Strichartz estimates for the magnetic Schrödinger equation with potentials $V$ of critical decay (1602.00789v2)

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.