Difference equations and pseudo-differential operators onZn

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 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 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 .Contents 1991 Mathematics Subject Classification. 58J40, 35S05, 35S30, 42B05, 47G30.

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

“…for all β with |β| = 2κ, then T σ extends to a bounded operator from L p (Z n ) into L p (Z n ) provided that 1 ≤ p ≤ ∞, and κ > n/2. This implies the L p -boundedness of pseudo-differential operators associated to the discrete Hörmander class S 0 0,0 (Z n × T n ) introduced in Botchway, Kibiti and Ruzhansky [5].…”

“…Let us mention that Corollary 3.11 implies the L p -boundedness of pseudodifferential operators associated to the discrete Hörmander class S 0 0,0 (Z n × T n ) introduced in Botchway, Kibiti and Ruzhansky [5].…”

“…In this subsection we provide some results on the boundedness of discrete pseudodifferential operators in the multilinear context. By following Botchway Kibiti and Ruzhansky [5], a symbol σ belongs to the discrete Hörmander class S m ρ,δ (Z n × T n ), if it satisfies discrete inequalities of the type…”

“…Pseudo-differential operators on Z n (discrete pseudo-differential operators) were introduced by Molahajloo in [47], and some of their properties were developed in the last few years, see [8,16,50,51,52,53,55]. However, Botchway, Kibiti, and Ruzhansky in their recent fundamental work [5] investigated the discrete pseudo-differential calculus and its applications to difference equations.…”

Pseudo-Differential Operators on $${\mathbb {Z}}^n$$ Z n with Applications to Discrete Fractional Integral Operators

“…e-mail: cardonaduvan@javeriana.edu.co ℓ p -estimates are results of boundedness in ℓ p -spaces of suitable linear operators. Although the L p -estimates for multipliers on R n , are motivated by its applications to PDE's, (see, e.g., Hörmander [13]) ℓ p -bounds for multipliers on Z n can be applied to discrete problems, ones arising from difference equations and others from number theory (see [18,20]). Taking this into account, multipliers on Z n have been studied principally from three sources.…”