On fractional Laplacians

Abstract

We compare two natural types of fractional Laplacians $(-\Delta {)}^s$, "Navier" and "Dirichlet" ones. We show that for $0<s<1$ their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.

14 pages; this version is considerably enlarged