Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 529-544

SOBOLEV AND JAWERTH EMBEDDINGS FOR SPACES WITH VARIABLE SMOOTHNESS AND INTEGRABILITY

Jan Vybíral

Universität Jena, Mathematisches Institut
Ernst-Abbe-Platz 2, 07740 Jena, Germany; vybiral 'at' mathematik.uni-jena.de

Abstract. We consider the Triebel-Lizorkin spaces Fs(.)p(.),q(.)(Rn) of variable smoothness and integrability as introduced recently by Diening, Hästö and Roudenko in [9]. Under certain regularity conditions on the function parameters involved we show that

Fs_0(.)p_0(.),q(.)(Rn) \hookrightarrow Fs_1(.)p_1(.),q(.)(Rn)

if

s0(x) \ge s1(x) and s0(x) - n/p0(x) = s1(x) - n/p1(x) for all x \in Rn

with embeddings of Sobolev and Bessel potential spaces included as special cases.

If infx \in R^n (s0(x) - s1(x)) > 0 we recover also the analogue of the Jawerth embedding

Fs_0(.)p_0(.),q_0(.)(Rn) \hookrightarrow Fs_1(.)p_1(.),q_0(.)(Rn)

for any q0, q1. The proofs are based on the decomposition techniques of [9] and work exclusively with the associated sequence spaces fs(.)p(.),q(.).

2000 Mathematics Subject Classification: Primary 46E35, 46E30.

Key words: Triebel-Lizorkin spaces, variable smoothness, variable integrability, Jawerth embedding, Sobolev embedding.

Reference to this article: J. Vybíral: Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability. Ann. Acad. Sci. Fenn. Math. 34 (2009), 529-544.

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