Pointwise and grand maximal function characterizations of Besov-type and Triebel-Lizorkin-type spaces

TL;DR

This paper provides new pointwise and maximal function characterizations of Besov-type and Triebel-Lizorkin-type spaces, extending previous results for classical spaces to more general parameter settings.

Contribution

It introduces fractional Haj ext{l}asz gradients and grand Littlewood-Paley maximal functions to characterize these spaces for a wider range of parameters.

Findings

Characterizations via fractional Haj ext{l}asz gradients for 0 < s < 1.

Grand Littlewood-Paley maximal function characterizations for all parameters.

Extension of previous characterizations to Besov-type and Triebel-Lizorkin-type spaces.

Abstract

In this note, we establish characterizations for the homogeneous Besov-type spaces $\dot{B}^{s,\tau}_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}(\mathbb{R}^n)$, introduced by Yang and Yuan, through fractional Haj\l asz-type gradients for suitable values of the parameters $p$, $q$ and $\tau$ when $0 < s < 1$, and through grand Littlewood-Paley-type maximal functions for all admissible values of the parameters. These characterizations extend the characterizations obtained by Koskela, Yang and Zhou for the standard homogeneous Besov and Triebel-Lizorkin spaces.