Difference equations and pseudo-differential operators on ℤⁿ

2017/05/22 by Botchway, Linda N. A., Kibiti, P. Gaël, Ruzhansky, Michael
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1705.07564

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 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 ℓ², weighted ℓ², and ℓ^p boundedness of operators and for their compactness on ℓ^p. We describe a link to the toroidal quantization on the torus \mathbbTⁿ, 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 Garding and sharp Garding inequalities, with an application to the unique solvability of parabolic equations on the lattice ℤⁿ.

Related

• Semi-classical pseudo-differential operators on ℏℤⁿ and applications
• Symbolic calculus and M-ellipticity of pseudo-differential operators on ℤⁿ
• The essential adjointness of pseudo-differential operators on ℤⁿ
• Difference operators and difference equations on lattices, or grids, up to the elliptic hypergeometric case
• Ellipticity and Fredholmness of pseudo-differential operators on\n \ℓ²( Zn)