Mixed multiplication of Besov and Triebel-Lizorkin spaces

TL;DR

This paper investigates embeddings involving products of Besov and Triebel-Lizorkin spaces, establishing new results under specific parameter restrictions and improving existing multiplication theorems.

Contribution

It introduces new embedding results for products of Besov and Triebel-Lizorkin spaces, enhancing previous pointwise multiplication theorems.

Findings

Established new embeddings for products of function spaces.

Improved existing results on pointwise multiplication in Triebel-Lizorkin spaces.

Utilized advanced tools like Franke-Jawerth embeddings and the $ ext{±}$-method.

Abstract

This paper is concerned with proving some embeddings of the form \begin{equation*} F_{p_{1},q}^{s_{1}}\cdot B_{p_{2},\infty }^{s_{2}}\cdot ...\cdot B_{p_{m},\infty }^{s_{m}}\hookrightarrow F_{p,q}^{s_{1}},\quad m\geq 2. \end{equation*} The different embeddings obtained here are under certain restrictions on the parameters. In particular, we improve some results of pointwise multiplication on Triebel-Lizorkin spaces. Franke-Jawerth embeddings, the $\pm $ - method of Gustaffson-Peetre and the relation between Hardy spaces and Triebel-Lizorkin spaces are the main tools.