On the function spaces of general weights

Abstract

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 $q=\infty$. 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 ${\dot{A}}_{p,q}({\mathbb{R}}^n,t_i),i\in \{1,2\}$.

We add Theorem 3.34 and corollaries 3.37 and 3.42. arXiv admin note: substantial text overlap with arXiv:2009.12223, arXiv:2106.00621, arXiv:2009.03636