An extension theorem for some pseudo-differential operators

TL;DR

This paper establishes an extension theorem for fractional powers of certain pseudo-differential operators, enabling their local realization and exploring properties of the relativistic Schrödinger operator.

Contribution

It introduces a new extension theorem for fractional pseudo-differential operators, generalizing the Caffarelli-Silvestre approach to a broader class.

Findings

Fractional powers of local pseudo-differential operators can be realized locally.

Provides properties of the relativistic Schrödinger operator $ ext{D}_m^eta$.

Extends the extension method to new classes of pseudo-differential operators.

Abstract

In this paper we provide an extension theorem for fractional powers of some pseudo-differential operators $P(D)$. These extensions yields realization of the fractional powers of some pseudo-differential operators in the spirit of Caffarelli and Silvestre \cite{CSilv}. In particular every fractional power of a local pseudo-differential operator can be realized locally. We also give some properties of the relativistic Schr\"odinger operator $\Dsm$.