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Lebesgue points of functions in the complex Sobolev space

Gabriel Vigny, Duc-Viet Vu

Abstract

Let $\varphi$ be a function in the complex Sobolev space $W^*(U)$, where $U$ is an open subset in $\mathbb{C}^k$. We show that the complement of the set of Lebesgue points of $\varphi$ is pluripolar. The key ingredient in our approach is to show that $|\varphi|^α$ for $α\in [1,2)$ is locally bounded from above by a plurisubharmonic function.

Lebesgue points of functions in the complex Sobolev space

Abstract

Let be a function in the complex Sobolev space , where is an open subset in . We show that the complement of the set of Lebesgue points of is pluripolar. The key ingredient in our approach is to show that for is locally bounded from above by a plurisubharmonic function.
Paper Structure (5 sections, 119 equations)

This paper contains 5 sections, 119 equations.

Table of Contents

  1. Introduction
  2. Plurisubharmonic upper bound for functions in $W^*(\mathbb{B})$
  3. Energy estimates
  4. Capacity estimates
  5. Proof of Theorem \ref{['th:Lebesgues']}

Theorems & Definitions (4)

  • proof
  • proof
  • proof : End of the proof of Theorem \ref{['tm:main?']}
  • proof : End of the proof of Theorem \ref{['th:Lebesgues']}