Ap weights for nondoubling measures in Rn and applications

Abstract

We study an analogue of the classical theory of Ap(µ) weights in Rn 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 Calderón-Zygmund operators. We also consider commutators of those Calderón-Zygmund operators with bounded mean oscillation functions (BMO), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. Finally, we study self-improving properties of Poincaré-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 a higher Lp integrability result for f, assuming a certain simple geometric condition on the functional a.