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2007 Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions
Xavier Tolsa
Publ. Mat. 51(2): 397-456 (2007).

Abstract

Let $\mu$ be a Borel measure on ${\mathbb R}^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq Cr^n$ for all $x\in \mathbb {R}^d$, $r>0$ and for some fixed $n$ with $0<n\leq d$. In this paper we introduce a maximal operator $N$, which coincides with the maximal Hardy-Littlewood operator if $\mu(B(x,r))\approx r^n$ for $x\in \rm{supp}(\mu)$, and we show that all $n$-dimensional Calderón-Zygmund operators are bounded on $L^p(w\,d\mu)$ if and only if $N$ is bounded on $L^p(w\,d\mu)$, for a fixed $p\in(1,\infty)$. 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 $f \in \mathit{RBMO}(\mu)$ and ${\varepsilon}>0$ is small enough, then $e^{{\varepsilon} f}$ belongs to this class of weights.

Citation

Download Citation

Xavier Tolsa. "Weighted norm inequalities for Calderón-Zygmund operators without doubling conditions." Publ. Mat. 51 (2) 397 - 456, 2007.

Information

Published: 2007
First available in Project Euclid: 31 July 2007

zbMATH: 1136.42303
MathSciNet: MR2334796

Subjects:
Primary: 42B20
Secondary: 42B25

Rights: Copyright © 2007 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.51 • No. 2 • 2007
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