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 $\Omega\subset\mathbbR^N$ from a complex vector space $E$ to a complex vector space $F$, then the Hardy-Sobolev inequality

$$
\int_\mathbbR^N\frac|D^\nu-\ellu(x)||x-x_0|^\ell\,dx\leq C \int_\mathbbR^N|A(x,D)u|dx, \quad u \in C_c^\infty(B ;E),
$$

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

$$
\bigcap_\xi\in\mathbbR^N\setminus{0}a_\nu(x_0,\xi)[E]={0}.
$$

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