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

Abstract

In this note, we establish characterizations for the homogeneous Besov-type spaces ${\dot{B}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ and Triebel-Lizorkin-type spaces ${\dot{F}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$, introduced by Yang and Yuan, through fractional Hajłasz-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.

arXiv comment15 pages; fixed typos