Toroidal pseudo-differential operators and scalar quantization on Lie groups

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 |Δ_ξ^α ∂_g^γσ(g, ξ)| ≤ c(α,γ) (1 + |ξ|)^−|α|, then Op(σ) extends to a bounded linear operator L^p(Tⁿ) → L^p(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′ sup_{X′ ∈ 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).

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 |Δ_ξ^α ∂_g^γσ(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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Yaremenko, M.I. (2026). Toroidal pseudo-differential operators and scalar quantization on Lie groups. Asia Mathematika, 10(1), 22–34. https://doi.org/10.5281/zenodo.20996294

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