Kato type smoothing estimates for magnetic Schrödinger equations with rough potentials

Abstract

We consider the magnetic Schrödinger equation $iD_0 u = D_k D^k u$ with electro-magnetic potential $A = (A_\mu)$ in $\boldsymbol{R}^{1+n}$, where $n \ge 3$ and $D_\mu = \partial_\mu + iA_\mu$. Under the assumption $A \in \cap_{j=0}^1 C^j(0,T;H^{1-j}_n)$ with $\sum_\alpha \| F_{\mu\nu};L^\infty (0,T; L^n(Q_\alpha))\|^2 \lt \infty$, we prove the Kato type smoothing estimate $$\sup_\alpha \| u; L^2(0,T; H^{1/2} (Q_\alpha)\| \lesssim \| u(0,\cdot)\|_2,$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $\{Q_\alpha\}_{\alpha \in \boldsymbol{Z}^n}$ is the family of unit cubes.