SOBOLEV AND JAWERTH EMBEDDINGS FOR SPACES WITH VARIABLE SMOOTHNESS AND INTEGRABILITY

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Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences (2011, background): “Furthermore, this was complemented in [59] by showing, that the classical Sobolev embedding theorem F s0(·) p0(·),q(·)(R ) ↪→ F s1(·) p1(·),q(·)(R ) holds also in this scale of function spaces if the usual condition is replaced by its point-wise analogue s0(x)− n/p0(x) = s1(x)− n/p1(x), x ∈ R.”
Besov spaces with variable smoothness and integrability (2010, background): “These studies were all restricted to bounded exponents p and q. Vyb́ıral [43] and Kempka [23] also considered Besov spaces B α(·) p(·),q—note that only the case of constant q was included.”
2-Microlocal Besov and Triebel-Lizorkin Spaces of Variable Integrability (2009, background): “Quite recently, Vybiral in [35] has proved embeddings of Sobolev and Jawerth type for F s(·) p(·),q(·)(R).”