Besov spaces with variable smoothness and integrability

Abstract

In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder-Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.

Lorentz spaces with variable exponents

“…The first statement follows from Lemma 4 and the embedding ℓ q0(·) (L p(·) ) ֒→ ℓ q1(·) (L p(·) ) which has been proven in [1].…”

“…It was shown in [1] that it always constitutes a quasi-norm if p − , q − > 0. Further it is known (see [20]) that it is a norm if either …”

“…If q(·) = q is a constant function, then the Proposition 3.3 in [1] shows that ℓ q(·) (L p(·) ) is really an iterated space, i.e.…”

Vector-valued singular integral operators on Morrey type spaces and variable Triebel-Lizorkin-Morrey spaces

“…The variable exponent analysis recently gains a lot of attentions [4,5,14,15,17,18,19,20,21,36,41,44,49,50,57,75] because the boundedness of several important linear operators in analysis, especially, the Hardy-Littlewood maximal operator, can be established for the variable Lebesgue spaces [14,17,19]. In addition, it provides a theoretical background on the studies of electrorheological fluid dynamics [1,2,61,63,64] and imaging processing [11].…”

“…For the rest of this paper, we use the lifting principle to introduce and study some new Morrey type spaces related to several Banach function spaces appeared in [4,5,20,24,25,26,32,35,41,42,43,44,47,52,56,66,67,71,73,74,75].…”