Singular integrals on regular curves in the Heisenberg group
Fässler, K., & Orponen, T. (2021). Singular integrals on regular curves in the Heisenberg group. Journal de Mathematiques Pures et Appliquees, 153, 30-113. https://doi.org/10.1016/j.matpur.2021.07.004
Published inJournal de Mathematiques Pures et Appliquees
© 2021 the Authors
Let be the first Heisenberg group, and let be a kernel which is either odd or horizontally odd, and satisfies The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel . We prove that convolution with k, as above, yields an -bounded operator on regular curves in . This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all 3-dimensional horizontally odd kernels yield bounded operators on Lipschitz flags in . This is needed for solving sub-elliptic boundary value problems on domains bounded by Lipschitz flags via the method of layer potentials. The details are contained in a separate paper. Finally, our technique yields new results on certain non-negative kernels, introduced by Chousionis and Li.
ISSN Search the Publication Forum0021-7824
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Related funder(s)Academy of Finland
Funding program(s)Academy Research Fellow, AoF
Additional information about fundingK.F. is supported by the Academy of Finland via the project Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, grant No. 321696.
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