L^p harmonic analysis for differential-reflection operators

We introduce and study differential-reflection operators Λ_A,ε acting on smooth functions defined on R. Here A is a Sturm-Liouville function with additional hypotheses and ε∈R. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). As, by construction, the operators Λ_A,ε are mixture of d/dx and reflection operators, we prove the existence of an operator V_A,ε so that Λ_A,εV_A,ε=V_A,εd/dx. The positivity of the intertwining operator V_A,ε is also established. Via the eigenfunctions of Λ_A,ε, we introduce a generalized Fourier transform F_A,ε. For -1≤ε≤1 and 0<p≤2/1+√1-ε², we develop an L^p-Fourier analysis for F_A,ε, and then we prove an L^p-Schwartz space isomorphism theorem for F_A,ε. Details of this paper will be given in other articles [3] and [4].