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Capacity and Hausdorff measure in Musielak-Orlicz-Sobolev spaces

Ankur Pandey, Nijjwal Karak, Debarati Mondal

TL;DR

This work investigates how zero Sobolev capacity in variable exponent spaces and Musielak–Orlicz–Sobolev spaces implies vanishing generalized Hausdorff measures, extending classical capacity–Hausdorff results to more flexible growth conditions. By employing log-Hölder and related regularity assumptions on the exponent functions and gauge constructions $h(\cdot,t)$, it establishes both upper bounds of $\mathcal{H}^{h(\cdot)}(E)$ in terms of capacity and zero-measure implications under $C_{p(\cdot)}(E)=0$ or $C_{\Phi(\cdot,\cdot)}(E)=0$. The main contributions include precise gauge choices $h(\cdot,t)$ with logarithmic refinements and their dependence on $p(\cdot)$ and $q(\cdot)$, extending the link between fractal-type measures and capacities to variable exponent and Musielak–Orlicz growth. These results enhance understanding of fine properties of functions in $W^{1,p(\cdot)}$ and $W^{1,\Phi(\cdot,\cdot)}$ spaces and have potential implications for PDEs in nonstandard growth frameworks.

Abstract

In this paper, we show that sets with zero Sobolev $p(\cdot)$-capacity have generalized Hausdorff $h(\cdot)$-measure zero, for some gauge function $h(\cdot).$ We also prove that sets with zero Musielak-Orlicz-Sobolev $Φ(\cdot,\cdot)$-capacity, for a particular class of functions $Φ(\cdot,\cdot),$ have generalized Hausdorff $h(\cdot)$-measure zero, for a suitable gauge function $h(\cdot).$

Capacity and Hausdorff measure in Musielak-Orlicz-Sobolev spaces

TL;DR

This work investigates how zero Sobolev capacity in variable exponent spaces and Musielak–Orlicz–Sobolev spaces implies vanishing generalized Hausdorff measures, extending classical capacity–Hausdorff results to more flexible growth conditions. By employing log-Hölder and related regularity assumptions on the exponent functions and gauge constructions , it establishes both upper bounds of in terms of capacity and zero-measure implications under or . The main contributions include precise gauge choices with logarithmic refinements and their dependence on and , extending the link between fractal-type measures and capacities to variable exponent and Musielak–Orlicz growth. These results enhance understanding of fine properties of functions in and spaces and have potential implications for PDEs in nonstandard growth frameworks.

Abstract

In this paper, we show that sets with zero Sobolev -capacity have generalized Hausdorff -measure zero, for some gauge function We also prove that sets with zero Musielak-Orlicz-Sobolev -capacity, for a particular class of functions have generalized Hausdorff -measure zero, for a suitable gauge function
Paper Structure (3 sections, 16 theorems, 74 equations)

This paper contains 3 sections, 16 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Main Results

Key Result

Theorem 1.2

Suppose that $p:\mathbb{R}^n\rightarrow [1,n]$ satisfies both the log-Hölder continuity and the log-Hölder decay condition and let $s:\mathbb{R}^n\rightarrow (0,n]$ be continuous with $s(\cdot)>n-p(\cdot).$ If $C_{p(\cdot)}(E)=0,$ then $H^{s(\cdot)}(E)=0.$

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2: Theorem 10.4.7, DHHR11
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 27 more