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

— 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 Lw p0 space for some p0 and for all Muckenhoupt Ap0 weights, then the one-weight inequality is also true in weighted grand Morrey space Lw pÞ; r; y with Ap 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 di¤erential equations with discontinuous coecients. © 2019 European Mathematical Society Publishing House. All rights reserved.