Uniform two-weight norm inequalities for Hankel transform partial sum operators

Proved are two-weight, uniform with respect to the order of the involved Bessel function, norm inequalities for the Hankel transform partial sum operators. The proof heavily relies on uniform pointwise asymptotic estimates for the Bessel functions done by Barcelo and Cordoba. Also, a technique used earlier by Muckenhoupt in the (discrete) Laguerre case is applied. The conditions appearing in the main theorem are then proved to be necessary, except some singular cases. The result is applied to obtain uniform estimates for the partial sum operators of Fourier-Neumann expansions. This generalizes former results in this direction done by Barcelo and Cordoba, and Ciaurri, Guadalupe, Perez and Varona.