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On fractional Orlicz boundary Hardy inequalities

Subhajit Roy

TL;DR

The article studies fractional boundary Hardy inequalities in Orlicz spaces on bounded Lipschitz domains, replacing the power nonlinearity with a Δ2-Young function $\Phi$ and addressing critical regimes with logarithmic corrections. The authors develop a framework of Orlicz-Poincaré inequalities, dyadic decompositions, and Lipschitz-change-of-variable techniques to prove Hardy-type bounds of the form $\int_{\Omega} \Phi\bigl(|u|/\delta_\Omega^{s}\bigr)dx \le C \iint_{\Omega}\Phi\bigl(|D_su(x,y)|\bigr)d\mu$, with log-corrections when $sp^+_\Phi=1$ or $sp^-_\Phi=N$ in certain domain classes $\mathcal{A}_i$. The results cover domains above Lipschitz graphs and exterior domains, and include weighted variants in $\mathbb{R}^N_+$, thereby extending previous scalar-power results to general Δ2-Young functions. These findings advance nonlocal boundary analysis in Orlicz spaces and have potential applications in stochastic processes and boundary behavior of fractional operators.

Abstract

We investigate the fractional Orlicz boundary Hardy-type inequality for bounded Lipschitz domains. Further, we establish fractional Orlicz boundary Hardy-type inequalities with logarithmic corrections for specific critical cases across various domains, such as bounded Lipschitz domains, domains above the graph of a Lipschitz function, and the complement of a bounded Lipschitz domain.

On fractional Orlicz boundary Hardy inequalities

TL;DR

The article studies fractional boundary Hardy inequalities in Orlicz spaces on bounded Lipschitz domains, replacing the power nonlinearity with a Δ2-Young function and addressing critical regimes with logarithmic corrections. The authors develop a framework of Orlicz-Poincaré inequalities, dyadic decompositions, and Lipschitz-change-of-variable techniques to prove Hardy-type bounds of the form , with log-corrections when or in certain domain classes . The results cover domains above Lipschitz graphs and exterior domains, and include weighted variants in , thereby extending previous scalar-power results to general Δ2-Young functions. These findings advance nonlocal boundary analysis in Orlicz spaces and have potential applications in stochastic processes and boundary behavior of fractional operators.

Abstract

We investigate the fractional Orlicz boundary Hardy-type inequality for bounded Lipschitz domains. Further, we establish fractional Orlicz boundary Hardy-type inequalities with logarithmic corrections for specific critical cases across various domains, such as bounded Lipschitz domains, domains above the graph of a Lipschitz function, and the complement of a bounded Lipschitz domain.
Paper Structure (6 sections, 17 theorems, 135 equations)

This paper contains 6 sections, 17 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Properties of Young functions
  4. Some function spaces:
  5. Orlicz Hardy inequality in a domain with flat boundaries
  6. Fractional Orlicz boundary Hardy-type inequalities

Key Result

Theorem 1.3

Let $N\geq 1,\,s\in (0,1),$ and $\Omega\in \mathcal{A}_1$. Let $R_\Omega:=\sup\{\delta_\Omega(x):x\in\Omega\}$ and $R\ge R_\Omega$. For any Young function $\Phi\in \bigtriangleup_2,$ if $sp^+_\Phi=1,$ then there exists $C=C(s,N,\Phi,\Omega)>0$ so that

Theorems & Definitions (31)

  • Definition 1.1: Young function
  • Definition 1.2: The $\bigtriangleup_2$-condition
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 21 more