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

Abstract

Let $\mu$ be a Borel measure on $R^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq C r^n$ 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 $L^p(\mu)$. 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 $L^2(\mu)$ functions, as in the classical case of homogeneous spaces.