CONJUGATE EXPANSIONS FOR ULTRASPHERICAL FUNCTIONS

Abstract

We define and investigate the Hilbert transform for expansions with respect to the system of ultraspherical functions. Using deep estimates done by Muckenhoupt and Stein in the polynomial expansion case we prove the existence of boundary values of the conjugate Poisson integrals of integrable functions. The limit function then satisfies usual L^pand weak type ( 1, 1 ) estimates.