Calderón–Zygmund Singular Operators in Extrapolation Spaces

We study the boundedness of the Hardy–Littlewood maximal operator in abstract extrapolation Banach function lattices and their Köthe dual spaces. The extrapolation spaces are generated by compatible families of Banach function lattices on quasi-metric measure spaces with doubling measure. These results combined with a variant of the integral Coifman–Fefferman inequality imply that every Calderón–Zygmund singular operator is bounded in considered extrapolation spaces. We apply these results to extrapolation spaces determined by compatible families of Calderón–Lozanovskii spaces, in particular to compatible families of Orlicz spaces that are interpolation of weighted Lp-spaces (1 < p < ∞) with Ap weights defined on spaces of homogeneous type.