SPRING 2021 Nonlinear Analysis Seminar Series


Wednesday, March 24, 2021 - 18:00 to 19:00


on Zoom



Professor Haim Brezis, National Academy of Sciences

Title: "New perspectives on Sobolev norms"


The classical Sobolev spaces involve L^p norms of the gradient of a function u. We present an original point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space L^p is replaced by the slightly larger Marcinkiewicz space M^p (aka weak L^p space) --- a popular tool in Harmonic Analysis. Surprisingly, these spaces coincide with the standard Sobolev spaces, a fact which sheds a new light onto these classical objects and should have numerous applications. It allows e.g., to rectify some well-known “irregularities” occurring in the theory of fractional Sobolev spaces. In particular, we may derive alternative estimates in some exceptional cases (involving W^{1,1}) where the anticipated fractional Sobolev and Gagliardo-Nirenberg inequalities fail.
Part of the central argument relies on an innocuous looking new calculus inequality which might be useful in other situations. The current proof of this inequality is more complicated than expected and it would be desirable to find a simpler one.
The lecture is based on a recent joint work with Jean Van Schaftingen and Po-Lam Yung (PNAS Feb 2021).


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