Lebesgue Points of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness

Abstract

In this article, the authors study the Lebesgue point of functions from Hajłasz–Sobolev, Besov, and Triebel–Lizorkin spaces with generalized smoothness on doubling metric measure spaces and prove that the exceptional sets of their Lebesgue points have zero capacity via the capacities related to these spaces. In case these functions are not locally integrable, the authors also consider their generalized Lebesgue points defined via the γ-medians instead of the classical ball integral averages and establish the corresponding zero-capacity property of the exceptional sets.

Keywords

• Hajłasz–Sobolev space
• Hajłasz–Besov space
• Hajłasz–Triebel–Lizorkin space
• generalized smoothness
• Lebesgue point
• capacity