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Rectifiable measures in $\bold R\sp n$ and existence of principal values for singular integrals. (English) Zbl 0880.28002
The authors studies relations between rectifiability properties of Borel measures in ${\bbfR}^{n}$ and some singular integrals with respect to them. Let $|\cdot |$ be the norm in ${\bbfR^{n}}$ and $B(x,r)$ the closed ball with center $x$ and radius $r$, and let $\Phi$ be a Borel regular measure. Let $$K_{m}\Phi (a)=\lim_{\varepsilon \downarrow 0}\int_{{\bbfR}^{n}\setminus B(a,\varepsilon)}|y-x|^{m-1}(y-x)d\Phi(y)$$ The m-dimensional lower density of the measure $\Phi$ at the point $a$ is defined by $$\underline{D}_{m}(\Phi,a)=\liminf_{r\downarrow 0}r^{-m}\Phi (B(a,r))$$ A measure $\Phi$ is said to be $m$ rectifiable if there exist $m$ dimensional $C^{1}$ submanifolds $M_{i}$ such that $\Phi({\bbfR}^{n}\setminus \bigcup_{i=1}^{\infty}M_{i})=0$, and $\Phi$ should be absolutely continuous with respect to the $m$ dimensional Hausdorff measure ${\cal H}^{m}$. One of the main result states that, if $\Phi$ is a finite Borel measure in ${\bbfR^{n}}$ and if $0<\underline{D}_{m}(\Phi,a)<\infty$ and $K_{m}\Phi(a)$ exist for $\Phi$ almost all $a\in{\bbfR}^{n}$, then $\Phi$ is $m$ rectifiable.

28A75Length, area, volume, other geometric measure theory
42B20Singular and oscillatory integrals, several variables
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