Lp harmonic analysis for differential-reflection operators

Abstract

We introduce and study differential-reflection operators ΛA,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 ΛA,ε are mixture of d/dx and reflection operators, we prove the existence of an operator VA,ε so that ΛA,ε∘VA,ε=VA,ε∘d/dx. The positivity of the intertwining operator VA,ε is also established.Via the eigenfunctions of ΛA,ε, we introduce a generalized Fourier transform FA,ε. For −1≤ε≤1 and 0<p≤21+1−ε2, we develop an Lp-Fourier analysis for FA,ε, and then we prove an Lp-Schwartz space isomorphism theorem for FA,ε.Details of this paper will be given in other articles [3] and [4].