Uniform resolvent estimates for magnetic operators

Abstract

We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schr\"{o}dinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic potentials, and excluding a threshold resonance at zero, we show that for all $z \in \mathbb{C}\setminus[0,+\infty)$, \begin{equation*} \|(H-z)^{-1}\phi\|_{L^{q}}\lesssim|z|^{\theta(p,q)} (1+|z|^{\gamma}) \|\phi\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac1p,\frac1q)\in\Delta(n)$, where $\theta(p,q)=\frac n2(\frac1p-\frac1q)-1$. Here $\gamma=\frac 12\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or $\gamma=\frac{n-1}{4n}$ under a different decay assumption on $A(x)$; for the second case we use a weak endpoint estimate of Frank--Simon type \begin{equation*} \|R_{0}(z)\phi\| _{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_{\omega}} \lesssim |z|^{-\frac12} \|\phi\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_{\omega}}. \end{equation*} The result extends the known electromagnetic estimates from fixed frequency and a smaller exponent region to all frequencies and the full Kenig--Ruiz--Sogge range. We also prove a variant with weaker local assumptions in a smaller range $\Delta_1(n)$. As applications, we obtain $L^p-L^{p'}$ restriction type estimates for the density of the spectral measure of magnetic Schr\"{o}dinger operators, and an eigenvalue enclosure result for complex scalar perturbations.