On the Function Spaces of General Weights

The aim of this paper is twofold. Firstly, we establish the $$\varphi $$ -transform characterizations of 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 $$q = \infty $$ , in the sense of Frazier and Jawerth. Secondly, under some suitable assumptions on the $$p$$ -admissible weight sequence $$\{ {{t}_{k}}\} $$ , we prove that $${{\dot {A}}_{{p,q}}}({{\mathbb{R}}^{n}},\{ {{t}_{k}}\} ) = {{\dot {A}}_{{p,q}}}({{\mathbb{R}}^{n}},{{t}_{j}}),\quad j \in \mathbb{Z},$$ 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 $${{\dot {A}}_{{p,q}}}({{\mathbb{R}}^{n}},{{t}_{i}})$$ , $$i \in \{ 1,2\} $$ .