Boundedness of singular integrals on C^{1,alpha} intrinsic graphs in the Heisenberg group

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Chousionis , V , Faessler , K & Orponen , T 2019 , ' Boundedness of singular integrals on C^{1,alpha} intrinsic graphs in the Heisenberg group ' , Advances in Mathematics , vol. 354 , 106745 . https://doi.org/10.1016/j.aim.2019.106745

Title: Boundedness of singular integrals on C^{1,alpha} intrinsic graphs in the Heisenberg group
Author: Chousionis, Vasileios; Faessler, Katrin; Orponen, Tuomas
Contributor: University of Helsinki, University of Connecticut
University of Helsinki, Department of Mathematics and Statistics
Date: 2019-10-01
Language: eng
Number of pages: 45
Belongs to series: Advances in Mathematics
ISSN: 0001-8708
URI: http://hdl.handle.net/10138/307139
Abstract: 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.
Subject: Singular integrals
Heisenberg group
Removable sets for harmonic functions
LIPSCHITZ GRAPHS
RIESZ TRANSFORM
RECTIFIABILITY
OPERATORS
111 Mathematics
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