# Analysis Seminar

#### Transport of BMO-Type Spaces by a Measure-Preserving Map and Applications to 2D Euler Equation

**Speaker:**
Frederic Bernicot, University of Nantes

**Location:**
Warren Weaver Hall 1314

**Date:**
Tuesday, December 4, 2012, 10 a.m.

**Synopsis:**

We will present some sharp estimates about the transport of BMO-type spaces, via a bi-Lipschitz measure preserving map in the Euclidean space. More precisely, we are interested in inequalities of the following type: $$ \| f(\phi) \|_X \lesssim C(\phi) \|f\|_X $$ where \(X\) is a space like BMO, Lipschitz space, Hardy space, Carleson measure spaces .... and \(\phi\) is a bi-Lipschitz measure preserving map. The aim is to prove such inequalities with a sharp constant \(C(\phi)\). We want to emphasize how the "measure preserving" property allows us to get some improved inequalities. Then, we will explain how we can use this argument to describe a new framework for 2D Euler equations. We can define a space strictly containing \(L^\infty\) where global well-posedness results can be proved with the vorticity living in this new space.