Department of Mathematics

Speaker: Francesco Di Plinio
Title: Fractional Leibniz Rules and Banach-valued Singular Integrals
Affiliation: Brown University
Date: Friday, January 22, 2016
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. Fractional Leibniz rules, that is to say, L^p bounds involving the fractional derivative of a product, play a fundamental role in the well-posedness theory of nonlinear dispersive PDEs such as the generalized Korteweg-deVries equation and, in the form of commutator estimates, in the study of the Euler and Navier-Stokes equations. I will describe the history of the subject and its intimate connection with the theory of Coifman-Meyer multilinear singular integrals. I will also present a new bi-parameter Leibniz rule of mixed norm type, extending previous results by Muscalu, Pipher, Tao and Thiele and answering a question posed by Kenig. Our mixed norm Leibniz rule finds application to nonhomogeneous nonlinear dispersive PDEs. This result is obtained as a particular case of a novel Coifman-Meyer theorem for operator-valued multilinear multipliers acting on Banach-valued functions.

No background in the subject will be assumed. Partly joint work with Yumeng Ou.