Andrea Nahmod
Department of Mathematics
University of Massachusetts

Abstract: In past few decades harmonic analysis has developed beyond the linear framework into the multilinear one where we find pioneer work by J.M. Bony and by R. Coifman and Y. Meyer. Still, questions remained about operators which could have (highly) singular multipliers or (multilinear) symbols with 'non-standard' decay conditions. In this talk we describe recent work in this direction and some open questions. We start with translation invariant bilinear operators with non-smooth multipliers and a comprehensive criterion in one dimensions ensuring their boundedness. The approach relies on time frequency analysis as pioneered by C. Fefferman and further developed by Lacey and Thiele. We continue by considering the non-translation invariant case; i.e bilinear pseudo-differential operators with x-dependent symbols. We describe a new bilinear T(1,1)-theorem for a class of these under modulation invariance. A natural example is the x-dependent bilinear Hilbert transform.