Littlewood-Paley Analysis on Non Homogeneous Spaces

It is well know that the doubling property of the underlying measure is a basic hypothesis in the classical Calderón-Zygmund theory. A measure μ on ℝⁿ is said to be doubling if there exists some constant C such that μ(B(x, 2r)) ≤ cμ (B(x, r)) for all x ϵ supp(μ),r > 0, where $B\left( {x,r} \right) = \left\{ {y \in R^n :\left| {y - x} \right| < r} \right\}$. Recently it has been shown that many results of the classical Calderón-Zygmund theory also hold without assuming the doubling property. See [GM], [MMNO], [NTV1], [NTV2], [NTV3], [T1], [T2] and [T3] for more material.

Suppose that μ is a Radon measure on ℝⁿ, which may be non-doubling and only satisfies the growth condition, namely there is a constant C > 0 such that for all x ϵ supp(μ) and r > 0,

$$\mu \left( {B\left( {x,r} \right)} \right) \le C_0 r^d $$

(5.1)

where 0 > d ≤ n.