Extrapolation in New Weighted Grand Morrey Spaces Beyond the Muckenhoupt Classes

Rubio de Francia's extrapolation theorem for new weighted grand Morrey spaces $\mathcal{M}_{w}^{p),\lambda,\theta}(X)$ with weights $w$ beyond the Muckenhoupt $A_p$ classes is established. This result, in particular, implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. Necessary conditions for the boundedness of the Hardy–Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are also investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.