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

Ziwei Li,Wen Yuan,Dachun Yang

doi:10.3390/math9212724

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.

Highlights

The study of function spaces on the Euclidean space Rn and its subsets with generalized smoothness started from the middle of the 1970s, and has found various applications in interpolations, embedding properties of function spaces [5,6,7,8], fractal analysis ([9], Chapters 18–23), and many other fields such as probability theory and stochastic processes [10,11]
A lot of attention has been paid to Besov and Triebel–Lizorkin spaces on Rn with logarithmic smoothness; see, for instance [17,18,19,20,21,22,23,24,25,26,27]
Using Hajłasz gradient sequences, the authors [28] introduced Hajłasz–Besov and Hajłasz–Triebel–Lizorkin spaces with generalized smoothness on a given metric space

Summary

Introduction

The study of function spaces on the Euclidean space Rn and its subsets with generalized smoothness started from the middle of the 1970s (see, for instance, [1,2,3,4]), and has found various applications in interpolations, embedding properties of function spaces [5,6,7,8], fractal analysis ([9], Chapters 18–23), and many other fields such as probability theory and stochastic processes [10,11]. To overcome the difficulties caused by this, we borrow the notion of the modulus of continuity and, for certain φ that satisfies such assumptions, find a dense subset of M φ,p (X) consisting of generalized Lipschitz functions Applying these dense properties, we obtain the boundedness of discrete maximal operators on these Hajłasz-type spaces and a weak-type capacitary estimate for restricted maximal functions, which is further used to prove that the exceptional sets of Lebesgue points φ φ of functions from M φ,p (X), N p,q (X), and M p,q (X) have zero Cap M φ,p (X) , Cap N φ (X) , and p,q. Hausdorff contents to measure the exceptional set of Lebesgue points of functions from these Hajłasz-type spaces

Hajłasz–Besov and Hajłasz–Triebel–Lizorkin Spaces with Generalized Smoothness

Lebesgue Points of φ-Hajłasz-Type Functions

Generalized Lebesgue Points of φ-Hajłasz-Type Functions