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On a trilinear singular integral form with determinantal kernel


Authors: Philip Gressman, Danqing He, Vjekoslav Kovač, Brian Street, Christoph Thiele and Po-Lam Yung
Journal: Proc. Amer. Math. Soc. 144 (2016), 3465-3477
MSC (2010): Primary 42B20
Published electronically: April 13, 2016
MathSciNet review: 3503714
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Abstract: We study a trilinear singular integral form acting on two-
dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calderón commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the $ \textup {L}^p$ estimates hold for the considered form.


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Additional Information

Philip Gressman
Affiliation: Department of Mathematics, David Rittenhouse Lab, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: gressman@math.upenn.edu

Danqing He
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: dhd27@mail.missouri.edu

Vjekoslav Kovač
Affiliation: Faculty of Science, Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
Email: vjekovac@math.hr

Brian Street
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: street@math.wisc.edu

Christoph Thiele
Affiliation: Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: thiele@math.uni-bonn.de

Po-Lam Yung
Affiliation: The Chinese University of Hong Kong, Ma Liu Shui, Hong Kong
Email: plyung@math.cuhk.edu.hk

DOI: https://doi.org/10.1090/proc/13007
Keywords: Multilinear form, bilinear Hilbert transform, Calder\'on commutator, invariant trilinear form
Received by editor(s): October 5, 2015
Published electronically: April 13, 2016
Additional Notes: The first author was partially supported by the NSF grant DMS-1361697 and an Alfred P. Sloan Research Fellowship
The second author was partially supported by the Simons Foundation Grant Number 315380
The third author was partially supported by the Croatian Science Foundation under the project 3526
The fourth author was partially supported by the NSF grant DMS-1401671
The fifth author was partially supported by the Hausdorff Center for Mathematics
The sixth author was partially supported by a direct grant for research from the Chinese University of Hong Kong (4053120)
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2016 American Mathematical Society