Realization of homogeneous Triebel–Lizorkin spaces with $p=\infty $ and characterizations via differences

In this paper, via the decomposition of Littlewood–Paley, the homogeneous Triebel-Lizorkin space $\dot{F}_{\infty,q}^{s}$ is defined on $\mathbb{R}^n$ by distributions modulo polynomials in the sense that $\|f\|=0$ ($\|\cdot\|$ the quasi-seminorm in $\dot F^{s}_{\infty,q}$) if and only if $f$ is a polynomial on $\mathbb{R}^n$. We consider this space as a set of “true” distributions and we are lead to examine the convergence of the Littlewood-Paley sequence of each element in $\dot F^{s}_{\infty,q}$. First we use the realizations and then we obtain the realized space $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ of $\dot{F}_{\infty,q}^{s}$.
Our approach is as follows. We first study the commuting translations and dilations of realizations in $\dot{F}_{\infty,q}^{s}$, and employing distributions vanishing at infinity in the weak sense, we construct $\dot{\widetilde{F}}{^{s}_{\infty,q}}$. Then, as another possible definition of $\dot{F}_{\infty,q}^{s}$, in the case $s>0$, we make use of the differences and describe $\dot{\widetilde{F}}{^{s}_{\infty,q}}$ as $s>\max(n/q-n,0)$.