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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.

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