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

Let mu be a Radon measure on R-d which may be non-doubling. The only condition that mu must satisfy is mu(B(x, r)) less than or equal to Cr-n, for all x is an element of R-d, r > 0, and for some fixed 0 < n less than or equal to d. In this paper, Littlewood Paley theory for functions in L-p(mu) is developed. One of the main difficulties to be solved is the construction of "reasonable" approximations of the identity in order to obtain a Calderon type reproducing formula. Moreover, it is shown that the T(l) theorem for n-dimensional Calderon Zygmund operators, without doubling assumptions, can be proved using the Littlewood Paley type decomposition that is obtained for functions in L-2(mu), as in the classical case of homogeneous spaces.} (C) 2001 Elsevier Science.