On Removable Sets for Weighted Sobolev Functions

2025/05/26 by Esmayli, Behnam, Mishra, Riddhi
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E35 #Secondary 46E36

paper · doi:10.48550/arxiv.2505.20555

Abstract

We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted \mathbb Rⁿ, 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.

Citations

• Capacitary density and removable sets for Newton-Sobolev functions in metric spaces
• Removability of product sets for Sobolev functions in the plane
• Removable sets for weighted Orlicz-Sobolev spaces
• A density problem for Sobolev spaces on Gromov hyperbolic domains
• The Poincaré inequality is an open ended condition
• Modulus and the Poincar� inequality on metric measure spaces

Related

• Removable sets for weighted Orlicz-Sobolev spaces
• Capacitary density and removable sets for Newton-Sobolev functions in metric spaces
• Removable sets for Orlicz-Sobolev spaces
• Removable singularities for div v = f in weighted Lebesgue spaces
• Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces