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

Tolsa, 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 &gt; 0 and for some fixed n with 0 &lt; 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 &gt; 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 &gt; 0 and for some fixed n with 0 &lt; 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 &gt; 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 -