Annales Academiæ Scientiarum Fennicæ
Volumen 43, 2018, 187-209


Helena F. Gonçalves, Henning Kempka and Jan Vybíral

TU Chemnitz, Faculty of Mathematics
Reichenhainer Str. 39, 09126 Chemnitz, Germany; helena.goncalves 'at'

University of Applied Sciences, Department of Fundamental Sciences
PF 100314, 07703 Jena, Germany; henning.kempka 'at'

Charles University, Department of Mathematical Analysis
Sokolovská 83, 186 00, Prague 8, Czech Republic; vybiral 'at'

Abstract. The classical Jawerth and Franke embeddings

Fs0p0,q(Rn) → Bs1p1,p0(Rn) and Bs0p0,p1(Rn) → Fs1p1,q(Rn)

are versions of Sobolev embedding between the scales of Besov and Triebel–Lizorkin function spaces for s0 > s1 and s0n/p0 = s1n/p1. We prove Jawerth and Franke embeddings for the scales of Besov and Triebel–Lizorkin spaces with all exponents variable

Fs0(⋅)p0(⋅),q(⋅)(Rn) → Bs1(⋅)p1(⋅),p0(⋅)(Rn) and Bs0(⋅)p0(⋅),p1(⋅)(Rn) → Fs1(⋅)p1(⋅),q(⋅)(Rn)

respectively, if infxRn(s0(x) – s1(x)) > 0 and

s0(x)n/p0(x) = s1(x) – n/p1(x), xRn.

We work exclusively with the associated sequence spaces bs(⋅)p(⋅),q(⋅)(Rn) and fs(⋅)p(⋅),q(⋅)(Rn), 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 Bwp(⋅),q(⋅)(Rn) and Fwp(⋅),q(⋅)(Rn) which unify the smoothness scales of spaces of variable smoothness and generalized smoothness spaces.

2010 Mathematics Subject Classification: Primary 42B35, 46E35.

Key words: Besov spaces, Triebel–Lizorkin spaces, variable smoothness, variable integrability, Franke–Jawerth embedding, 2-microlocal spaces.

Reference to this article: H. F. Gonçalves, H. Kempka and J. Vybíral: Franke–Jawerth embeddings for Besov and Triebel–Lizorkin spaces with variable exponents. Ann. Acad. Sci. Fenn. Math. 43 (2018), 187-209.

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