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Harmonic analysis and the geometry of subsets of $${\mathbb{R}}^ n$$. (English) Zbl 0734.42012
Let E be a closed d-dimensional subset of $${\mathbb{R}}^ n$$, how do we measure the “smoothness” of E? In other words, how do we decide whether E can be locally approximated by d-dimensional planes? The present paper presents a sketch of the work of the authors on how to build a theory deciding this question, that theory resembles the use of the Paley-Littlewood methods to attack similar questions for functions.
To give a flavor of what is discussed let us introduce some notations. Let $$\mu$$ stand for d-dimensional Hausdorff measure restricted to E. E is said to be regular if there is $$C>0$$ such that $C^{-1} R^ d\leq \mu (E\cap B(x,R))\leq CR^ d,$ for all $$x\in E$$, $$R>0$$, B(x,R) $$=$$ ball of center x and radius R. One says that E has big Lipschitz subsets if there are $$K>0,\eta >0$$ so that for every $$x\in E,$$ $$R>0$$ there exist $$F\subseteq E\cap B(x,R),$$ $$G\subseteq {\mathbb{R}}^ d,\rho: G\to F$$ surjective so that $$\mu (F)\geq \eta R^ d,$$ diameter of $$G\leq R,$$ and $$| \rho (z)-\rho (w)| \leq K| z-w|,$$ for all $$z,w\in G.$$
Theorem. If E is regular then the following two conditions are equivalent: (i) E has big Lipschitz subsets. (ii) For each odd $$\psi \in C^{\infty}_ 0({\mathbb{R}}^ n),$$ the measure on $$E\times {\mathbb{R}}_ t$$ $\sum^{\infty}_{j=-\infty}| \int_{E}z^{-jd} \psi ((x- y)/2^ j)d\mu (y)|^ 2 d\mu (x)\delta_{2^{-j}}(t)$ is a Carleson measure.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 28A78 Hausdorff and packing measures
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