Smoothing properties of the discrete fractional maximal operator on Besov and Triebel--Lizorkin spaces

TL;DR

This paper investigates how the discrete fractional maximal operator enhances smoothness in fractional Hajlasz, Besov, and Triebel-Lizorkin spaces on metric measure spaces, extending known results and unifying various cases.

Contribution

It demonstrates that the discrete fractional maximal operator increases smoothness in these function spaces, generalizing previous results and applying to a broad setting including Euclidean spaces.

Findings

The operator maps fractional Hajlasz spaces to higher smoothness spaces.

Results unify and extend previous findings in the literature.

Applicable to general metric measure spaces, including Euclidean spaces.

Abstract

Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show that the discrete fractional maximal operator maps these spaces to the spaces of the same type with higher smoothness. Our results extend and unify aforementioned results. We present our results in general setting, but they are new already in the Euclidean case.