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

Abstract

Let m be a Radon measure on R d which may be non-doubling. The only condition that m must satisfy is m (B(x, r), r > 0, and for some fixedOne of the main difficulties to be solved is the construction of ''reasonable'' approximations of the identity in order to obtain a Calderó n 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 L 2 (m), as in the classical case of homogeneous spaces.}

Besov spaces with non-doubling measures

“…We use the same notation and definitions as in Tolsa [27] (also [28]), and for the reader's convenience, we recall some basic notation and definitions here; see [27,28] for more details.…”

“…We first recall the definition of doubling cubes of Tolsa in [26,27]. Given α > 1 and β > α n , we say that the cube Q ⊂ R d is (α, β)-doubling if µ(αQ) ≤ βµ(Q); see [26,27] for the existence and some other basic properties of the doubling cubes.…”

“…So, for x, y ∈ supp (µ) and some cube Q, the notations δ(x, Q) and δ(x, y) make sense; see [28,27] for some useful properties of δ(·, ·).…”

“…Recently more attention has been paid to non-doubling measures. It has been shown that many results of this theory still hold without assuming the doubling property; see [16,17,18,19,23,24,25,29,5,6] for some results on Calderón-Zygmund operators, [15,26,27,28] for some other results related to the spaces BM O(µ) and H 1 (µ), and [7,8,20] for the vector-valued inequalities on the Calderón-Zygmund operators and weights. …”

“…More recently, Tolsa constructed a "reasonable" approximation to the identity. To be precise, Tolsa in [27] constructed a sequence of integral operators…”