Weighted extrapolation in grand Morrey spaces and applications to partial differential equations

Abstract

In this paper we derive weighted extrapolation results in grand Morrey spaces. In particular, the main statements say that if for a class of pairs of measurable functions the one-weight inequality holds in the classical weighted $L_w^{p_0}$ space for some $p_0$ and for all Muckenhoupt $A_{p_0}$ weights, then the one-weight inequality is also true in weighted grand Morrey space $L_w^{p),r,\theta }$ with $A_p$ weights for all $p$. The spaces under consideration are defined on quasi-metric measure spaces. The obtained results are applied to derive one-weight estimates for some operators of Harmonic Analysis and to study regularity properties of solutions of second order partial differential equations with discontinuous coefficients.