Let μ be a Borel measure on Rd which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Crn for all x ∈ ℝd, r > 0 and for some fixed n with 0 < n ≤ d. In this paper we introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if μ(B(x, r)) ≈ rn for x ∈ supp(μ), and we show that all n-dimensional Calderón-Zygmund operators are bounded on Lp(w dμ) if and only if N is bounded on Lp(w d μ), for a fixed p ∈ (1, ∞). 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 Holder inequality, in general, but some kind of self improving property still holds. On the other hand, if f ∈ RBMO(μ) and ε > 0 is small enough, then eε f belongs to this class of weights.
- Calderón-Zygmund operators
- Non doubling measures