$M$-Ellipticity of Fredholm Pseudo-Differential Operators on $L^p(\mathbb{R}^n)$ and G\r{a}rding's Inequality

TL;DR

This paper investigates the $M$-ellipticity of Fredholm pseudo-differential operators on $L^p$ spaces and establishes G {a}rding's inequality for these operators and their hybrid SG$M$ class, advancing the theory of elliptic operators.

Contribution

It introduces the concept of $M$-ellipticity for Fredholm pseudo-differential operators on $L^p$ spaces and proves G {a}rding's inequality for these and SG$M$-elliptic operators, extending existing elliptic theory.

Findings

Established $M$-ellipticity for Fredholm pseudo-differential operators.

Proved G {a}rding's inequality for $M$-elliptic and SG$M$-elliptic operators.

Extended elliptic theory to weighted symbol classes on $L^p$ spaces.

Abstract

In this paper, we study the $M$-ellipticity of Fredholm pseudo-differential operators associated with weighted symbols on $L^p(\mathbb{R}^n)$, $1 < p < \infty$. We also prove the G\r{a}rding's inequality for $M$-elliptic operators and the hybrid class of pseudo-differential operators, namely SG$M$-elliptic operators.