Generalized Besov-type and Triebel-Lizorkin-type spaces

TL;DR

This paper introduces new generalized Besov-type and Triebel-Lizorkin-type function spaces with specific growth conditions, establishing their properties and atomic decompositions to extend classical analysis tools.

Contribution

It defines novel generalized function spaces with specific growth conditions and proves their fundamental properties and atomic decompositions.

Findings

Established embedding properties of the new spaces.

Derived atomic decompositions for the spaces.

Extended classical function space theory to more general settings.

Abstract

Let $0<p<\infty$, $0<q\leq\infty$, and $s\in\mathbb{R}$. We introduce a new type of generalized Besov-type spaces $B_{p,q}^{s,\varphi}(\mathbb{R}^d)$ and generalized Triebel-Lizorkin-type spaces $F_{p,q}^{s,\varphi}(\mathbb{R}^d)$, where $\varphi$ belongs to the class $\mathcal{G}_p$, that is, $\varphi:(0,\infty) \rightarrow (0,\infty)$ is nondecreasing and $t^{-d/p}\varphi(t)$ is nonincreasing in $t>0$. We establish several properties, including some embedding properties, of these spaces. We also obtain the atomic decomposition of the spaces $B_{p,q}^{s,\varphi}(\mathbb{R}^d)$ and $F_{p,q}^{s,\varphi}(\mathbb{R}^d)$.