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Littlewood-Paley theory and the T(1) theorem with non doubling measures

Xavier Tolsa
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Abstract

Let $μ$ be a Borel measure on $R^d$ which may be non doubling. The only condition that $μ$ must satisfy is $μ(B(x,r))\leq C r^n$ for all $x\in R^d$, $r>0$, and for some fixed $0<n\leq d$. In this paper, we develop Littlewood-Paley theory for functions in $L^p(μ)$. One of the main difficulties is the construction of reasonable approximations of the identity for obtaining a Calderon type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderon-Zygmund operators, without doubling assumptions, can be proved using the Littlewood-Paley decomposition that is obtained for $L^2(μ)$ functions, as in the classical case of homogeneous spaces.

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Xavier Tolsa. 2000-06-05. Littlewood-Paley theory and the T(1) theorem with non doubling measures. https://arxiv.org/abs/math/0006039

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