A pointwise inequality for fractional laplacians

TL;DR

This paper extends a key pointwise inequality for fractional Laplacians, which are important in models involving Lévy processes and fluid interfaces, providing new insights and proof sketches for related operators.

Contribution

It introduces an extended pointwise inequality for fractional Laplacians and offers proof sketches for fractional Laplace-Beltrami and Dirichlet-Neumann operators.

Findings

Extended inequality for fractional Laplacians.

Applicable to Laplace-Beltrami and Dirichlet-Neumann operators.

Provides proof sketches and underlying principles.

Abstract

The fractional laplacian is an operator appearing in several evolution models where diffusion coming from a L\'evy process is present but also in the analysis of fluid interphases. We provide an extension of a pointwise inequality that plays a r\^ole in their study. We begin recalling two scenarios where it has been used. After stating the results, for fractional Laplace-Beltrami and Dirichlet-Neumann operators, we provide an sketch of their proofs, unravelling the underlying principle to such inequalities.