ON THE NORMS OF GENERALIZED TRANSLATION OPERATORS GENERATED BY THE JACOBI-DUNKL OPERATORS

摘要

We establish an integral representation and improve norm estimation of generalized translation operators generated by the Jacobi-Dunkl operators A_(α,β)f(x) = f'(x) + A'_(α,β)(x)/A_(α,β)(x) f(x)-f(-x)/2. where A_(α,β)(x) = (1 — cosx)~α(l + cosx)~β|sinx|, in the spaces L_p[—π,π] with the weight A_(α,β). For α≥ β)≥ -1/2 we prove that these norms do not exceed 2.