On the growth of Sobolev norms for Hartree equation

This paper is devoted to the growth of high-order Sobolev norms of solutions in time in the context of the Hartree equation in 2- or 3-dimensional spaces. By combining modified energies with appropriate Strichartz estimates, we derive polynomial-time bounds for the Hs(s >= 2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s(s\ge 2)$$\end{document} norms. Compared to the bound in 2 dimensions due to harmonic analysis techniques in the work (Sohinger in Discr Cont Dyn A 32(10):3733-3771, 2012), we obtain a more refined growth result of Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>s$$\end{document} norm for s <= 21/5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \le 21/5$$\end{document}.