Browsing by Subject "RECTIFIABILITY"

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  • Orponen, Tuomas (2019)
    Let mu be Radon measures on R, with mu nonatomic and nu doubling, and write mu = mu(a) + mu(s) for the Lebesgue decomposition of mu relative to nu. For an interval I subset of R, define alpha(mu,nu) (I) := W-1 (mu(I), nu(I)), the Wasserstein distance of normalised blow-ups of mu and nu restricted to I. Let S nu be the square function S-nu(2) (mu) = Sigma alpha(2)(mu,nu)(I) chi(1), where D is the family of dyadic intervals of side-length at most 1. I prove that S-nu(mu) is finite mu(a) almost everywhere and infinite mu(s) almost everywhere. I also prove a version of the result for a nondyadic variant of the square function S-nu(mu). The results answer the simplest "n = d = 1" case of a problem of J. Azzam, G. David and T. Toro.
  • Chousionis, Vasileios; Faessler, Katrin; Orponen, Tuomas (2019)
    We study singular integral operators induced by 3-dimensional Calderon-Zygmund kernels in the Heisenberg group. We show that if such an operator is L (2) bounded on vertical planes, with uniform constants, then it is also L-2 bounded on all intrinsic graphs of compactly supported C-1,C-alpha functions over vertical planes. In particular, the result applies to the operator R, induced by the kernel K(z) = del(H )parallel to z parallel to(-2), z is an element of H \ {0}, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L-2 boundedness of R, is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite 3-dimensional measure. (C) 2019 Elsevier Inc. All rights reserved.
  • Chousionis, Vasileios; Fässler, Katrin; Orponen, Tuomas (2019)
    The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group H. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 1990s. The theory in H has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in R-3. Our main object of study are the intrinsic Lipschitz graphs in H, introduced by B. Franchi, R. Serapioni, and F. Serra Cassano in 2006. We claim that these 3-dimensional sets in H, if any, deserve to be called quantitatively 3-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical beta-numbers. Conversely, extending a result of David and Semmes from R-n, we prove that a 3-Ahlfors-David regular subset in H, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.