Estimates for multiparameter maximal operators of Schr\"odinger type

Multiparameter maximal estimates are considered for operators of Schr\"odinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT* argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schr\"odinger equation and make some comments on an open problem.