Triebel-Lizorkin-Type Spaces with Variable Exponents

In this article, the authors first introduce the Triebel-Lizorkin-type space $F_{p(\cdot ),q(\cdot )}^{s(\cdot ),\varphi }({\mathbb{R}}^n)$ with variable exponents, and establish its $\phi$-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 $F_{p(\cdot ),q(\cdot )}^{s(\cdot ),\varphi }({\mathbb{R}}^n)$ are also obtained, which are used to prove a trace theorem of $F_{p(\cdot ),q(\cdot )}^{s(\cdot ),\varphi }({\mathbb{R}}^n)$. The authors also characterize the space $F_{p(\cdot ),q(\cdot )}^{s(\cdot ),\varphi }({\mathbb{R}}^n)$ via Peetre maximal functions.