What does this research mean for the field?

The Besov spaces ${{ ext{dot }B}_{p,q}}({ ext{R}}^{n}, ext{t}_k)$ and the Triebel–Lizorkin spaces ${{ ext{dot }F}_{p,q}}({ ext{R}}^{n}, ext{t}_k)$ can be characterized using the $ ext{phi}$-transform for $q = ext{infinity}$. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The paper aims to establish characterizations of Besov and Triebel–Lizorkin spaces and prove equivalence conditions for spaces defined by weight sequences.

March 1, 2026

On the Function Spaces of General Weights

Key Points

The paper aims to establish characterizations of Besov and Triebel–Lizorkin spaces and prove equivalence conditions for spaces defined by weight sequences.
Employ phi-transform characterizations for Besov and Triebel–Lizorkin spaces.
Investigate conditions for equivalence of quasi-norms under weight sequences.
Demonstrate necessary and sufficient conditions for the coincidence of specific function spaces.
Characterizations of Besov and Triebel–Lizorkin spaces for q = ∞ are established.
Equivalence of spaces under p-admissible weight sequences is proven.
Necessary and sufficient conditions for the coincidence of specific function spaces are identified.

Abstract

The aim of this paper is twofold. Firstly, we establish the -transform characterizations of the Besov spaces { {B}₏, ₐ} ({R^n}, \ {{t₊}\}) and the Triebel–Lizorkin spaces { {F}₏, ₐ} ({R^n}, \ {{t₊}\}) for q =, in the sense of Frazier and Jawerth. Secondly, under some suitable assumptions on the p -admissible weight sequence \ {{t₊}\}, we prove that { {A}₏, ₐ} ({R^n}, \ {{t₊}\}) = { {A}₏, ₐ} ({R^n}, {t₉}), j Z, in the sense of equivalent quasi-norms, with A \ {B, F\}. Moreover, we find a necessary and sufficient conditions for the coincidence of the spaces { {A}₏, ₐ} ({R^n}, {t₈}), i \ 1, 2\.

Bookmark

On the Function Spaces of General Weights

Key Points

Abstract

Cite This Study