Uniform resolvent estimates for magnetic operators

Abstract

We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\partial +A(x){)}^2+V(x)$ on ${\mathbb{R}}^n$, $n\ge 3$. 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}ϕ\|_{L^{q}}\lesssim|z|^{θ(p,q)} (1+|z|^γ) \|ϕ\|_{L^{p}} \end{equation*} throughout the full free resolvent range $(\frac{1}{p},\frac{1}{q})\in \hat{I}”(n)$, where $\hat{I}¸(p,q)=\frac{n}{2}(\frac{1}{p}-\frac{1}{q})-1$. Here ${\hat{I}}^3=\frac{1}{2}\frac{n-1}{n+1}$ under the basic magnetic decay hypothesis, or ${\hat{I}}^3=\frac{n-1}{4n}$ 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)ϕ\| _{L^{\frac{2n}{n-1},\infty}_{r}L^{2}_ω} \lesssim |z|^{-\frac12} \|ϕ\|_{L^{\frac{2n}{n+1},1}_{r}L^{2}_ω}. \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 $\hat{I}{”}_1(n)$. As applications, we obtain $L^p-L^{p^{\prime }}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations.