Difference equations and pseudo-differential operators on $\mathbb{Z}^n$

Abstract

In this paper we develop the calculus of pseudo-differential operators on the lattice $\mathbb{Z}^n$, which we can call pseudo-difference operators. An interesting feature of this calculus is that the phase space is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable. We establish formulae for composition, adjoint, transpose, and for parametrix for the elliptic operators. We also give conditions for the $\ell^2$, weighted $\ell^2$, and $\ell^p$ boundedness of operators and for their compactness on $\ell^p$. We describe a link to the toroidal quantization on the torus $\mathbb{T}^n$, and apply it to give conditions for the membership in Schatten classes on $\ell^2(\mathbb{Z}^n)$. Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their $\ell^2$-boundedness. The results are applied to give estimates for solutions to difference equations on the lattice $\mathbb{Z}^n$. Moreover, we establish Garding and sharp Garding inequalities, with an application to the unique solvability of parabolic equations on the lattice $\mathbb{Z}^n$.