Asia Mathematika, Vol 10, issue 1, pages: 22--34.

Toroidal pseudo-differential operators and scalar quantization on Lie groups

Mykola Ivanovich Yaremenko
National Technical University of Ukraine, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine

Received: 22 Feb 2026 | Accepted: 04 May 2026 | Final Version: 30 May 2026

Abstract

Toroidal pseudo-differential operators on tori Tn = Rn/Zn are studied, and global pseudo-differential calculus for symbols defined on Tn × Zn is introduced on tori. We establish the condition on symbols associated with toroidal pseudo-differential operators under which these operators map into certain functional spaces. Thus, assume Op(σ) is the pseudo-differential operator associated with σ(g, ξ) on (Tn × Zn) that is continuous in g for each ξ and satisfies |Δξα ∂gγσ(g, ξ)| ≤ c(α,γ) (1 + |ξ|)−|α|, then Op(σ) extends to a bounded linear operator Lp(Tn) → Lp(Tn) for all p ∈ (1, ∞). We consider a simply connected Lie group G with Haar measure μ, and g is the Lie algebra associated with G. Assume that a is a smooth function satisfying ∫g′ supX′ ∈ g′ |â(X′, Z′)| dη̂(Z′) < ∞, where â denotes the Fourier transform of a. Then the mapping Op(a) is given by Op(a)f(g) = ∫g′ ∫G exp(iX′(log(gh−1))) a(g, X′) f(h) dμ(h) dη̂(X′) and is a bounded linear operator L2(G) → L2(G).

Keywords:
Periodic pseudodifferential operators, periodic integral operators, symbol analysts, Fourier transform, Lie group, Lie algebra, dynamic system, quantization.

1. Introduction

This article is dedicated to the theory of pseudo-differential operators on tori Tn = Rn/Zn with symbols on Tn × Zn. Our goal is to investigate the general regularity properties of toroidal operators by elementary means employing methods of classical harmonic analysis. The symbols σ(g, ξ) of toroidal pseudo-differential operators are defined on Tn × Zn and are continuous in g for each ξ ∈ Zn, and satisfy the condition |Δξα ∂gγσ(g, ξ)| ≤ c(α, s, γ) (1 + |ξ|)s − ρ|α| − ϑ|γ|, for all g ∈ Tn, ξ ∈ Zn, and all multi-indices α and γ. Thus, the classical restriction σ ∈ C∞(Tn × Zn) can be mitigated.

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