Magnetic uniform resolvent estimates
Piero D'Ancona, Zhiqing Yin
TL;DR
This work establishes uniform $L^{p}-L^{q}$ resolvent bounds for magnetic Schrödinger operators $H=(i\partial+A)^2+V$ in $n\ge3$, with precise $z$-dependence and the same optimal index range as the free Laplacian, including weak endpoints. By decomposing the free resolvent into frequency-localized pieces and carefully handling the perturbed resolvent via a Lippmann-Schwinger-type analysis under decay assumptions on $A$ and $V$, the authors obtain uniform estimates for all frequencies and derive $L^{p}-L^{p'}$ bounds for the spectral measure. They also prove a variant with weaker decay for a restricted parameter range and extend the framework to more general perturbations, demonstrating robustness of the method. The results have broad implications for harmonic analysis, spectral theory, and dispersive PDEs, enabling spectral-measure bounds and refined resolvent control in magnetic settings.
Abstract
We establish uniform $L^{p}-L^{q}$ resolvent estimates for magnetic Schrödinger 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}φ\|_{L^{q}}\lesssim|z|^{θ(p,q)} (1+|z|^{\frac 12 \frac{n-1}{n+1}}) \|φ\|_{L^{p}} \end{equation*} where $θ(p,q)=\frac{n}{2}(\frac{1}{p}-\frac{1}{q})-1$. 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 Schrödinger operators.