  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.
