A Spectral Multiplier Theorem Associated with a Schrödinger Operator

Abstract

We establish a Hörmander type spectral multiplier theorem for a Schrödinger operator $H=-\Delta +V(x)$ in $\mathbb {R}^3$, provided V is contained in a large class of short range potentials. This result does not require the Gaussian heat kernel estimate for the semigroup $e^{-tH}$, and indeed the operator H may have negative eigenvalues. As an application, we show local well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.

Appendix: Lorentz Spaces and Interpolation Theorem

Following [21], we summarize useful properties of the Lorentz spaces. Let $(X,\mu )$ be a measure space. The Lorentz (quasi) norm is defined by

(7.30)

Lemma 7.4

(Properties of the Lorentz spaces) Let $1\le p\le \infty$ and $1\le q, q_1, q_2\le \infty$.

• (i) $L^{p,p}=L^p$, and $L^{p,\infty }$ is the weak $L^p$-space.

• (ii) If $q_1\le q_2$, $L^{p,q_1}\subset L^{p,q_2}$.

Lemma 7.5

(Hölder inequality) If $1\le p, p_1, p_2, q,q_1,q_2\le \infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}$, then

$$\begin{aligned} \Vert fg\Vert _{L^{p,q}}\lesssim \Vert f\Vert _{L^{p_1,q_1}}\Vert fg\Vert _{L^{p_2,q_2}}. \end{aligned}$$

(7.31)

Lemma 7.6

(Dual characterization of $L^{p,q}$) If $1<p<\infty$ and $1\le q\le \infty$, then

$$\begin{aligned} \Vert f\Vert _{L^{p,q}}\sim \sup _{\Vert g\Vert _{L^{p',q'}}\le 1}\Big |\int _X f\bar{g} d\mu \Big |. \end{aligned}$$

(7.32)

A measurable function f is called a sub-step function of height H and width W if f is supported on a set E with measure $\mu (E)=W$ and $|f(x)|\le H$ almost everywhere. Let T be a linear operator that maps the functions on a measure space $(X,\mu _X)$ to functions on another measure space $(Y,\mu _Y)$. We say that T is restricted weak-type $(p,\tilde{p})$ if

$$\begin{aligned} \Vert Tf\Vert _{L^{\tilde{p},\infty }}\lesssim HW^{1/p} \end{aligned}$$

(7.33)

for all sub-step functions f of height H and width W.

Theorem 7.7

(Marcinkiewicz interpolation theorem) Let T be a linear operator such that

$$\begin{aligned} \langle Tf,g\rangle _{L^2}=\int _Y Tf \bar{g}d\mu _Y \end{aligned}$$

(7.34)

is well-defined for all simple functions f and g. Let $1\le p_0,p_1, \tilde{p}_0, \tilde{p}_1\le \infty$. Suppose that T is restricted weak-type $(p_i, \tilde{p}_i)$ with constant $A_i>0$ for $i=0,1$. Then,

$$\begin{aligned} \Vert Tf\Vert _{L^{\tilde{p}_\theta ,q}}\lesssim A_0^{1-\theta }A_1^\theta \Vert f\Vert _{L^{p_\theta ,q}}, \end{aligned}$$

(7.35)

where $0<\theta <1$, $\frac{1}{p_\theta }=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}$, $\frac{1}{\tilde{p}_\theta }=\frac{1-\theta }{\tilde{p}_0}+\frac{\theta }{\tilde{p}_1}$, $\tilde{p}_\theta >1$ and $1\le q\le \infty$.

In this paper, we use the interpolation theorem of the following form.

Corollary 7.8

(Marcinkiewicz interpolation theorem) Let T be a linear operator. Let $1\le p_1<p_2\le \infty$. Suppose that for $i=0,1$, T is bounded from $L^{p_i,1}$ to $L^{p_i,\infty }$. Then T is bounded on $L^p$ for $p_1<p<p_2$.

Proof

The corollary follows from Theorem 7.7, since T is restricted weak-type $(p_i, p_i)$:

$$\begin{aligned} \Vert f\Vert _{L^{p_i,1}}=p_i\int _0^\infty \mu (|f|\ge \lambda )^{1/p_i} d\lambda \le p_i\int _0^H W^{1/p_i}d\lambda =p_i HW^{1/p_i}, \end{aligned}$$

(7.36)

for a sub-step function f of height H and width W. $\square$

Corollary 7.9

(Fractional integration inequality in the Lorentz spaces)

$$\begin{aligned} \Big \Vert \int _{\mathbb {R}^d}\frac{f(y)}{|x-y|^{d-s}}dy\Big \Vert _{L^{q,r}(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^{p,r}}, \end{aligned}$$

(7.37)

where $1<p<q<\infty$, $1\le r\le \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{s}{d}$. At the endpoints, we have

$$\begin{aligned} \Big \Vert \int _{\mathbb {R}^d}\frac{f(y)}{|x-y|^{d-s}}dy\Big \Vert _{L^{\frac{d}{d-s},\infty }(\mathbb {R}^d)}&\lesssim \Vert f\Vert _{L^1}, \Big \Vert \int _{\mathbb {R}^d}\frac{f(y)}{|x-y|^{d-s}}dy\Big \Vert _{L^\infty (\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^{d/s,1}}. \end{aligned}$$

(7.38)

Proof

(7.38) follows from [18, Theorem 1, p. 119] and duality. Then, (7.37) follows from Corollary 7.8. $\square$