Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients

Abstract

We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg-flat domains. The coefficients are supposed to be only measurable in one of the space variables and small BMO with respect to the others. We obtain boundedness of the Hardy-Littlewood maximal operator in the generalized Morrey spaces W p,ϕ , p ∈ (1, ∞) and weight ϕ satisfying certain supremum condition. This permits us to obtain Calderón-Zygmund type estimate for the gradient of the weak solution of the problem.

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

“…In the next paper, we shall introduce the generalized weighted Nikolskii-Morrey spaces and will study some embedding theorems. We will also investigate the boundedness of fractional multilinear integral operators with rough kernels T A 1 ;A 2 ;:::;A k ;˛o n the generalized weighted Nikolskii-Morrey spaces, see for example, [30]. These results may be applicable to some problems of partial differential equations; see for example [6,7,19,20,26,28,30,43].…”

Weighted Sobolev-Morrey Regularity of Solutions to Variational Inequalities

“…holds with c 2 = c 2 (n, q, κ, ϕ, w, Ω) (see [20]). We will need also the following standard measure theory results regarding weighted spaces.…”

Hardy-Littlewood, Bessel-Riesz, and Fractional Integral Operators in Anisotropic Morrey and Campanato Spaces

“…Morrey spaces for non-Euclidean distances find their applications in many problems, see e.g. [GS15a,GS15b] and [GS16].…”