\(A_p\) weights for nondoubling measures in \(\mathbb R^n\) and applications (Q2781396)

The authors study an analogue of the classical theory of \(A_p(\mu)\) weights in \(\mathbb R^n\) without assuming the measure \(\mu\) to be doubling, but the measure of the boundary of any cube with sides parallel to the axes should be zero. For \(A_\infty (\mu)\) (defined as the union of all \(A_p(\mu)\)) several equivalent characterizations are proved, these are similar to those in the classical case. And if a weight is in \(A_p(\mu)\) then the centered Hardy-Littlewood maximal operator is bounded in \(L^p(\mu)\). Then the authors consider Calderón-Zygmund operators and prove boundedness in \(L^p(\omega\, d\mu)\) when \(\omega\in A_\infty (\mu)\) and \(\mu\) is a ``\(d\)-dimensional'' Borel measure and whenever the Hardy-Littlewood maximal operator is bounded. As an application they consider commutators of these Calderón-Zygmund operators with functions in BMO. Finally, the authors study self-improving properties of Poincaré-BMO type: if \(f\) is a locally integrable function satisfying NEWLINE\[NEWLINE {1\over\mu (Q)}\int_{Q}|f-f_{Q}|\,d\mu\leq a(Q) NEWLINE\]NEWLINE for all cubes \(Q\) then it is possible to deduce a higher \(L^p\) integrability result for \(f\), assuming that the functional \(a\) satisfies a simple geometric condition.