Abstract
The classical Jawerth and Franke embeddings F-p0,q(s0)(R-n) hooked right arrow B-p1,p0(s1)(R-n) and B-p0,p1(s0)(R-n) hooked right arrow F-p1,q1(s1)(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) - n/p(0) = s(1) - n/p(1). We prove Jawerth and Franke embeddings for the scales of Besov and Triebel-Lizorkin spaces with all exponents variable F-p0(.),q(.)(s0(.))(R-n) hooked right arrow B-p1(.),p0(.)(s1(.))(R-n) and B-p0(.),q(.)(s0(.))(R-n) hooked right arrow F-p1(.),p0(.)(s1(.))(R-n), respectively, if inf(x is an element of Rn)(s(0)(x) - s(1)(x)) > 0 and s(0)(x) = n/P-0(x) = s(1)(x) - n/p(1)(x), x is an element of R-n.
We work exclusively with the associated sequence spaces b(P(.),q(.))(s(.))(R-n) and f(P(.),q(.))(s(.))(R-n), 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(.),q(.)(omega)(R-n) and F-P(.),q(.)(omega)(R-n) which unify the smoothness scales of spaces of variable smoothness and generalized smoothness spaces.