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

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 $$s_0(x)-\frac{n}{p_0(x)}=s_1(x)-\frac{n}{p_1(x)},x\in {\mathbb{R}}^n.$$ We work exclusively with the associated sequence spaces $b_{p(\cdot ),q(\cdot )}^{s(\cdot )}$ and $f_{p(\cdot ),q(\cdot )}^{s(\cdot )}$, 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 $B_{p(\cdot ),q(\cdot )}^{\mathbf{w}}({\mathbb{R}}^n)$ and $F_{p(\cdot ),q(\cdot )}^{\mathbf{w}}({\mathbb{R}}^n)$ which unify the smoothness scales of spaces of variable smoothness and generalized smoothness spaces.