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

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.