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Littlewood-Paley Analysis on Non Homogeneous Spaces

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1966)

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.

Keywords

Homogeneous Space Besov Space Radon Measure Singular Integral Operator Homogeneous Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2009

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