Local Hardy-Sobolev inequalities for canceling elliptic differential operators

Mardi 10 octobre 2017 14:00-15:00 - Tiago H. Picon - Université de São Paulo

Résumé : In this lecture we show that if A(x,D) is a linear differential operator of order \nu with smooth complex coefficients in \Omega\subset\mathbb{R}^N from a complex vector space E to a complex vector space F, then the Hardy-Sobolev inequality

<br class='autobr' />
\int_{\mathbb{R}^{N}}\frac{|D^{\nu-\ell}u(x)|}{|x-x_0|^{\ell}}\,dx\leq C \int_{\mathbb{R}^{N}}|A(x,D)u|dx, \quad u \in C_{c}^{\infty}(B<small class="fine"> </small>;E),<br class='autobr' />

for \ell \in \left\{ 1,...,\min\left\{\nu,N-1 \right\} \right\} holds locally at any point x_0\in\Omega if and only if A(x,D) is elliptic and the constant coefficients homogeneous operator A_\nu(x_0,D) is canceling in the sense of Van Schaftingen for every x_0\in \Omega, which means that

<br class='autobr' />
\bigcap_{\xi\in\mathbb{R}^N\setminus\{0\}}a_\nu(x_0,\xi)[E]=\{0\}.<br class='autobr' />

Here A_\nu(x,D) is the homogeneous part of order \nu of A(x,D) and a_\nu(x,\xi) is the principal symbol of A(x,D).
This is joint work with Jorge Hounie (UFSCar, Brazil).

Lieu : Salle 113-115 (Bâtiment 425)

Local Hardy-Sobolev inequalities for canceling elliptic differential operators  Version PDF