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<p>We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, defined as the closure of smooth functions in the weighted Sobolev norm. Our porosity conditions are in terms of suitable coverings by cubes. The weights are assumed to be doubling and satisfy a Poincaré inequality, which includes, but is not equal to, the famous class of Muckenhoupt weights. Our proofs use ideas and techniques from the theory of analysis on metric spaces.</p>

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On Removable Sets for Weighted Sobolev Functions

  • 期刊论文

  • 发表日期：

    2025年12月10日

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  • Behnam Esmayli,
  • Riddhi Mishra

摘要

We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted \(\mathbb {R}^n\) R n , defined as the closure of smooth functions in the weighted Sobolev norm. Our porosity conditions are in terms of suitable coverings by cubes. The weights are assumed to be doubling and satisfy a Poincaré inequality, which includes, but is not equal to, the famous class of Muckenhoupt weights. Our proofs use ideas and techniques from the theory of analysis on metric spaces.