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Fractional Hardy inequality with singularity on submanifold

Adimurthi, Prosenjit Roy, Vivek Sahu

TL;DR

This work extends fractional Hardy inequalities to bounded domains in $\mathbb{R}^d$ with singularities supported on smooth submanifolds $K$ of codimension $k$, using the weight $\delta_K(x)^{-\alpha}\ln^{-*}(2R/\delta_K(x))$ where $\delta_K$ is the distance to $K$. It establishes the complete range of inequalities for $sp<k$, the critical case $sp=k$ with optimal logarithmic corrections, and the supercritical regime $sp>k$, providing explicit relations for $\alpha$ and the logarithmic exponent $\beta$ depending on $\tau$ and the dimension, and proving optimality in the critical case for $\tau=p$. The proofs decrease to a flat-boundary model, develop a dyadic decomposition and a slicing lemma, and then patch these local estimates to general Lipschitz domains via a partition of unity and coordinate charts, thus enabling a comprehensive, geometry-aware nonlocal Hardy theory. The results have potential applications to regional fractional $p$-Laplacian problems with singular weights and provide sharp tools for studying weighted nonlocal PDEs on manifolds of higher codimension. Overall, the paper advances fractional Sobolev inequalities in geometries with singularities beyond boundaries, revealing how distance-to-set geometry interacts with nonlocal interaction in critical and subcritical regimes.

Abstract

We establish fractional Hardy inequality on bounded domains in $\mathbb{R}^{d}$ with inverse of distance function from smooth boundary of codimension $k$, where $k=2, \dots,d$, as weight function. The case $sp=k$ is the critical case, where optimal logarithmic corrections are required. All the other cases of $sp<k$ and $sp>k$ are also addressed.

Fractional Hardy inequality with singularity on submanifold

TL;DR

This work extends fractional Hardy inequalities to bounded domains in with singularities supported on smooth submanifolds of codimension , using the weight where is the distance to . It establishes the complete range of inequalities for , the critical case with optimal logarithmic corrections, and the supercritical regime , providing explicit relations for and the logarithmic exponent depending on and the dimension, and proving optimality in the critical case for . The proofs decrease to a flat-boundary model, develop a dyadic decomposition and a slicing lemma, and then patch these local estimates to general Lipschitz domains via a partition of unity and coordinate charts, thus enabling a comprehensive, geometry-aware nonlocal Hardy theory. The results have potential applications to regional fractional -Laplacian problems with singular weights and provide sharp tools for studying weighted nonlocal PDEs on manifolds of higher codimension. Overall, the paper advances fractional Sobolev inequalities in geometries with singularities beyond boundaries, revealing how distance-to-set geometry interacts with nonlocal interaction in critical and subcritical regimes.

Abstract

We establish fractional Hardy inequality on bounded domains in with inverse of distance function from smooth boundary of codimension , where , as weight function. The case is the critical case, where optimal logarithmic corrections are required. All the other cases of and are also addressed.
Paper Structure (12 sections, 21 theorems, 217 equations, 4 figures)

This paper contains 12 sections, 21 theorems, 217 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The flat boundary case
  4. Proof of the main results
  5. Proof of Theorem \ref{['theorem 1 sp=k']} without optimality of the weight function
  6. Proof of Theorem \ref{['theorem 2 sp<k']}
  7. Proof of Theorem \ref{['theorem 3 sp>k']}
  8. Proof of Corollary \ref{['cor 1']}
  9. Optimality of Weight function in Theorem \ref{['theorem 1 sp=k']}
  10. Appendix
  11. Some inequality
  12. Slicing inequality

Key Result

Theorem 1

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$ with $d \geq 2$, and let $K \subset \Omega$ be a compact set of codimension $k$ of class $C^{0,1}$, where $1 < k \leq d$ and $k \in \mathbb{N}$. Assume that $sp = k$, $\delta_{K}(x) < R$ for all $x \in \Omega$, for some $R > 0$, and $\al where $C=C(d,p,s, \tau, R, K,\Omega)>0$, holds true in each of the following cases: The weight fun

Figures (4)

  • Figure 1: $Q= B^{k}_{1}(0) \times (0,1)^{d-k}$.
  • Figure 2: $A_{\ell}$.
  • Figure 3: $A^{i}_{\ell}= \{ x=(x_{k},x_{d-k}) \in A_{\ell} : x_{d-k} \in C^{i}_{\ell} \}$.
  • Figure 4: $A^{i}_{\ell}$ and $A^{j}_{\ell +1}$ .

Theorems & Definitions (42)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Lemma 2.1
  • ...and 32 more