On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P (0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.