Triebel-Lizorkin-Type Spaces with Variable Exponents

TL;DR

This paper introduces a new class of Triebel-Lizorkin-type function spaces with variable exponents, providing their characterizations, trace theorem, and maximal function description, advancing the understanding of variable exponent function spaces.

Contribution

It establishes the definition, characterizations, and properties of Triebel-Lizorkin-type spaces with variable exponents, including transform, atomic, molecular, and maximal function characterizations.

Findings

Defined Triebel-Lizorkin-type spaces with variable exponents.

Proved $oldsymbol{ extit{ ext{φ}}}$-transform characterization.

Established atomic and molecular characterizations.

Abstract

In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ with variable exponents, and establish its $\varphi$-transform characterization in the sense of Frazier and Jawerth, which further implies that this new scale of function spaces is well defined. The smooth molecular and the smooth atomic characterizations of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$. The authors also characterize the space $F_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb R^n)$ via Peetre maximal functions.