Probabilistic Strichartz estimates in Schatten classes and their applications to the Hartree equation

Abstract: In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this type of inequality. In this paper, we introduce two randomizations of operators and show that they allow us to treat strictly bigger Schatten exponents than the sharp exponents of the deterministic orthonormal Strichartz estimates. In the proofs, the orthogonality does not have any essential role, and randomness works instead of it. We also prove that our randomizations of operators never change the Schatten classes to which they originally belong. Moreover, we give some applications of our results to the Hartree and linearized Hartree equation for infinitely many particles. First, we construct local solutions to the Hartree equation with initial data in wide Schatten classes. By only using deterministic orthonormal Strichartz estimates, it is impossible to give any solution in our settings. Next, we consider the scattering problem of the linearized Hartree equation. One of our randomizations allows us to treat wide Schatten classes with some Sobolev regularities, and by the other randomization, we can remove all Sobolev regularities.