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 dµ), 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 Hölder inequality, in general, but some kind of self improving property still holds. On the other hand, if ∈ RBMO(µ) and ε > 0 is small enough, then eεf belongs to this class of weights.
@article{urn:eudml:doc:41900,
title = {Weighted norm inequalities for Calder\'on-Zygmund operators without doubling conditions.},
journal = {Publicacions Matem\`atiques},
year = {2007},
pages = {397-456},
zbl = {1136.42303},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41900}
}
Tolsa, Xavier. Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions.. Publicacions Matemàtiques, (2007), pp. 397-456. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41900/