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Spectral multipliers for Schrödinger operators. (English) Zbl 1235.42008

This paper is concerned with a sharp Hörmander multiplier theorem on \(L^p\) spaces for Schrödinger operators: \(H=-\Delta+V\) on \(\mathbb{R}^n\), where \(\Delta\) is the usual Laplacian operator and \(V\) is a general real-valued operator where the heat kernel estimates may not hold. For a Borel measurable function \(\phi:\mathbb{R}\to \mathbb{C}\), define \(\phi(H)=\int \phi(\lambda)dE_\lambda\) by functional calculus, where \(H=\int\lambda dE_\lambda\) is the spectral resolution of the self-adjoint operator \(H\) acting in \(L^2(\mathbb{R}^n)\). The spectral multiplier problem is to find sufficient conditions on a bounded function \(\mu\) on \(\mathbb{R}\) so that \(\mu(H)\) is bounded on \(L^p(\mathbb{R}^n),\;1<p<\infty\).
Throughout the paper, the potential \(V\) is assumed to satisfy the following estimates:
(1) (Weighted \(L^2\) estimate) There exists some \(s>n/2\) so that for all \(j\) and \(\phi\in W_2^s\left([\frac14,1]\cup[-1,-\frac14]\right)\), \[ \sup_y\left\||x-y|^s\phi(\lambda_j^2H)(x,y)\right\|_{L^2_x}\leq c\lambda_j^{s-n/2}\|\phi\|_{W^s_2}. \]
(2) (Weighted \(L^\infty\) estimate) There exist a finite measure \(d\xi\) and \(0<\varepsilon\leq 1\) so that for all \(x,y,j\) and \(\phi\in W_2^{n+\varepsilon}\left([-1,1]\right)\), \[ \left|\phi(\lambda_j^2H)(x,y)\right|\leq c\lambda_j^{-n}\int_{\mathbb{R}^n}\left(1+\lambda_j^{-1}|x-y-u|\right)^{-n-\varepsilon}d\xi(u), \] where \(c=c(\|\phi\|_{W_2^{n+\varepsilon}})\).
The following theorems are the main results of the paper
Theorem 1. Suppose \(H\) satisfies the above assumptions for some \(s>n/2\). If \(\|\mu\|_{W^s_{2,sloc}}<\infty\), then \(\mu(H)\) is bounded on \(L^p(\mathbb{R}^n),\;1<p<\infty\), and has weak type \((1,1)\). Moreover, \[ \|\mu(H)\|_{L^1\to \mathrm{weak}-L^1}\leq c\|\mu\|_{W^s_{2,\mathrm{sloc}}}. \]
Theorem 2. In one dimension, when the potential \(V\in L^1_1\) (namely, \(\int(1+|x|)|V(x)|dx\) is finite) and \(H\) has no resonance at zero, or \(V\in L_2^1(\mathbb{R})\), then the above result holds as well.

MSC:

42B15 Multipliers for harmonic analysis in several variables
35J10 Schrödinger operator, Schrödinger equation
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References:

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