Uniform resolvent estimates for magnetic operators

We prove Kenig–Ruiz–Sogge type uniform resolvent estimates for selfadjoint magnetic Schrödinger operators $H=(i\mathrm{\partial }+A(x){)}^2+V(x)$ 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 $z\in \mathbb{C}\setminus [0,+\mathrm{\infty })$, \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 $(\frac{1}{p},\frac{1}{q})\in \mathrm{\Delta }(n)$, where $\theta (p,q)=\frac{n}{2}(\frac{1}{p}-\frac{1}{q})-1$. Here $\gamma =\frac{1}{2}\frac{n-1}{n+1}$ 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)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 ${\mathrm{\Delta }}_1(n)$. As applications, we obtain $L^p-L^{p^{\mathrm{\prime }}}$ restriction type estimates for the density of the spectral measure of magnetic Schrödinger operators, and an eigenvalue enclosure result for complex scalar perturbations.