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We study an analogue of the classical theory of Ap(µ) weights in R n without assuming that the underlying measure µ is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderon-Zygmund operators (in the sense of (NTV1)). We also consider commutators of those Calderon-Zygmund operators with bounded mean oscillation functions (BMO), extending the main result from (CRW). Finally, we study self-improving properties of Poincare-B.M.O. type inequalities within this context, more precisely we show that if f is a locally integrable function satisfying 1 µ(Q) R Q |f fQ|dµ a(Q) for all cubes Q, then it is possible to deduce higher L p integrability result of f assuming certain simple geometric condition on the functional a.

𝐴_{𝑝} weights for nondoubling measures in 𝑅ⁿ and applications