An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

Abstract

In this note we prove that every classical 1-periodic pseudodifferential operator A of order $\alpha \in \mathbb R \\mathbb N_0$ can be represented in the form

where $\alpha_±$ and $a$ are $C^{\infty}$-smooth 1-periodic functions and $\kappa_{\alpha}^±$ are 1-periodic functions or distributions with Fourier coefficients $\kappa_{\alpha}^+(n) = |n|^{\alpha}$ and $\kappa_{\alpha}^–(n) = |n|^{\alpha}$ sign$(n)$ $(0 \neq n \in \mathbb Z)$ with respect to the trigonometric orthonormal basis $\{e^{in2xt}\}_{n \in \mathbb Z}$ of $L^2 (0,1)$. Some explicit formulae for $\kappa_{\alpha}^±$ are given. The case of operators of order $\alpha \in \mathbb N_0$ is discussed, too.