Seminars & Colloquia Calendar

Haim Brezis – Rutgers University and Technion

Wednesday, February 3, 10:45AM

Title:

"A surprising formula for Sobolev norms"

Abstract:

The classical Sobolev spaces involve Lp norms of the gradient of a function f. We present an original point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space Lp is replaced by the slightly larger Marcinkiewicz space Mp (aka weak Lp 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. In particular, it rectifies some well-known irregularities occurring in the theory of fractional Sobolev spaces. For example, we may derive alternative estimates in some exceptional cases (involving \(W^{1,1}\)) where the "anticipated'' fractional Sobolev and Gagliardo-Nirenberg inequalities fail.

The argument relies on an "innocuous looking'' new calculus inequality which might be useful in other situations.

The lecture is based on a recent joint work with Jean Van Schaftingen and Po-Lam Yung (to appear in PNAS).