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.
Categories: math.AP math-ph math.MP