Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures

Abstract

Let μ be a Radon 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>0, and for some fixed 0<n⩽d. In this paper, Littlewood–Paley theory for functions in Lp(μ) is developed. One of the main difficulties to be solved is the construction of “reasonable” approximations of the identity in order to obtain a Calderón type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderón–Zygmund operators, without doubling assumptions, can be proved using the Littlewood–Paley type decomposition that is obtained for functions in L2(μ), as in the classical case of homogeneous spaces.}