  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.
