Strichartz estimates for the magnetic Schrödinger equation with potentials $V$ of critical decay

Abstract

We study the Strichartz estimates for the magnetic Schrödinger equation in dimension $n\ge 3$. More specifically, for all Schrödinger admissible pairs $(r,q)$, we establish the estimate $$\Vert e^{itH}f{\Vert }_{L_t^q(\mathbb{R};L_x^r({\mathbb{R}}^n))}\le C_{n,r,q,H}\Vert f{\Vert }_{L^2({\mathbb{R}}^n)}$$ when the operator $H=-{\Delta }_A+V$ satisfies suitable conditions. In the purely electric case $A\equiv 0$, we extend the class of potentials $V$ to the Fefferman-Phong class. In doing so, we apply a weighted estimate for the Schrödinger equation developed by Ruiz and Vega. Moreover, for the endpoint estimate of the magnetic case in ${\mathbb{R}}^3$, we investigate an equivalence $$\Vert H^{\frac{1}{4}}f{\Vert }_{L^r({\mathbb{R}}^3)}\approx C_{H,r}\Vert (-\Delta {)}^{\frac{1}{4}}f{\Vert }_{L^r({\mathbb{R}}^3)}$$ and find sufficient conditions on $H$ and $r$ for which the equivalence holds.

References in corpus (2)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Smoothing - Strichartz Estimates for the Schrodinger Equation with small Magnetic Potential