Weighted periodic and discrete Pseudo-Differential Operators

TL;DR

This paper develops symbolic calculus for weighted pseudo-differential operators on the torus, including asymptotic formulas, parametrices, and criteria for compactness, with applications to elliptic operators and PDE solutions.

Contribution

It introduces a new symbolic calculus framework for weighted pseudo-differential operators on the torus, including asymptotic, composition, and adjoint formulas, and establishes criteria for ellipticity and compactness.

Findings

Derived asymptotic sum formulas for operators.

Constructed parametrices for M-elliptic operators.

Established Gohberg's lemma and Gårding's inequalities.

Abstract

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class $M_{\rho, \Lambda}^m(\mathbb{ T}\times \mathbb{Z})$ (associated to a suitable weight function $\Lambda$ on $\mathbb{Z}$) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of $M$-elliptic pseudo-differential operators on $\mathbb{ T}$. Further, we prove a version of Gohberg's lemma for pseudo-differetial operators with weighted symbol class $M_{\rho, \Lambda}^0(\mathbb{ T}\times \mathbb{Z})$ and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on $L^2(\mathbb{T})$. Finally, we provide G\r{a}rding's and Sharp G\r{a}rding's inequality for $M$-elliptic operators on $\mathbb{Z}$ and $\mathbb{T}$, respectively, and present an application in the context of strong solution of the pseudo-differential equation $T_{\sigma} u=f$ in $L^{2}\left(\mathbb{T}\right)$.