Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions

Abstract

Let $\mu$ be a Borel measure on ${\mathbb R}^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq Cr^n$ for all $x\in \mathbb {R}^d$, $r>0$ and for some fixed $n$ with $0<n\leq d$. In this paper we introduce a maximal operator $N$, which coincides with the maximal Hardy-Littlewood operator if $\mu(B(x,r))\approx r^n$ for $x\in \rm{supp}(\mu)$, and we show that all $n$-dimensional Calderón-Zygmund operators are bounded on $L^p(w\,d\mu)$ if and only if $N$ is bounded on $L^p(w\,d\mu)$, for a fixed $p\in(1,\infty)$. Also, we prove that this happens if and only if some conditions of Sawyer type hold. We obtain analogous results about the weak $(p,p)$ estimates. This type of weights do not satisfy a reverse Hölder inequality, in general, but some kind of self improving property still holds. On the other hand, if $f \in \mathit{RBMO}(\mu)$ and ${\varepsilon}>0$ is small enough, then $e^{{\varepsilon} f}$ belongs to this class of weights.