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Stein-Weiss, and power weight Korn type Hardy-Sobolev Inequalities in $L^1$ norm

Wen Qi Zhang

TL;DR

The paper extends endpoint $L^1$ Stein‑Weiss inequalities by replacing the cocanceling requirement with a vanishing moment condition for noninteger $a$, obtaining $L^1(|x|^a dx)$ bounds for all $0<a<1$ and mean zero data. It also establishes a Korn type Hardy‑Sobolev inequality in ${\bf R}^2$ for integer $a$ by exploiting extra cancellations of the symmetric gradient and its Green function, with an explicit construction that illustrates improvement beyond the standard duality estimates. The approach hinges on technical reductions of convolution kernels, moment vanishing, and refined cancellation mechanisms that enlarge admissible exponent regions, and it connects endpoint estimates to elasticity type inequalities in two dimensions. Together these results broaden the reach of endpoint Stein‑Weiss and Hardy‑Sobolev inequalities under weaker structural assumptions on the data and operators, highlighting the role of vanishing moments and Korn type cancellations in weighted contexts.

Abstract

We extend the $L^1$ Stein-Weiss inequalities studied by De Nápoli and Picon [4] in two ways: First we address an open question posed by the authors about whether the cocanceling condition was necessary for some of their Stein-Weiss inequalities. We replace the cocanceling condition with a weaker vanishing moment assumption, and under this assumption extend the $L^1$ Stein-Weiss inequalities to $L^1(|x|^{a } dx)$ data for all positive, non-integer exponents $a$. Second, in relation to integer exponents, while [4] showed that Stein-Weiss fails for $L^1(|x| dx)$ data, we prove a weaker Korn type Hardy-Sobolev inequality. These inequalities were previously inaccessible due to the growth of $|x|$, and we demonstrate a specific example on $\mathbb{R}^2$ of where the original duality estimate by Bousquet and Van Schaftingen [2] for canceling operators can be improved.

Stein-Weiss, and power weight Korn type Hardy-Sobolev Inequalities in $L^1$ norm

TL;DR

The paper extends endpoint Stein‑Weiss inequalities by replacing the cocanceling requirement with a vanishing moment condition for noninteger , obtaining bounds for all and mean zero data. It also establishes a Korn type Hardy‑Sobolev inequality in for integer by exploiting extra cancellations of the symmetric gradient and its Green function, with an explicit construction that illustrates improvement beyond the standard duality estimates. The approach hinges on technical reductions of convolution kernels, moment vanishing, and refined cancellation mechanisms that enlarge admissible exponent regions, and it connects endpoint estimates to elasticity type inequalities in two dimensions. Together these results broaden the reach of endpoint Stein‑Weiss and Hardy‑Sobolev inequalities under weaker structural assumptions on the data and operators, highlighting the role of vanishing moments and Korn type cancellations in weighted contexts.

Abstract

We extend the Stein-Weiss inequalities studied by De Nápoli and Picon [4] in two ways: First we address an open question posed by the authors about whether the cocanceling condition was necessary for some of their Stein-Weiss inequalities. We replace the cocanceling condition with a weaker vanishing moment assumption, and under this assumption extend the Stein-Weiss inequalities to data for all positive, non-integer exponents . Second, in relation to integer exponents, while [4] showed that Stein-Weiss fails for data, we prove a weaker Korn type Hardy-Sobolev inequality. These inequalities were previously inaccessible due to the growth of , and we demonstrate a specific example on of where the original duality estimate by Bousquet and Van Schaftingen [2] for canceling operators can be improved.

Paper Structure

This paper contains 16 sections, 11 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Outline
  4. Preliminaries
  5. Classical Results
  6. An Adjacent Result
  7. Canceling Framework
  8. Cocanceling Condition
  9. Further Cancellations
  10. Exponent Diagrams
  11. A Trivial Inequality
  12. Technical Reductions
  13. Stein-Weiss Inequalities for Zero-Moment Functions
  14. Korn Type Hardy-Sobolev Inequalities on ${\mathbb R}^2$
  15. Key Improvement
  16. ...and 1 more sections

Key Result

Theorem 1.1

$$ Let $E$ be a finite dimensional vector space, $n\geq 2$, $0<k<n$ and suppose that $0<a$, $q\in[1,\frac{n}{n-k})$ and $b$ are the exponents satisfying $\frac{n+b}{q}=n-k+a$. If $a\notin {\mathbb N}$ then the inequality holds for all $f\in C^{\infty}_c({\mathbb R}^n,E)$ (with implied constant independent of $f$) satisfying the vanishing moment assumption: for all multi-indices $\gamma\leq \lfloo

Figures (3)

  • Figure 1: Permissible exponents under the cocanceling condition.
  • Figure 2: Permissible exponents under the vanishing moment assumption.
  • Figure 3: Permissible exponents for power weight Hardy-Sobolev inequalties for $\textup{D}_{\textup{sym}}u$ on ${\mathbb R}^2$

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: A.P. Calderón and A. Zygmund, 1952 calderonzygmund
  • Theorem 2.2: D. Ornstein, 1962 Ornstein
  • Theorem 2.3: E. M. Stein and G. Weiss, 1958 sw58
  • Definition 2.4: Canceling
  • Definition 2.5: Cocanceling
  • Theorem 2.6: NapPic Theorem 1.2
  • Lemma 2.6
  • Lemma 3.1
  • ...and 8 more