Maximal estimates of the Schrödinger operators with quadratic potential for orthonormal systems

Abstract

We establish an equivalence principle for local mixed-norm estimates for generalized Schrödinger propagators acting on orthonormal systems. Under suitable mixed-norm conditions, estimates for generalized propagators follow from the corresponding free Schrödinger estimates for every summability exponent $1\leβ\le2$. Applications include maximal estimates for Schrödinger equations with time-dependent quadratic potentials and for the special Hermite operator, as well as almost everywhere convergence of the associated densities. For the special Hermite operator, we also obtain a single-function maximal estimate that is sharp up to the endpoint. We construct counterexamples for the Schrödinger operator, by combining single particle and many particle constructions in complementary dimensions, yielding new necessary conditions in an intermediate range of summability exponents.