##### Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 12, 15:30 ~ 15:50

## Nonlocal operators of order near zero

### de Pablo, Arturo

We study nonlocal operators $\mathcal{L}u(x)=\int_{\mathbb{R^N}}(u(x)-u(y))J(x,y)\,dy$, driven by non integrable Lévy kernels $J$ in the limit of integrability. We look at the Sobolev type spaces associated to those operators and show some important properties, like Hardy inequalities, a symmetrization result, or the inclusion in a Lorentz space and the compact inclusion in $L^2$. We also study the effect of applying the operators to continuous functions or the characteristic function of a set. We finally characterize the smoothing effect for the solutions of the elliptic problem $\mathcal{L}u=f$ and the parabolic problem $u_t+\mathcal{L}u=0$.

Autores: de Pablo, Arturo / Correa, Ernesto.