Magnetic uniform resolvent estimates

Abstract: We establish uniform $L^{p}-L^{q}$ resolvent estimates for magnetic Schr\"odinger operators $H=(i\partial+A(x))^2+V(x)$ in dimension $n \geq 3$. Under suitable decay conditions on the electromagnetic potentials, we prove that for all $z \in \mathbb{C}\setminus[0,+\infty)$ with $|\Im z| \leq 1$, the resolvent satisfies \begin{equation*} |(H-z)^{-1}\phi|_{L^{q}}\lesssim|z|^{\theta(p,q)} (1+|z|^{\frac 12 \frac{n-1}{n+1}}) |\phi|_{L^{p}} \end{equation*} where $\theta(p,q)=\frac{n}{2}(\frac{1}{p}-\frac{1}{q})-1$. This extends previous results by providing estimates valid for all frequencies with explicit dependence on $z$, covering the same optimal range of indices as the free Laplacian case, and including weak endpoint estimates. We also derive a variant with less stringent decay assumptions when restricted to a smaller parameter range. As an application, we establish the first $L^p-L^{p'}$ bounds for the spectral measure of magnetic Schr\"odinger operators.