On the function spaces of general weights

The aim of this paper is twofold. Firstly, we chatacterize the Besov spaces ${\dot{B}}_{p,q}({\mathbb{R}}^n,\{t_k\})$ and the Triebel-Lizorkin spaces ${\dot{F}}_{p,q}({\mathbb{R}}^n,\{t_k\})$ for $q=\mathrm{\infty }$. Secondly, under some suitable assumptions on the $p$-admissible weight sequence $\{t_k\}$, we prove that \begin{equation*} \dot{A}{p,q}($\mathbb{R}$^{n},{t{k}})=\dot{A}{p,q}($\mathbb{R}$ ^{n},t{j}),\quad j\in $\mathbb{Z}$, \end{equation*} in the sense of equivalent quasi-norms, with $\dot{A}$ $\in \{\dot{B},\dot{F}\}$. Moreover, we find a necessary and sufficient conditions for the coincidence of the spaces ${\dot{A}}_{p,q}({\mathbb{R}}^n,t_i),i\in \{1,2\}$.