$ norm with the energy of the inverse-square potential, and extends to Lorentz norms $L^{p^*,r}$. It provides endpoint results in the radial case with explicit optimizer families $u_\eta$ (for $L^{2^*}$) and $u_c$ (for $L^{2^*,\infty}$), linking to Terracini's radial solutions. In the general case, it proves two main endpoint results: (i) the special case $d=3$, $\theta=1/3$ and (ii) a Lorentz-norm interpolation with sharp $\theta$-range $\frac{p}{\min(r,p^*)} \le \theta \le \frac{1}{p}-\frac{1}{r}$, including sharpness via bubble-type counterexamples. The work clarifies the role of inverse-square energies in critical embeddings and lays groundwork for optimizer existence questions beyond radial symmetry.

Abstract

We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers.

Hardy-Sobolev interpolation inequalities

TL;DR

The paper develops Hardy–Sobolev interpolation inequalities that extend Hardy's inequality to the Sobolev critical exponent by replacing the norm with the energy of the inverse-square potential, and extends to Lorentz norms . It provides endpoint results in the radial case with explicit optimizer families (for ) and (for ), linking to Terracini's radial solutions. In the general case, it proves two main endpoint results: (i) the special case , and (ii) a Lorentz-norm interpolation with sharp -range , including sharpness via bubble-type counterexamples. The work clarifies the role of inverse-square energies in critical embeddings and lays groundwork for optimizer existence questions beyond radial symmetry.

Abstract

We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers.
Paper Structure (5 sections, 3 theorems, 85 equations)

This paper contains 5 sections, 3 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Radial case
  3. General case
  4. Proof of Theorem \ref{['thm:1']}
  5. Proof of Theorem \ref{['thm:2']}

Key Result

Theorem 1

If $d=3$ and $\theta=1/3$, then the inequality holds with a constant $C=C(d,\theta)>0$ independent of $u\in \dot H^1(\mathbb{R}^d)$. Moreover, eq:Hardy-improved-1 does not hold if $d\ge 4$ or if $\theta \ne 1/3$.

Theorems & Definitions (12)

  • Theorem 1: Hardy-Sobolev interpolation inequality
  • Remark 1
  • Remark 2
  • Theorem 2: Hardy-Sobolev inequalities with Lorentz norms
  • Theorem 3: Radial optimizers
  • Remark 3
  • Remark 4
  • Remark 5
  • proof : Proof of \ref{['eq:Hardy-improved-1']} for $d=3=1/\theta$
  • proof : Proof of the necessity of $d=3=1/\theta$
  • ...and 2 more