L^p estimates for angular maximal functions associated with Stieltjes and Laplace transforms

Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing over argument. Compositions of the so defined maximal angular operator (in suitable sectors) with the Poisson, Stieltjes and Laplace transforms are shown to be bounded (nonlinear) operators from $L^p$ to $L^q$ for the same values of $p$ and $q$ as their standard counterparts.