Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

Abstract

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.

Mixed Sobolev-like Inequalities in Lebesgue spaces of variable exponents and in Orlicz spaces

“…Remark 1. 1 The condition (1.3) above can be easily deduced from inequality (1.2) by homogeneity with respect to dilations: indeed consider the function f λ (x) = f (λx) with λ > 0, by a change of variables one obtains…”

“…Let us remark that in order to obtain (1.16), the authors of [5] develop a very interesting theory related to fractional maximal functions. In particular, they use some local 1 Hedberg-like inequalities for this type of operators (see the details in Proposition 3.3 of [5]) since unbounded versions of these arguments introduce some problems as mentioned in Example 3.4 of the cited article.…”

“…In the classical setting, the representation of these spaces using dyadic blocs is well known (see the books [2] and [13]). For spaces of variable exponents, this theory is given in [1], see also Chapter 12 of the book [10]. Let us point out that the identification of the spaces given by a Littlewood-Paley decomposition with the spaces used here (in the non-homogeneous case) is done in the article [11].…”

Compactness in the spaces of variable integrability and summability

“…In particular, the Sobolev inequalities have been shown for variable exponent spaces on Euclidean spaces (see [6,8,9]) and on Riemannian manifolds [11]. Furthermore, other types of spaces with variable exponent have been considered, for example, Hölder, Hardy, Campanato [28], Besov [1,2].…”

“…The variable exponent Besov space is defined via the mixed Lebesgue-sequence space 1 𝓁 𝑞(⋅) (𝐿 𝑝(⋅) ). Some basic properties of spaces 𝓁 𝑞(⋅) (𝐿 𝑝(⋅) ) are known.…”