Toroidal Pseudo-Differential Operators

• Toroidal pseudo-differential operators are global quantizations on the n-torus that extend classical symbolic calculus to discrete frequency settings.
• They are constructed via discrete Fourier analysis with defined symbol classes, composition laws, and classical asymptotic expansions for precise operator analysis.
• They exhibit robust mapping properties in Lp, Sobolev, Besov, and weighted spaces, making them vital for analyzing periodic magnetic operators and manifold random fields.

Toroidal pseudo-differential operators are global quantizations acting on functions over the $n$-torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$ or its variants, with symbolic calculus and mapping properties paralleling those on Euclidean spaces but adapted to the discrete frequency setting inherent to compact abelian groups. The calculus is developed via discrete Fourier analysis, with symbol classes and operator constructions that facilitate the study of periodic, global, and quantum systems, manifold random fields, and evolutionary PDEs.

1. Definitions and Symbol Classes

A toroidal pseudo-differential operator is defined by the action

$\Op(\sigma)\,u(x) = \sum_{k \in \mathbb{Z}^n} e^{i x \cdot k} \sigma(x,k) \widehat{u}(k)$

for $u \in C^\infty(\mathbb{T}^n)$, with $$\widehat{u}(k) = \int_{\mathbb{T}^n} e^{- i x \cdot k} u(x)\,dx$$. The kernel representation is

$$K(x,y) = \sum_{k \in \mathbb{Z}^n} e^{i (x-y) \cdot k} \sigma(x,k)$$

so that $\Op(\sigma)u(x) = \int_{\mathbb{T}^n} K(x,y) u(y)\,dy$ (Cardona et al., 27 Feb 2025).

The symbol classes $$S^m_{\rho,\delta}(\mathbb{T}^n \times \mathbb{Z}^n)$$, for parameters $m \in \mathbb{R}$ and $0 \leq \delta \leq \rho \leq 1$, consist of functions $\sigma(x,k)$ with smoothness in $x$ and finite-difference regularity in $k$: $$|\partial_x^\beta \Delta_k^\alpha \sigma(x,k)| \leq C_{\alpha,\beta} \langle k \rangle^{m - \rho |\alpha| + \delta |\beta|}$$ where $\langle k \rangle = (1 + |k|^2)^{1/2}$, $\Delta_k$ denotes discrete difference operators in $k$, and $\partial_x$ corresponds to ordinary derivatives in $x$ (Cornean et al., 27 Dec 2025, Velasquez-Rodriguez, 2019, Cardona et al., 27 Feb 2025).

2. Symbolic Calculus: Composition, Adjoint, and Classical Expansion

The global symbolic calculus on $\mathbb{T}^n$ mirrors the Hörmander–type calculus in the Euclidean setting:

• Composition: If $\sigma_j \in S^{m_j}_{\rho, \delta}$ for $j=1,2$, then

$\Op(\sigma_1)\Op(\sigma_2) = \Op(\sigma_1 \# \sigma_2)$

where

$$\sigma_1 \# \sigma_2(x,k) \sim \sum_{|\alpha| \geq 0} \frac{1}{\alpha!} \Delta_k^\alpha \sigma_1(x,k) \partial_x^\alpha \sigma_2(x,k)$$

modulo lower-order remainders (Cardona et al., 27 Feb 2025, Escobar-Velasquez, 12 Nov 2025).

• Adjoint: $\Op(\sigma)^* = \Op(\sigma^*)$ has symbol

$$\sigma^*(x,k) \sim \sum_{|\alpha|\geq 0} \frac{1}{\alpha!} \overline{\partial_x^\alpha \Delta_k^\alpha \sigma(x,k)}$$

• Classical Expansion: For $S^m_{1,0}$, classical symbols admit asymptotic expansions

$$\sigma(x, k) \sim \sum_{j=0}^\infty \sigma_{m-j}(x,k)$$

with each $\sigma_{m-j}(x,k)$ homogeneous of degree $m-j$ in $k$.

This calculus is globally defined, not depending on local coordinate patches or charts, which streamlines remainder calculations and facilitates quantization on arbitrary compact Lie groups (Cornean et al., 27 Dec 2025, Martínez et al., 2017).

3. Mapping and Continuity Properties: $L^p$, Sobolev, Besov, and Weighted Spaces

Toroidal pseudo-differential operators exhibit a range of continuity properties:

• $L^p$-boundedness: Fefferman-type and Hardy-space estimates extend to the torus. For $\sigma \in S^m_{\rho, \delta}$, boundedness on $L^p(\mathbb{T}^n)$ holds if

$$m \leq -n \left( (1-p) | 1/p -1/2 | + \max\{0, (\delta-p)/2 \} \right)$$

These results recover the classical Fefferman range when $\delta \leq \rho$ and cover borderline cases $\delta \geq \rho$ not attainable by local manifold theory (Cardona et al., 27 Feb 2025).

• Weighted $L^p(w)$: Sharp maximal function estimates imply that, under symbol smoothness and Muckenhoupt $A_p$ weights, $T_\sigma$ is bounded on $L^p(w)$ (Cardona et al., 18 Aug 2025).
• Sobolev and Besov regularity: If $\sigma \in S^m_{\rho,\delta}$ and $\rho > \delta$, then $\Op(\sigma): H^s \to H^{s-m}$ is bounded for all $s$. Besov space continuity,

$$T_\sigma: B^s_{p,q}(\mathbb{T}^n) \to B^{s-\mu}_{q,r}(\mathbb{T}^n)$$

holds under explicit index and symbol order constraints (Martínez et al., 2017, Cardona et al., 18 Aug 2025, Escobar-Velasquez, 12 Nov 2025).

4. Kernel Estimates, Spectral Theory, and Infinite Matrix Representation

The discrete Fourier structure leads to sharp kernel bounds. Cardona-Martínez kernel estimates characterize off-diagonal decay and Hölder continuity for Schwartz kernels $K(x, y)$: $$|x-y|^N |\partial_x^\alpha \partial_y^\beta K(x, y)| \lesssim \langle x-y\rangle^{-M}$$ guaranteeing Calderón–Zygmund–type behaviour (Escobar-Velasquez, 12 Nov 2025, Cornean et al., 27 Dec 2025).

Spectral analysis proceeds via infinite-matrix representations in Fourier space. The matrix $M_\sigma$ with entries $a_{jk} = \hat{\sigma}(j-k, k)$ yields spectral and boundedness characterizations; Gershgorin theory localizes the spectrum of $T_\sigma$ to unions of discs determined by diagonal and off-diagonal entries. Necessary and sufficient conditions for Riesz, strictly singular, or compactness properties reduce to decay of $\sup_x |\sigma(x,k)|$ as $|k| \to \infty$ (Velasquez-Rodriguez, 2019).

5. Weighted Estimates, Maximal Regularity, and Banach Space Extensions

Park–Tomita’s maximal-function technique, extended to the toroidal setting, delivers pointwise bounds in terms of the Fefferman–Stein sharp maximal operator and establishes continuity in weighted Lebesgue spaces $L^p(w)$ and vector-valued Besov spaces $B^s_{p,q}(\mathbb{T}^n; E)$ for arbitrary Banach spaces $E$: $\|T_\sigma f\|_{L^p(w)} \leq C \|f\|_{L^p(w)}$ provided $w \in A_{p/r}$, $1<r \leq 2 \leq p < \infty$ (Cardona et al., 18 Aug 2025, Martínez et al., 2017). Dyadic Littlewood–Paley decomposition adapts convolution block kernel techniques for operator-valued symbols of limited smoothness, with boundedness controlled by symbolic growth and x–Hölder continuity.

6. Applications: Periodic and Magnetic Operators, Manifold Random Fields

Toroidal pseudo-differential operators naturally arise in global analysis of periodic differential operators, Bloch–Floquet theory, and mathematical physics. Periodic magnetic Schrödinger operators, under Bloch–Floquet–Zak decomposition, yield fibre operators acting as toroidal ΨDOs with explicit symbols depending on quasi-momenta. The kernel and matrix representations facilitate spectral analysis and Fredholm characterizations (Cornean et al., 27 Dec 2025).

In probability and spatial statistics, toroidal ΨDO theory describes regularity thresholds for random fields (e.g., Matérn processes), with critical dimension-dependent regularity: smoothness parameter $\nu > 3d/2$ is required on $\mathbb{T}^d$, in contrast to the Euclidean case $\nu > 0$ (Escobar-Velasquez, 12 Nov 2025). Constructing canonical Matérn fields with symbols shifted by $|k|^{-2}$ achieves two orders higher regularity.

7. Examples, Counterexamples, and Further Developments

• Order-zero operators: All $$\sigma \in S^0_{1,0}(\mathbb{T}^n \times \mathbb{Z}^n)$$ yield bounded operators on $L^p(\mathbb{T}^n)$, $1 \leq p \leq \infty$ (Cardona et al., 27 Feb 2025).
• Non-compact Riesz operator: Explicit construction on the circle $T$ shows strictly singular, non-compact toroidal ΨDOs with symbols decaying to zero but not in compact operator class (Velasquez-Rodriguez, 2019).
• Operator-valued and limited smoothness symbols: The theory encompasses symbols $a(x, k)$ valued in $L(E)$, allowing continuity results on $B^s_{p,q}(T^n; E)$ even for limited $x$-regularity and finitely many frequency differences (Martínez et al., 2017).

The calculus is adaptable to other compact abelian groups, noncommutative tori, and quantum systems, with the discrete setting streamlining analysis and providing new tools for global regularity, maximal regularity, and spectral theory.

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References:

(Velasquez-Rodriguez, 2019): Velásquez-Rodríguez, "On some spectral properties of pseudo-differential operators on T" (Cardona et al., 27 Feb 2025): Cardona–Martínez, "Estimates for pseudo-differential operators on the torus revisited. I" (Cardona et al., 18 Aug 2025): Cardona–Martínez, "Estimates for pseudo-differential operators on the torus revisited. III" (Martínez et al., 2017): Denk et al., "Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces" (Escobar-Velasquez, 12 Nov 2025): Azencott et al., "Pseudo-Differential Operators and Generalized Random Fields over Tori" (Cornean et al., 27 Dec 2025): Cornean–Helffer–Purice, "The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators"