Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 475-503.

The convolution algebra of Schwartz kernels along a singular foliation

Authors: Iakovos Androulidakis (1), Omar Mohsen (2), Robert Yuncken (3)
Author institution: (1) National and Kapodistrian University of Athens, Department of Mathematics, Panepistimiopolis, GR-15784 Athens, Greece
(2) Mathematisches Institut der WWU Muenster, Einsteinstrasse 62, 48149 Muenster, Germany
(3) Laboratoire de Mathematiques Blaise Pascal, Universite Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Summary: Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.

DOI: http://dx.doi.org/10.7900/jot.2019nov12.2291
Keywords: pseudodifferential operators, groupoids, foliations, singular foliations, convolution

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