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

Abstract:

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 $\frac {1}{\mu (Q)}\int _{Q} |f-f_{Q}| d\mu \le a(Q)$ 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 $a$.