## 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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