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 Lp_0_w space for some p_0 and for all Muckenhoupt Ap_0 weights, then the one-weight inequality is also true in weighted grand Morrey space Lp), r, θ_w 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.