CHARACTERIZATION OF n-RECTIFIABILITY IN TERMS OF JONES’ SQUARE FUNCTION: PART II
نویسنده
چکیده
We show that a Radon measure μ in R which is absolutely continuous with respect to the n-dimensional Hausdorff measure H is n-rectifiable if the so called Jones’ square function is finite μ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to H. Further, in this paper we also investigate the relationship between the Jones’ square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.
منابع مشابه
CHARACTERIZATION OF n-RECTIFIABILITY IN TERMS OF JONES’ SQUARE FUNCTION: PART I
In this paper it is shown that if μ is a finite Radon measure in R which is n-rectifiable and 1 ≤ p ≤ 2, then ∫ ∞ 0 β μ,p(x, r) 2 dr r < ∞ for μ-a.e. x ∈ R, where β μ,p(x, r) = inf L ( 1 rn ∫ B̄(x,r) ( dist(y, L) r )p dμ(y) )1/p , with the infimum taken over all the n-planes L ⊂ R. The β μ,p coefficients are the same as the ones considered by David and Semmes in the setting of the so called unif...
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