Lp-Fourier analysis for certain differential-reflection operators.

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