Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions. (English) Zbl 1136.42303

Tolsa, Xavier

Publ. Mat., Barc. 51, No. 2, 397-456 (2007).

Summary: 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 \(x\in\operatorname{supp}(\mu)\), 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 \(f \in RBMO(\mu)\) and \({\varepsilon}>0\) is small enough, then \(e^{{\varepsilon} f}\) belongs to this class of weights.

MSC:

42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25	Maximal functions, Littlewood-Paley theory