Franke-Jawerth embeddings for Besov and Triebel-Lizorkin spaces with variable exponents

TL;DR

This paper extends classical Sobolev embeddings between Besov and Triebel-Lizorkin spaces to variable exponent spaces, providing new proofs and encompassing 2-microlocal spaces, thus broadening the scope of functional analysis in variable smoothness contexts.

Contribution

It proves Franke-Jawerth embeddings for variable exponent Besov and Triebel-Lizorkin spaces, including new proofs and generalizations to 2-microlocal spaces.

Findings

Embeddings hold under conditions on variable exponents

Results include spaces of variable and generalized smoothness

Provides alternative proof avoiding duality and interpolation

Abstract

The classical Jawerth and Franke embeddings $$ F^{s_0}_{p_0,q}({\mathbb R}^n)\hookrightarrow B^{s_1}_{p_1,p_0}({\mathbb R}^n) \quad \mbox{and} \quad B^{s_0}_{p_0,p_1}({\mathbb R}^n)\hookrightarrow F^{s_1}_{p_1,q}({\mathbb R}^n) $$ are versions of Sobolev embedding between the scales of Besov and Triebel-Lizorkin function spaces for $s_0>s_1$ and $$ s_0-\frac{n}{p_0} = s_1-\frac{n}{p_1}.$$ We prove Jawerth and Franke embeddings for the scales of Besov and Triebel-Lizorkin spaces with all exponents variable $$ F^{s_0(\cdot)}_{p_0(\cdot),q(\cdot)}\hookrightarrow B^{s_1(\cdot)}_{p_1(\cdot),p_0(\cdot)} \quad \mbox{and} \quad B^{s_0(\cdot)}_{p_0(\cdot),p_1(\cdot)}\hookrightarrow F^{s_1(\cdot)}_{p_1(\cdot),q(\cdot)}, $$ respectively, if $\inf_{x\in\mathbb{R}^n}(s_0(x)-s_1(x))>0$ and $$ s_0(x) -\frac{n}{p_0(x)} = s_1(x) -\frac{n}{p_1(x)}, \quad x \in {\mathbb R}^n. $$ We work exclusively with the associated sequence spaces $b^{s(\cdot)}_{p(\cdot),q(\cdot)}$ and $f^{s(\cdot)}_{p(\cdot),q(\cdot)}$, which is justified by well known decomposition techniques. We give also a different proof of the Franke embedding in the constant exponent case which avoids duality arguments and interpolation. Our results hold also for 2-microlocal function spaces $B^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$ and $F^{\mathbf{w}}_{p(\cdot),q(\cdot)}({\mathbb R}^n)$ which unify the smoothness scales of spaces of variable smoothness and generalized smoothness spaces.