$A_p$ weights for nondoubling measures in $R^n$ and applications

We study an analogue of the classical theory of $A_p(\mu)$weights in $\mathbb{R} ^n$ without assuming that the underlying measure $\mu$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 $\frac{1}{\mu(Q)}\int_{Q} \vert f-f_{Q}\vert d\mu \le a(Q)$ for all cubes $Q$, then it is possible to deduce a higher $L^p$ integrability result for $f$, assuming a certain simple geometric condition on the functional $a$.