78 research outputs found
Characterization of n-rectifiability in terms of Jones' square function: Part II
We show that a Radon measure in which is absolutely
continuous with respect to the -dimensional Hausdorff measure is
-rectifiable if the so called Jones' square function is finite -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
-rectifiable measures which are absolutely continuous with respect to
. 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.Comment: A corollary regarding analytic capacity and a few new references have
been adde
High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures
This is the second of two papers wherein we estimate multiscale least squares
approximations of certain measures by Menger-type curvatures. More
specifically, we study an arbitrary d-regular measure on a real separable
Hilbert space. The main result of the paper bounds the least squares error of
approximation at any ball by an average of the discrete Menger-type curvature
over certain simplices in in the ball. A consequent result bounds the
Jones-type flatness by an integral of the discrete curvature over all
simplices. The preceding paper provided the opposite inequalities. Furthermore,
we demonstrate some other discrete curvatures for characterizing uniform
rectifiability and additional continuous curvatures for characterizing special
instances of the (p, q)-geometric property. We also show that a curvature
suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit
within our framework.Comment: 32 pages, no figure
An Analyst's Traveling Salesman Theorem For Sets Of Dimension Larger Than One
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable
curves in the plane via a multiscale sum of -numbers. These
-numbers are geometric quantities measuring how far a given set deviates
from a best fitting line at each scale and location. Jones' result is a
quantitative way of saying that a curve is rectifiable if and only if it has a
tangent at almost every point. Moreover, computing this square sum for a curve
returns the length of the curve up to multiplicative constant. K. Okikiolu
extended his result from subsets of the plane to subsets of Euclidean space. G.
David and S. Semmes extended the discussion to include sets of (integer)
dimension larger than one, under the assumption of Ahlfors regularity and using
a variant of Jones' numbers. In this paper we give a version of P.
Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean
space. We estimate the -dimensional Hausdorff measure of a set in terms of
an analogous sum of -type numbers. There is no assumption of Ahlfors
regularity, but rather, only of a lower bound on the Hausdorff content. We
adapt David and Semmes' version of Jones' -numbers by redefining them
using a Choquet integral. A key tool in the proof is G. David and T. Toro's
parametrization of Reifenberg flat sets (with holes).Comment: Corrected more typos. There are still several typos and small
mistakes in the published version of the paper, so the authors will maintain
an up-to-date version on their webpages as we continue to correct the
REMOVABLE SETS FOR LIPSCHITZ HARMONIC FUNCTIONS ON CARNOT GROUPS
Abstract. Let G be a Carnot group with homogeneous dimension Q ≥ 3 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is (Q − 1). Moreover we construct self-similar sets with positive and finite H Q−1 measure which are removable. 1
An endpoint estimate for the commutators of singular integrals with non doubling measures
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