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Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions
Tolsa Domènech, Xavier (Institució Catalana de Recerca i Estudis Avançats)

Data: 2007
Resum: 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 2 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 2 supp(μ), and we show that all n-dimensional Calder´on-Zygmund operators are bounded on Lp(w dμ) if and only if N is bounded on Lp(w dμ), for a fixed p 2 (1,1). 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 H¨older inequality, in general, but some kind of self improving property still holds. On the other hand, if f 2 RBMO(μ) and " > 0 is small enough, then e"f belongs to this class of weights.
Drets: Tots els drets reservats.
Llengua: Anglès.
Document: article ; recerca ; publishedVersion
Publicat a: Publicacions Matemàtiques, V. 51 n. 2 (2007) p. 397-456, ISSN 0214-1493

DOI: 10.5565/PUBLMAT_51207_06

60 p, 467.9 KB

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