Abstract In this paper we develop the calculus of pseudo-differential operators on the lattice Z n , which we can call pseudo-difference operators. An interesting feature of this calculus is that the global frequency space ( T n ) 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 l 2 , weighted l 2 , and l p boundedness of operators and for their compactness on l p . We describe a link to the toroidal quantization on the torus T n , and apply it to give conditions for the membership in Schatten classes on l 2 ( Z n ) . Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their l 2 -boundedness. The results are applied to give estimates for solutions to difference equations on the lattice Z n . Moreover, we establish Garding and sharp Garding inequalities, with an application to the unique solvability of parabolic equations on the lattice Z n .

中文翻译：

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Zn 上的微分方程和伪微分算子

摘要 在本文中，我们开发了格 Z n 上的伪微分算子的演算，我们可以将其称为伪差分算子。该演算的一个有趣特征是全局频率空间 (Tn) 是紧凑的，因此符号类是根据与晶格变量相关的行为来定义的。我们为椭圆算子建立了组合、伴随、转置和参数的公式。我们还给出了算子的 l 2 、加权 l 2 和 lp 有界性以及它们在 lp 上的紧致性的条件。我们描述了环面 T n 上环形量化的链接，并应用它来给出 l 2 (Z n ) 上 Schatten 类的成员资格条件。此外，我们讨论了格子上的傅立叶积分算子的一个版本，并给出了它们的 l 2 有界的条件。应用结果来估计晶格 Z n 上的差分方程的解。此外，我们建立了 Garding 不等式和尖锐 Garding 不等式，并将其应用于晶格 Z n 上抛物线方程的唯一可解性。