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 (Au)(t) = \int^1_0 [\kappa_{\alpha}^+(t - s)a_+(t,s) + \kappa_{\alpha}^– (t-s)a\_(t,s) + a(t,s)]u(s)ds 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.