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Rectifiable measures in 𝐑 n and existence of principal values for singular integrals. (English) Zbl 0880.28002

The authors studies relations between rectifiability properties of Borel measures in n and some singular integrals with respect to them. Let |·| be the norm in n and B(x,r) the closed ball with center x and radius r, and let Φ be a Borel regular measure. Let

K m Φ(a)=lim ε0 n B(a,ε) |y-x| m-1 (y-x)dΦ(y)

The m-dimensional lower density of the measure Φ at the point a is defined by

D ̲ m (Φ,a)=lim inf r0 r -m Φ(B(a,r))

A measure Φ is said to be m rectifiable if there exist m dimensional C 1 submanifolds M i such that Φ( n i=1 M i )=0, and Φ should be absolutely continuous with respect to the m dimensional Hausdorff measure m . One of the main result states that, if Φ is a finite Borel measure in n and if 0<D ̲ m (Φ,a)< and K m Φ(a) exist for Φ almost all a n , then Φ is m rectifiable.

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