Sobolev norm estimates for a class of bilinear multipliers

Frédéric Bernicot; Vjekoslav Kovač

doi:10.3934/cpaa.2014.13.1305

Article Contents

2014, Volume 13, Issue 3: 1305-1315. Doi: 10.3934/cpaa.2014.13.1305

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Sobolev norm estimates for a class of bilinear multipliers

Frédéric Bernicot^1, and
Vjekoslav Kovač^2,

1.
Laboratoire Paul Painlevé - CNRS, Université Lille 1, 59655 Villeneuve d’Ascq Cedex
2.
University of Zagreb, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb

Received: July 2013

Revised: September 2013

Published: May 2015

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Abstract

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing $\mathrm{L}^p$ estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order $0$.

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Mathematics Subject Classification: 42B15.

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References

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