  This paper provides maximal function characterizations of anisotropic
Triebel-Lizorkin spaces associated to general expansive matrices for the full
range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $\alpha \in
\mathbb{R}$. The equivalent norm is defined in terms of the decay of wavelet
coefficients, quantified by a Peetre-type space over a one-parameter dilation
group. As an application, the existence of dual molecular frames and Riesz
sequences is obtained; the wavelet systems are generated by translations and
anisotropic dilations of a single function, where neither the translation nor
dilation parameters are required to belong to a discrete subgroup. Explicit
criteria for molecules are given in terms of mild decay, moment, and smoothness
conditions.
