Strichartz estimates for the magnetic Schr\"odinger equation with potentials $V$ of critical decay

Abstract

We study the Strichartz estimates for the magnetic Schr\"odinger equation in dimension $n\geq3$. More specifically, for all Schr\"odinger admissible pairs $(r,q)$, we establish the estimate $$ \|e^{itH}f\|_{L^{q}_{t}(\mathbb{R}; L^{r}_{x}(\mathbb{R}^n))} \leq C_{n,r,q,H} \|f\|_{L^2(\mathbb{R}^n)} $$ when the operator $H= -\Delta_A +V$ satisfies suitable conditions. In the purely electric case $A\equiv0$, we extend the class of potentials $V$ to the Fefferman-Phong class. In doing so, we apply a weighted estimate for the Schr\"odinger equation developed by Ruiz and Vega. Moreover, for the endpoint estimate of the magnetic case in $\mathbb{R}^3$, we investigate an equivalence $$ \| H^{\frac{1}{4}} f \|_{L^r(\mathbb{R}^3)} \approx C_{H,r} \big\| (-\Delta)^{\frac{1}{4}} f \big\|_{L^r(\mathbb{R}^3)} $$ and find sufficient conditions on $H$ and $r$ for which the equivalence holds.