# Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions.

Publicacions Matemàtiques (2007)

- Volume: 51, Issue: 2, page 397-456
- ISSN: 0214-1493

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topTolsa, Xavier. "Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions.." Publicacions Matemàtiques 51.2 (2007): 397-456. <http://eudml.org/doc/41900>.

@article{Tolsa2007,

abstract = {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.},

author = {Tolsa, Xavier},

journal = {Publicacions Matemàtiques},

keywords = {Integrales singulares; Operadores de Calderón-Zygmund; Operador maximal de Hardy-Littlewood; Desigualdades},

language = {eng},

number = {2},

pages = {397-456},

title = {Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions.},

url = {http://eudml.org/doc/41900},

volume = {51},

year = {2007},

}

TY - JOUR

AU - Tolsa, Xavier

TI - Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions.

JO - Publicacions Matemàtiques

PY - 2007

VL - 51

IS - 2

SP - 397

EP - 456

AB - 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.

LA - eng

KW - Integrales singulares; Operadores de Calderón-Zygmund; Operador maximal de Hardy-Littlewood; Desigualdades

UR - http://eudml.org/doc/41900

ER -

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