Abstract
In this paper we develop the calculus of pseudo-differential operators on the lattice Zn, which we can call pseudo-difference operators. An interesting feature of this calculus is that the global frequency space (Tn) 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 ℓ2, weighted ℓ2, and ℓp boundedness of operators and for their compactness on ℓp. We describe a link to the toroidal quantization on the torus Tn, and apply it to give conditions for the membership in Schatten classes on ℓ2(Zn). Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their ℓ2-boundedness. The results are applied to give estimates for solutions to difference equations on the lattice Zn. Moreover, we establish Gårding and sharp Gårding inequalities, with an application to the unique solvability of parabolic equations on the lattice Zn.
Highlights
In this paper we develop the calculus of pseudo-differential operators on the lattice Zn, which we can call pseudodifference 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 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 2(Zn)
Summary
We develop elements of the symbolic calculus of pseudo-differential operators on Zn by deriving formulae for the composition, adjoint, transpose, as well as for the parametrix for elliptic operators. The composition Op(σ) ◦ Op(τ ) is a pseudo-differential operator with symbol ς ∈ Sρμ,1δ+μ2 (Zn × T n), which can be given as an asymptotic sum ς(k, x) ∼. The pseudo-differential operators with symbols σ and τ are given by e2πi(k−m)·xσ(k, x)f (m)dx, m∈ZnT n. Σj ∈ Sρμ,Nδ (Zn × T n), for all N ∈ N This statement immediately follows from [34, Theorem 4.4.1] since the symbol classes are the same modulo swapping the order of the variables. An operator A ∈ Op(Sρμ,δ(Zn × T n)) is elliptic if and only if there exists B ∈ Op(Sρ−,δμ(Zn × T n)) such that. By the composition formula in Theorem 3.1 we have σB0A = σB0 σA − σT 1 − σT , for some T ∈ Sρ−,δ(ρ−δ)(Zn × T n), B0A = I − T. The rest follows by using a similar argument to the proof of [34, Theorem 4.9.13], completing the proof. 2
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