We introduce and study the family of Besov–Köthe spaces which is a generalization of the Besov spaces, the Besov–Morrey spaces and the variable Besov spaces. As an application of the general results for the Besov–Köthe spaces, we identify a pre-dual of the Besov–Morrey space.

where Q <= Rm is a bounded domain, JS?W denotes m-dimensional Lebesgue measure, (p = (<p\..., (p) is a continuous transformation from Q to Rn (m ^ n) belonging to some Sobolev space W x p' (Q), and (px is the (almost everywhere defined) matrix function (d^/oxj). Here ƒ is a realvalued function o n Q x i ? n x M „ x m , where Mnxm is the space o f n x m matrices. Such integrals arise in the anal...

Let Ω be an open bounded set in R n (n ≥ 2), with C 2 boundary, and N p,λ (Ω) (1 < p < +∞, 0 ≤ λ < n) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem:

We investigate geometric curvature energies on closed curves involving integral versions of the Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding theorems, which may be viewed as counterparts to respective results on one-dimensional sets in the context of harmonic analysis. Mathematics Subject Classification (2010): 28A75 (primary); 53A04,...

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces L(Γ) on curves satisfying the arc-chord condition, for a class of ”radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces L(0, ...

‎Some functional inequalities‎ ‎in variable exponent Lebesgue spaces are presented‎. ‎The bi-weighted modular inequality with variable exponent $p(.)$ for the Hardy operator restricted to non‎- ‎increasing function which is‎‎$$‎‎int_0^infty (frac{1}{x}int_0^x f(t)dt)^{p(x)}v(x)dxleq‎‎Cint_0^infty f(x)^{p(x)}u(x)dx‎,‎$$‎ ‎is studied‎. ‎We show that the exponent $p(.)$ for which these modular ine...

We derive some of the basic properties of weighted variable exponent Lebesgue spaces L p(.) w (R) and investigate embeddings of these spaces under some conditions. Also a new family of Wiener amalgam spaces W (L p(.) w , L q υ) is defined, where the local component is a weighted variable exponent Lebesgue space L p(.) w (R) and the global component is a weighted Lebesgue space Lυ (R) . We inves...