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In this paper, we study the $M$-ellipticity of Fredholm pseudo-differentialoperators 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 operatorsand the hybrid class of pseudo-differential operators, namely SG$M$-ellipticoperators.

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