On fractional Laplacians

TL;DR

This paper compares two types of fractional Laplacians, showing their difference is positive definite and that they share the same Sobolev constants, contributing to the understanding of fractional differential operators.

Contribution

It establishes the positivity and equivalence of Sobolev constants for Navier and Dirichlet fractional Laplacians, clarifying their relationship.

Findings

Difference between the two fractional Laplacians is positive definite.

The Sobolev constants for both operators coincide.

Positivity preserving property of their difference.

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.