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

FRANKE–JAWERTH EMBEDDINGS FOR BESOV AND TRIEBEL–LIZORKIN SPACES WITH VARIABLE EXPONENTS

Helena F. Gonçalves, Henning Kempka and Jan Vybíral

TU Chemnitz, Faculty of Mathematics
Reichenhainer Str. 39, 09126 Chemnitz, Germany; helena.goncalves 'at' mathematik.tu-chemnitz.de

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

Charles University, Department of Mathematical Analysis
Sokolovská 83, 186 00, Prague 8, Czech Republic; vybiral 'at' karlin.mff.cuni.cz

Abstract. The classical Jawerth and Franke embeddings

F^s0_p0,q(Rⁿ) → B^s1_p1,p0(Rⁿ) and B^s0_p0,p1(Rⁿ) → F^s1_p1,q(Rⁿ)

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

F^s0(⋅)_{p0(⋅),q(⋅)}(Rⁿ) → B^s1(⋅)_{p1(⋅),p0(⋅)}(Rⁿ) and B^s0(⋅)_{p0(⋅),p1(⋅)}(Rⁿ) → F^s1(⋅)_{p1(⋅),q(⋅)}(Rⁿ)

respectively, if inf_x∈Rn(s₀(x) – s₁(x)) > 0 and

s_0(x) – n/p₀(x) = s₁(x) – n/p₁(x), x ∈ Rⁿ.

We work exclusively with the associated sequence spaces b^s(⋅)_{p(⋅),q(⋅)}(Rⁿ) and f^s(⋅)_{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 B^w_{p(⋅),q(⋅)}(Rⁿ) and F^w_{p(⋅),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.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4310