Uniform convergence with rates for smooth Poisson–Cauchy-type singular integral operators

In this article we continue the study of smooth Poisson–Cauchy-type singular integral operators on the line of very general kind. We establish their uniform convergence to the unit operator, with rates. The estimates are mostly sharp and they are pointwise or uniform. The inequalities established involve the higher order modulus of smoothness. To prove optimality we use mainly the geometric moment theory method.