L^p-Fourier analysis for certain differential-reflection operators

In a previous paper we introduced a family of differential-reflection operators Λ_{A , ϵ} acting on smooth functions defined on R, where the spectral problem for the operators Λ_A,ϵ has been studied. Here A is a Sturm-Liouville function with additional hypotheses and-1 ≤ ϵ ≤ 1. Via the eigenfunctions of Λ_A,ϵ, we introduce in this paper a generalized Fourier transform f_A,ϵ. An L^p-harmonic analysis for f_A,ϵ is developed when 0 < p ≤ 2/1 + 1-ϵ². In particular, an L^p-Schwartz space isomorphism theorem for f_A,ϵ is proved.