Weak type $(1,1)$ bounds for Schr\"odinger groups

TL;DR

This paper proves weak type (1,1) bounds for Schr"odinger groups associated with certain operators on spaces of homogeneous type, extending known $L^p$ bounds to the endpoint case $p=1$ with sharpness results.

Contribution

It establishes the weak type (1,1) boundedness of the operator $(1+L)^{-n/2} e^{itL}$ for a broad class of operators, including Schr"odinger groups, on spaces of homogeneous type.

Findings

Proves weak type (1,1) bounds for Schr"odinger groups.

Shows the index $n/2$ is sharp in Euclidean space.

Applicable to elliptic operators, Schr"odinger operators, and Laplacians on various structures.

Abstract

Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat kernel of $L$ satisfies a Gaussian upper bound. It is known that the operator $(I+L)^{-s } e^{itL}$ is bounded on $L^p(X)$ for $s> n|{1/ 2}-{1/p}| $ and $ p\in (1, \infty)$ (see for example, \cite{CCO, H, Sj}). The index $s= n|{1/ 2}-{1/p}|$ was only obtained recently in \cite{CDLY, CDLY2}, and this range of $s$ is sharp since it is precisely the range known in the case when $L$ is the Laplace operator $\Delta$ on $X=\mathbb R^n$ (\cite{Mi1}). In this paper we establish that for $p=1,$ the operator $(1+L)^{-n/2}e^{itL}$ is of weak type $(1, 1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that \begin{eqnarray*} \mu\Big(\Big\{x: \big|(I+L)^{-n/2 }e^{itL} f(x)\big|>\lambda \Big\}\Big) \leq C\lambda^{-1}(1+|t|)^{n/2} {\|f\|_{L^1(X)} }, \ \ \ t\in{\mathbb R} \end{eqnarray*} (for $\lambda > 0$ when $\mu (X) = \infty$ and $\lambda>\mu(X)^{-1}\|f\|_{L^1(X)}$ when $\mu (X) < \infty$). Moreover, we also show the index $n/2$ is sharp when $L$ is the Laplacian on ${\mathbb R^n}$ by providing an example. Our results are applicable to Schr\"odinger group for large classes of operators including elliptic operators on compact manifolds, Schr\"odinger operators with non-negative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces.

Equations

$$\displaystyle\mu\Big{(}\Big{\{}x:\big{|}(I+L)^{-n/2}e^{itL}f(x)\big{|}>\lambda\Big{\}}\Big{)}\leq C\lambda^{-1}(1+|t|)^{n/2}{\|f\|_{L^{1}(X)}},\ \ \ t\in{\mathbb{R}}$$

$$\displaystyle\left\{\begin{array}[]{ll}i{\partial_{t}u}+Lu=0,\\[4.0pt] u|_{t=0}=f\end{array}\right.$$

$$u(x,t)=e^{itL}f(x)={\int }_0^{\infty }{e}^{it\lambda }d{E}_L(\lambda )f(x),t\in \mathbb{R}$$

$$\displaystyle\left\|(1+\Delta)^{-s}e^{it\Delta}f\right\|_{L^{p}(\mathbb{R}^{n})}\leq C(1+|t|)^{s}\|f\|_{L^{p}(\mathbb{R}^{n})},\ \ \ t\in{\mathbb{R}},\ \ \ s=n\big{|}{1\over 2}-{1\over p}\big{|}$$

$$\displaystyle\left\|(I+L)^{-s}e^{itL}f\right\|_{L^{p}(X)}\leq C(1+|t|)^{s}\|f\|_{L^{p}(X)},\ \ \ t\in{\mathbb{R}},\ \ \ s>n\big{|}{1\over 2}-{1\over p}\big{|},$$

$$e^{-tL}f(x)={\int }_X{H}_t(x,y)f(y)d\mu (y),\mathrm{for}\mathrm{every}f\in L^2(X),\mathrm{and}a.e.x\in X,$$

$${\mathrm{GE}}_m$$

$$\displaystyle\left\|(I+L)^{-s}e^{itL}f\right\|_{L^{p}(X)}\leq C(1+|t|)^{s}\|f\|_{L^{p}(X)},\ \ \ t\in{\mathbb{R}},\ \ \ s=n\big{|}{1\over 2}-{1\over p}\big{|}.$$

$$\displaystyle\mu\Big{(}\Big{\{}x:|(I+L)^{-n/2}e^{itL}f(x)|>\lambda\Big{\}}\Big{)}\leq C\lambda^{-1}(1+|t|)^{n/2}{\|f\|_{L^{1}(X)}},\ \ \ t\in{\mathbb{R}}.$$

$$m(\xi )=\theta (\xi )|\xi {|}^{-1}{e}^{i|\xi {|}^2}\hat{f}(\xi )$$

$$\displaystyle\Big{|}\Big{\{}x:\big{|}(I-{d^{2}/dx^{2}})^{-1/2}e^{-i{d^{2}/dx^{2}}}f(x)\big{|}>\lambda\Big{\}}\Big{|}\leq{C\over\lambda}\|f\|_{L^{1}({\mathbb{R}})}.$$

$${\int }_{|x|\ge 2|y{|}^{1/(1-\theta )}}|K(x-y)-K(x)|dx\le C<\infty$$

$$\displaystyle\int_{d(x,y)>c_{1}\sqrt{1+|t|}2^{k}}\big{|}K_{e^{itL}F_{k}(L)}(x,y)\big{|}d\mu(y)\leq C(1+|t|)^{n/2}$$

$$V(x,2r)\le CV(x,r)\forall r>0,x\in X.$$

$$V(x,\lambda r)\le C{\lambda }^{n}V(x,r).$$

$$V(y,r)\leq C\Big{(}1+\frac{d(x,y)}{r}\Big{)}^{D}V(x,r)$$

$$\hat{f}(\xi )=\frac{1}{(2\pi {)}^{n/2}}{\int }_{{\mathbb{R}}^n}f(x)e^{-ix\xi }dx,\xi \in {\mathbb{R}}^n.$$

$$\left|K_{(I+tL{)}^{-s}}(x,y)\right|\le a_t(x,y)=\frac{C}{V(x,t^{1/m})}\exp \left(-c{\left(\frac{d(x,y{)}^m}{t}\right)}^{\frac{1}{2(m-1)}}\right)$$

$$(I+tL{)}^{-s}=\frac{1}{\Gamma (s)}\underset{0}{\overset{\infty }{\int }}{e}^{-\lambda tL}{e}^{-\lambda }{\lambda }^{s-1}d\lambda .$$

$$\left|K_{(I+tL{)}^{-s}}(x,y)\right|$$

$${\lambda }^{1/(m-1)}+{\left(\frac{d(x,y{)}^m}{\lambda t}\right)}^{\frac{1}{m-1}}\ge 2{\left(\frac{d(x,y{)}^m}{t}\right)}^{\frac{1}{2(m-1)}},$$

$$\left|K_{(I+tL{)}^{-s}}(x,y)\right|$$

$$\displaystyle\int\limits_{0}^{\infty}e^{-\lambda+2^{-1}\lambda^{1/(m-1)}}\lambda^{s-1}\big{(}1+\lambda^{-1/m}\big{)}^{n}d\lambda$$

$$\underset{z\in B(y,r)}{\sup }{G}_t(x,z)\le C\underset{z\in B(y,r)}{\inf }{G}_{\theta t}(x,z)$$

$$\left|K_{(I+tL{)}^{-n/2}}(x,y)\right|\le P_t(x,y)$$

$$\underset{y}{\sup }{\int }_X|P_t(x,y)|d\mu (x)\le C,\mathrm{and}{\int }_X|P_t(x,y){|}^{2}d\mu (x)\le \frac{C}{V(y,t^{1/m})};$$

$$\underset{y:R\le d(x,y)\le 2R}{\sup }{P}_t(x,y)\le C\underset{y:R\le d(x,y)\le 2R}{\inf }{P}_t(x,y)$$

$$(I+tL{)}^{-n/2}=\frac{1}{\Gamma (n/2)}\underset{0}{\overset{\infty }{\int }}{e}^{-\lambda tL}{e}^{-\lambda }{\lambda }^{n/2-1}d\lambda .$$

$$|H_t(x,y)|$$

$$\left|K_{(I+tL{)}^{-n/2}}(x,y)\right|$$

Full text

[TABLE]

(for $\lambda>0$ when $\mu(X)=\infty$ and $\lambda>\mu(X)^{-1}\|f\|_{L^{1}(X)}$ when $\mu(X)<\infty$). Moreover, we also show the index $n/2$ is sharp when $L$ is the Laplacian on ${\mathbb{R}^{n}}$ by providing an example.

Our results are applicable to Schrödinger group for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with non-negative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces.

1. Introduction

Let $L$ be a non-negative self-adjoint operator on the Hilbert space $L^{2}(X),$ where $X$ is a metric measure space with a distance $d$ and a measure $\mu$. Consider the Schrödinger equation in $X\times{\mathbb{R}},$

[TABLE]

with initial data $f$. Then the solution can be formally written as

[TABLE]

for $f\in L^{2}(X)$, where $E_{L}$ denotes the resolution of the identity associated with $L.$ By the spectral theorem ([29]), the operator $e^{itL}$ is continuous on $L^{2}(X)$, and forms the Schrödinger group. A natural problem is to study the mapping properties of families of operators derived from the Schrödinger group on various functional spaces defined on $X$. This has attracted a lot of attention in the last decades, and has been a very active research topic in harmonic analysis and partial differential equations– see for example, [1, 3, 4, 5, 7, 12, 21, 22, 23, 24, 25, 27, 28, 30, 31, 33].

In 1960 Hörmander ([23]) addressed this problem for the Laplace operator $\Delta=-\sum_{i=1}^{n}\partial_{x_{i}}^{2}$ on the Euclidean space $\mathbb{R}^{n}$, and proved that the Schrödinger group $e^{it\Delta}$ is bounded on $L^{p}({\mathbb{R}}^{n})$ only for $p=2$. For $p\not=2,$ it is well-known (see for example, [4, 27, 33]) that for $s>n|{1/2}-{1/p}|$, the operator $e^{it\Delta}$ maps the Sobolev space $L^{p}_{2s}(\mathbb{R}^{n})$ into $L^{p}(\mathbb{R}^{n})$. Equivalently, this means that $(I+\Delta)^{-s}e^{it\Delta}$ is bounded on $L^{p}(\mathbb{R}^{n})$, and this is not the case if $0<s<n|{1/2}-{1/p}|$. The sharp endpoint $L^{p}$-Sobolev estimate is due to Miyachi ([30]), which states that for every $p\in(1,\infty)$,

[TABLE]

for some positive constant $C=C(n,p)$ independent of $t$. The estimate (1.3) is sharp both for the growth in $t$ and for the derivatives (see [30, p. 169-170]).

Parallel to the Laplacian on $\mathbb{R}^{n}$, the boundedness of the Schrödinger group was investigated for different operators in more general settings. Depending on the nature of the assumptions regarding the assumption of $e^{-tL}$ and the underlying space $X$, there are various nuances of the mapping properties of the Schrödinger group $e^{-tL}$ on $L^{p}$ spaces presently available in the literature. For example, when $L$ is an elliptic operator of order $m$ with constant coefficients on ${\mathbb{R}^{n}}$, the $L^{p}$ boundedness of $(1+L)^{-s}e^{itL}$ was studied in [2, 4, 22, 30, 33]) and the references therein; when $L$ are the sub-Laplacian on Lie groups with polynomial growth and the Laplace-Beltrami operator on manifolds with non-negative Ricci curvature, similar results as in (1.3) for $s>n\left|{1/2}-{1/p}\right|$ and $1<p<\infty$ were first announced by Lohoué in [28], then proved by Alexopoulos in [1]. In the abstract setting of operators on metric measure spaces, Carron, Coulhon and Ouhabaz [7] showed that for every $1<p<\infty,$

[TABLE]

provided the semigroup $e^{-tL}$, generated by $-L$ on $L^{2}(X)$ has the kernel $p_{t}(x,y)$ which satisfies the Gaussian upper bound (${\rm GE}_{m}$), i.e.

[TABLE]

where $H_{t}(x,y)$ satisfies a Gaussian upper bound

[TABLE]

for every $t>0,x,y\in X$, where $c,C$ are positive constants and $m\geq 2.$ Such estimate (${\rm GE}_{m}$) is typical for elliptic or sub-elliptic differential operators of order $m$ (see for example, [1, 5, 7, 12, 13, 15, 17, 20, 24, 25, 31, 32, 33, 35] and the references therein).

Recently, the problem whether estimate (1.4) holds with $s=n|{1/2}-{1/p}|$ was solved in [9, Theorem 1.1]. More specifically, if $L$ satisfies the Gaussian estimate (${\rm GE}_{m}$), then for every $p\in(1,\infty)$ there exists a constant $C=C(n,m,p)>0$ independent of $t$ such that

[TABLE]

For $p=1$, it was shown in [10] that the operator $(I+L)^{-n/2}e^{itL}$ is bounded from $H^{1}_{L}(X)$ into $L^{1}(X)$ where $H^{1}_{L}(X)$ is a class of Hardy spaces associated to the operator $L$. Based on this result, together with interpolation, a different proof of the estimate (1.5) was obtained.

The aim of this article is to investigate the Schrödinger groups $e^{itL}$ on $L^{1}$ spaces to show that under the assumption of a Gaussian upper bound (${\rm GE}_{m}$) of $L$, the operator $(I+L)^{-n/2}e^{itL}$ is of weak-type $(1,1)$. Our result can be stated as follows.

Theorem 1.1**.**

Suppose that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the property (${\rm GE}_{m}$). Then the operator $(I+L)^{-n/2}e^{itL}$ is of weak type $(1,1)$, that is, there is a constant $C$, independent of $t$ and $f$, so that

[TABLE]

for $\lambda>0$ when $\mu(X)=\infty$ and $\lambda>\mu(X)^{-1}\|f\|_{L^{1}(X)}$ when $\mu(X)<\infty$.

We would like to mention that in [8, Theorem 1.1], Chanillo, Kurtz and Sampason proved a weak type $(1,1)$ result for the multiplier operator ${\widehat{Tf}}(\xi)=m(\xi){\widehat{f}}(\xi)$ where $m$ is given by

[TABLE]

on ${\mathbb{R}}$, and $\theta$ is a $C^{\infty}$ function satisfying $\theta(\xi)=1$ for $|\xi|\geq 1$ and $\theta(\xi)=0$ for $|\xi|\leq 1/2$. This implies that the operator $(I-{d^{2}/dx^{2}})^{-1/2}e^{-i{d^{2}/dx^{2}}}$ is of weak type $(1,1)$, that is, for every $\lambda>0,$

[TABLE]

An examination of their proofs shows the dimension may be greater than $1$. As pointed out in [8, p. 129], standard arguments which work for the classical Calderón-Zygmund kernels can not be used to show the estimate (1.7) since they will fail for large intervals. Indeed, the kernels $K_{(I-{d^{2}/dx^{2}})^{-1/2}e^{-i{d^{2}/dx^{2}}}}(x)$ are not integrable away from the original, and they do not satisfy a regularity condition

[TABLE]

for any $0<\theta<1$ ([8, 26]). To overcome it, they based their proof on the argument used to prove the Bochner-Riesz multipliers in the work of Fefferman ([19, Theorem 3]).

Our proof of Theorem 1.1 is different from that of Chanillo-Kurtz-Sampson [8] where their result relies heavily on Fourier analysis. In our setting, we do not have Fourier transform at our disposal. We also do not assume that the heat kernel $p_{t}(x,y)$ satisfies the standard regularity condition, thus standard techniques of Calderón–Zygmund theory ([34]) are not applicable. The lack of smoothness of the kernel will be overcome in the proof by using the theory of singular integrals with rough kernels, which lies beyond the scope of the standard Calderón-Zygmund theory (see for example, [15, 17, 18] and the references therein). In the proof of Theorem 1.1, one of the main ingredients is to show that for $k\geq 0$ and $c_{1}>1,$

[TABLE]

with a constant $C>0$ independent of $k$ and $t,$ where $F_{k}(L)=(1+L)^{-n/2}(1-e^{-2^{mk}L})\varphi_{0}(2^{-m(k-k_{0})/(m-1)}L)$, and $\varphi_{0}$ is a smooth function with supp $\varphi_{0}\subset[0,1]$, $\varphi_{0}(\lambda)=1$ on $[0,1/2]$, and $2^{k_{0}}\sim\sqrt{1+|t|}$ (see Lemma 3.2 below), and this is a crucial estimate in the proof of Theorem 1.1.

The paper is organized as follows. In Section 2 we provide some preliminary results on the kernel estimates for the operators related to $(I+tL)^{-n/2}$, which play an important role in the proof of Theorem 1.1. The proof of Theorem 1.1 will be given in Section 3. In Section 4 we discuss some extensions of Theorem 1.1 to measurable subsets of a space of homogeneous type and obtain similar results for operators on irregular domains with Dirichlet boundary conditions.

2. Preliminary results

We start by introducing some notation and assumptions. Throughout this paper, unless we mention the contrary, $(X,d,\mu)$ is a metric measure space where $\mu$ is a Borel measure with respect to the topology defined by the metric $d$. Next, let $B(x,r)=\{y\in X,\,{d}(x,y)<r\}$ be the open ball with centre $x\in X$ and radius $r>0$. To simplify notation we often just use $B$ instead of $B(x,r)$ and given $\lambda>0$, we write $\lambda B$ for the $\lambda$-dilated ball which is the ball with the same centre as $B$ and radius $\lambda r$. Let $B^{c}$ be the set $X\backslash B$. We set $V(x,r)=\mu(B(x,r))$ the volume of $B(x,r)$ and we say that $(X,d,\mu)$ satisfies the doubling property (see Chapter 3, [11]) if there exists a constant $C>0$ such that

[TABLE]

If this is the case, there exist $C,n$ such that for all $\lambda\geq 1$ and $x\in X$

[TABLE]

In Euclidean space $\mathbb{R}^{n}$ with Lebesgue measure, the parameter $n$ corresponds to the dimension of the space, but in our more abstract setting, the optimal $n$ need not even be an integer. There also exist $C>0$ and $0\leq D\leq n$ so that

[TABLE]

uniformly for all $x,y\in X$ and $r>0$. Indeed, the property (2.2) with $D=n$ is a direct consequence of the triangle inequality for the metric $d$ and the strong homogeneity property (2.2).

For $1\leq p\leq+\infty$, we denote the norm of a function $f\in L^{p}(X,{\rm d}\mu)$ by $\|f\|_{p}$, by $\langle\cdot,\cdot\rangle$ the scalar product of $L^{2}(X,{\rm d}\mu)$, and if $T$ is a bounded linear operator from $L^{p}(X,{\rm d}\mu)$ to $L^{q}(X,{\rm d}\mu)$, $1\leq p,\,q\leq+\infty$, we write $\|T\|_{p\to q}$ for the operator norm of $T$. Given a subset $E\subseteq X$, we denote by $\chi_{E}$ the characteristic function of $E$. We denote the dilation of a function $F$ by $\delta_{r}F(\cdot):=F(r\cdot)$ and $\widehat{f}\,$ denotes the Fourier transform, i.e. of $f$,

[TABLE]

Sometimes we also use $\widehat{f}$ for ${\mathcal{F}}f$.

We now state the following auxiliary result (see also [31, Theorem 1]).

Lemma 2.1**.**

Assume that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the Gaussian upper bound (${\rm GE}_{m}$). Then for $s>n/m$, the kernels $K_{(I+tL)^{-s}}(x,y)$ of the operators $(I+tL)^{-s}$ satisfy

[TABLE]

for all $x,y\in{X}$ and $t>0$.

Proof.

Note that

[TABLE]

Hence the kernel $K_{(I+tL)^{-s}}(x,y)$ of $(I+tL)^{-s}$ is given by

[TABLE]

We use the doubling condition (2.2) to obtain that V(x,t^{1/m})\leq CV(x,(\lambda t)^{1/m})\big{(}1+\lambda^{-1/m}\big{)}^{n} for every $x\in X,\lambda,t>0.$ This, in combination with the fact that

[TABLE]

gives

[TABLE]

A simple calculation shows

[TABLE]

and the above integral (2.5) is finite for $m\geq 2$ and $s>n/m$. This shows the desired result (2.4). ∎

Following [18, Proposition 2.5], we say that a non-negative function $G_{t}(x,y)$ satisfies a Harnack-type inequality if for each $\mu>0$ there are $C,\theta\geq 1$ such that

[TABLE]

uniformly for $x,y\in X$ and $r,t>0$ with $r^{m}\leq\mu t.$ Examples of functions satisfying (2.6) include functions $h_{t}(x,y)$ in (${\rm GE}_{m}$) and $a_{t}(x,y)$ defined by (2.4) (see for example, [15, Lemma 1]).

In our Theorem 1.1, we will need some kernel estimates of the operator $(I+tL)^{-s}$ for $s=n/2$. Since this integral (2.5) is infinite for $m=2$ and $s=n/2$, estimate (2.6) may or may not hold for kernel of the operator $(I+tL)^{-n/2}$. Instead, we have the following observation, which will play a key role in the proof of Theorem 1.1 in Section 3.

Lemma 2.2**.**

Assume that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the Gaussian upper bound (${\rm GE}_{m}$). Then the kernels $K_{(I+tL)^{-n/2}}(x,y)$ of the operators $(I+tL)^{-n/2}$ satisfy

[TABLE]

for all $x,y\in{X}$ and $t>0$ where $P_{t}(x,y)$ is a function satisfying

• (i)

there exists a constant $C>0$ such that for $y\in X$ and $t>0$,

[TABLE]

• (ii)

There is a constant $C>0$ such that

[TABLE]

uniformly for $x\in X$ and $R,t>0$.

Proof.

Note that for every $t>0,$

[TABLE]

This, together with the fact (2.3) that

[TABLE]

yields that the kernel $K_{(I+tL)^{-n/2}}(x,y)$ of $(I+tL)^{-n/2}$ is given by

[TABLE]

for all $x,y\in{X}$ and $t>0$. A simple calculation shows that for all $y\in X,$

[TABLE]

Now we use the doubling condition (2.2) to obtain that V(y,t^{1/m})\leq CV(y,(\lambda t)^{1/m})\big{(}1+\lambda^{-1/m}\big{)}^{n} for every $y\in X,\lambda,t>0$. By Minkowski’s inequality,

[TABLE]

To show (ii), we observe that for all $x\in X,$ and $y_{1},y_{2}\in B(x,2R)\backslash B(x,R),R>0$, one has $d(x,y_{1})\leq d(x,y_{2})+d(y_{1},y_{2})\leq d(x,y_{2})+4R\leq 5d(x,y_{2}).$ This implies

[TABLE]

Hence, the desired estimate is valid with $C=5^{n-1}$ uniformly for $x\in X$ and $R,t>0$. ∎

Lemma 2.3**.**

Assume that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the property (${\rm GE}_{m}$). Let $\varphi_{1}$ be a smooth function with supp $\varphi_{1}\subset[1/2,\infty]$ and $\varphi_{1}(\lambda)=1$ on $[1,\infty]$. For every $c_{0}\leq 1$ and $k\in{\mathbb{N}}$, the kernels $K_{(I+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{1}(c_{0}L)}(x,y)$ of the operators $(I+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{1}(c_{0}L)$ satisfy

[TABLE]

for all $x,y\in{X}$ and $t>0$ where $Q(x,y)$ is a function satisfying

• (i)

there exists a constant $C>0$ such that

[TABLE]

uniformly for $c_{0}\leq 1$;

• (ii)

There is a constant $C>0$ such that

[TABLE]

uniformly for $x\in X$, $R>0$ and $c_{0}\leq 1$.

To prove Lemma 2.3, we recall that in [7, Proposition 4.1], Carron, Coulhon and Ouhabaz used some techniques introduced by Davies ([14]) to show that the upper Gaussian estimate (${\rm GE}_{m}$) on $e^{-tL},t>0,$ extends to a similar estimate on $e^{-zL}$ where $z$ belongs to the whole complex right half-plane $\mathbb{C}_{+}=\{z\in{\mathbb{C}}:{{\rm Re}z}>0\}$

[TABLE]

for and all $x,y\in X$, where $\theta={\rm arg}\,z$. The pointwise bounds have the following integrated form: for any $s\geq 0$, $R>0$ and $\tau\in{\mathbb{R}}$, the kernels $K_{e^{-(1+i\tau)R^{-m}L}}(x,y)$ of the operators ${e^{-(1+i\tau)R^{-m}L}}$ satisfy

[TABLE]

For a proof, see [17, Lemma 4.1].

Next we apply the estimate (2.8) to obtain the following result. For its proof, we refer to [17, Lemma 4.3] or [31, Theorem 7.15].

Lemma 2.4**.**

Suppose that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the property (${\rm GE}_{m}$). For any $R>0,s>0$ and $\epsilon>0$, there exists a constant $C=C(s,\epsilon)$ such that

[TABLE]

for all Borel functions $F$ such that supp $F\subseteq[R/4,R].$

Proof of Lemma 2.3.

Let $\phi$ be a non-negative $C_{c}^{\infty}$ function on $\mathbb{R}$ such that ${\rm supp}\ \phi\subseteq({1/4},1)$ and

[TABLE]

and let $\phi_{\ell}(\lambda)$ denote the function $\phi(2^{-\ell}\lambda)$. Since $c_{0}\leq 1,$ one has

[TABLE]

For every $j\in{\mathbb{Z}}$, we write $G_{j}(\lambda)=F_{j}(\lambda)e^{2^{-j}\lambda}$. Then we have $F_{j}(L)=G_{j}(L)e^{-2^{-j}L}$ and so

[TABLE]

Note that

[TABLE]

and from (2.3),

[TABLE]

We apply the heat kernel estimate for $e^{-2^{-j}L}$ and two estimates above to obtain that

[TABLE]

Then set $\widetilde{G}_{j}(\lambda)=G_{j}(\lambda^{m})$. By the Cauchy-Schwarz inequality and Lemma 2.4,

[TABLE]

On the other hand,

[TABLE]

and so it follows from (2.10) that

[TABLE]

Therefore the kernel $K_{(I+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{1}(c_{0}L)}(x,y)$ of $(I+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{1}(c_{0}L)$ satisfies

[TABLE]

for all $x,y\in{X}$ and $t>0$. Obviously,

[TABLE]

and the similar argument as in (ii) of Lemma 2.2 shows

[TABLE]

uniformly for $x\in X$ and $R>0$. ∎

3. Proof of Theorem 1.1

Fix a $f\in L^{1}$ and $\lambda>\mu(X)^{-1}\|f\|_{1}$, we apply the Calderón-Zygmund decomposition at height $\lambda$ to $|f|.$ There exist constants $C$ and $K$ so that

1. (i)

$f=g+b=g+\sum_{j}b_{j};$ 2. (ii)

$\|g\|_{1}\leq C\|f\|_{1},$ $\|g\|_{\infty}\leq C\lambda$; 3. (iii)

$b_{j}$ is supported in $B_{j}$ and $\#\{j:x\in{9\over 8}B_{j}\}\leq K$ for all $x\in X$; 4. (iv)

$\int_{X}\ |b_{j}|d\mu\leq C\lambda\mu(B_{j}),$ and $\sum_{j}\mu(B_{j})\leq C{\lambda}^{-1}\|f\|_{1}.$

It is easy to see that (ii) implies that $\|g\|_{2}^{2}\leq C\lambda\|f\|_{1}.$

Let $r_{B_{j}}$ be the radius of $B_{j}$ and let

[TABLE]

Write

[TABLE]

where $k_{0}$ is an integer such that $2^{k_{0}}\leq\sqrt{1+|t|}<2^{k_{0}+1}$. Then it is enough to show that there exists a constant $C>0$ independent of $\lambda$ and $t$ such that

[TABLE]

and such that for $i=1,2,$

[TABLE]

Note that by (ii) $\lambda^{-1}\|g\|_{2}^{2}\leq C\|f\|_{1}.$ Hence by spectral theory,

[TABLE]

which proves (3.1).

Proof of (3.2) for $i=1$. Since the Schrödinger group $e^{-itL}$ is bounded on $L^{2}(X)$, we have

[TABLE]

Recall that $h_{1}=\sum_{k\leq k_{0}}\sum_{j\in J_{k}}b_{j}$ with $\operatorname{supp}b_{j}\subset B_{j}$. In the following, we set $B^{\ast}_{j}={9\over 8}B_{j}$, a ball with the same center as $B_{j}$ but expand $9/8$ times. One has

[TABLE]

Let us estimate the term $I$. By Lemma 2.2, the kernels $K_{(I+L)^{-n/2}}(x,y)$ of the operators $(I+L)^{-n/2}$ satisfy

[TABLE]

for all $x,y\in{X}$ where $P_{1}(x,y)$ is a non-negative function satisfying the properties in Lemma 2.2. Since the $B_{j}^{\ast}$’s have bounded overlaps, we apply Minkowski’s inequality and (i) in Lemma 2.2 to obtain

[TABLE]

This, in combination with the doubling condition (2.2) that for every $y\in B_{j},$ with $r_{B_{j}}\leq 2^{k_{0}}<2\sqrt{1+|t|},$

[TABLE]

yields

[TABLE]

where in the last inequality we used (iv).

Next we estimate the term $II$. Let $x\in(B^{\ast}_{j})^{c}$ and $y\in B_{j}$, $P_{1}(x,y)$ is equivalent to $\inf_{y\in{B_{j}}}P_{1}(x,y)$ where $P_{t}(x,y)$ is the function given in Lemma 2.2. Thus, with $x_{B_{j}}$ is the center of $B_{j}$, it follows by Lemma 2.2 that

[TABLE]

This implies

[TABLE]

Combining the estimates for $I$ and $II$, we obtain (3.2) for $i=1$ with the term $h_{1},$ i.e.,

[TABLE]

Proof of (3.2) for $i=2$. Set $\Omega_{t}:=\cup_{j}\sqrt{1+|t|}B_{j}^{\ast}$. By (iv), we have

[TABLE]

Next we show that

[TABLE]

Since for every $j\in J_{k},k>k_{0}$, the function $b_{j}$ is supported in $B_{j}$, and the radius of the ball $B_{j}$ is equivalent to $2^{k}.$ We then decompose

[TABLE]

To analyse the term $(I+L)^{-n/2}e^{itL}e^{-2^{mk}L}b_{j},$ we need the following result.

Lemma 3.1**.**

With the notation above, there exists a constant $C>0$ such that

[TABLE]

Proof.

The proof is obtained by using the argument as in [15, estimate (10), p. 241]. We give a brief argument of this proof for completeness and convenience for the reader.

Since the functions $h_{t}(x,y)$ in (${\rm GE}_{m}$) satisfy a Harnack-type inequality (2.6), one has that for every $j\in J_{k},$

[TABLE]

Denoting by $M$ the Hardy-Littlewood maximal operator, we then have for every $u\in L^{2}(X)$

[TABLE]

Since $M$ is bounded on $L^{2}(X)$, we use the properties (iii) and (iv) to see that

[TABLE]

This finishes the proof of Lemma 3.1. ∎

Back to the proof of Theorem 1.1.  Now we apply Lemma 3.1 to obtain

[TABLE]

Concerning the term $(I+L)^{-n/2}e^{itL}(I-e^{-2^{mk}L})b_{j}$ in (3.9), we let $\varphi_{1}$ be a smooth function in Lemma 2.3 with supp $\varphi_{1}\subset[1/2,\infty]$ and $\varphi_{1}(\lambda)=1$ on $[1,\infty]$, and let $\varphi_{0}(\lambda)=1-\varphi_{1}(\lambda).$ One writes

[TABLE]

where

[TABLE]

and

[TABLE]

Hence

[TABLE]

As to be seen later, the term $III$ is the major one. To do it, we define a Besov type norm of $F$ by

[TABLE]

where ${\widehat{f}}$ denotes the Fourier transform of $f$. Since for every functions $F$ and $G$, it can be checked that

[TABLE]

and so by the Fubini theorem,

[TABLE]

We show the following estimate for the Schrödinger group $\{e^{itL}\}_{t\geq 0}$, which plays a key role in the proof of the term $III$.

Lemma 3.2**.**

Suppose that $(X,d,\mu)$ is a space of homogeneous type with a dimension $n$. Suppose that $L$ satisfies the property (${\rm GE}_{m}$). Let $\varphi_{0}$ be a smooth function with supp $\varphi_{0}\subset[0,1]$ and $\varphi_{0}(\lambda)=1$ on $[0,1/2]$. For $k>0$ and $t\in\mathbb{R}$, denote $F_{k}(L):=(1+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{0}(2^{-m(k-k_{0})/(m-1)}L)$ where $k_{0}$ in an integer such that $2^{k_{0}}\leq\sqrt{1+|t|}<2^{k_{0}+1}$. Then for some fixed $c_{1}>1$,

[TABLE]

with a constant $C>0$ independent on $k>k_{0}$ and $t$.

Proof.

Let $\phi$ be an auxiliary non-negative $C_{c}^{\infty}$ function on $\mathbb{R}$ as in (2.9). By the spectral theory, we write

[TABLE]

Set $F_{k,\ell}(L)=e^{itL}(I+L)^{-n/2}(I-e^{-2^{mk}L})\varphi_{0}(2^{-m(k-k_{0})/(m-1)}L)\phi(2^{-\ell}L).$ Then the kernels $K_{F_{k,\ell}(L)}(x,y)$ of the operators $F_{k,\ell}(L)$ satisfies

[TABLE]

where

[TABLE]

for some $n<s<n+1/2$. Next we set G(\lambda)=\big{(}\delta_{2^{\ell}}F_{k,\ell}(\lambda)\big{)}e^{\lambda}. Thus in virtue of the Fourier inversion formula

[TABLE]

By (2.8), we get

[TABLE]

To go on, we claim that

[TABLE]

Let us show the claim (3.14). Now we let $\eta\in C_{c}^{\infty}({\mathbb{R}})$ with $\operatorname{supp}\eta\subset[1/8,2]$ and $\eta(\lambda)=1$ for $\lambda\in[1/4,1]$. One has

[TABLE]

Note that $\ell\leq m(k-k_{0})/(m-1)$ and $\operatorname{supp}\eta\subset[1/8,2]$,

[TABLE]

and

[TABLE]

with $C_{s}$ independent of $k$ and $\ell$. Let us estimate $\|\phi(\lambda)e^{it2^{\ell}\lambda}(1+2^{\ell}\lambda)^{-n/2}\|_{{\bf B}^{s/2}}$. The Fourier transform {\mathcal{F}}\big{(}\phi e^{it2^{\ell}\cdot}(1+2^{\ell}\cdot)^{-n/2}\big{)}(\tau) of $\phi(\lambda)e^{it2^{\ell}\lambda}(1+2^{\ell}\lambda)^{-n/2}$ is given by

[TABLE]

Integration by parts gives for every $N\in\mathbb{N}$,

[TABLE]

which yields for $s>n$,

[TABLE]

Hence,

[TABLE]

This proves our claim (3.14). From this, we see that

[TABLE]

where we have used the fact that

[TABLE]

See [17, Lemma 4.4]. Therefore,

[TABLE]

The proof of Lemma 3.2 is complete. ∎

We now apply Lemma 3.2 to estimate the term $III$ to obtain

[TABLE]

Concerning the term $IV$, since the Schrödinger group $e^{-itL}$ is bounded on $L^{2}(X)$, we have

[TABLE]

We point out that the terms $IV_{1}$ and $IV_{2}$ can be handled similarly to the terms $I$ and $II$ with some modifications, respectively.

To estimate the term $IV_{1}$, we note that

[TABLE]

Since the $B_{j}^{\ast}$’s have bounded overlaps, we apply Minkowski’s inequality and the property (i) in Lemma 2.2 to obtain

[TABLE]

This, in combination with the doubling condition (2.2) that for every $y\in B_{j},$

[TABLE]

yields

[TABLE]

Now we prove the term $IV_{2}$. Let $x\in(B^{\ast}_{j})^{c}$ and $y\in B_{j}$, $Q(x,y)$ is equivalent to $\inf_{y\in{B_{j}}}Q(x,y),$ where $Q(x,y)$ is the function given in Lemma 2.3. Thus, with $x_{B_{j}}$ is the center of $B_{j}$, we have that for a fixed $x\in(B^{\ast}_{j})^{c}$, it follows from (ii) in Lemma 2.3 that

[TABLE]

This implies

[TABLE]

From the estimates for $IV_{1}$ and $IV_{2}$, $IV\leq C\lambda^{-1}(1+|t|)^{n/2}\|f\|_{1}.$ This combined with (3.7), (3.8), (3.9), (3.10), (3.11) and (3.15) yields (3.2) for $i=2$. The proof of the theorem now follows from (3.1) and (3.2) . $\square$

Recall that for every $p\in(1,\infty)$, Miyachi ([30]) showed that there exists a positive constant $C=C(n,p)$ independent of $t$ and $f$ so that

[TABLE]

The estimate (3.19) is sharp for the growth in $t,$ that is, the factor $(1+|t|)^{s}$ can not be improved. Following [30], we show that for $p=1,$ the factor $(1+|t|)^{n/2}$ in our Theorem 1.1 is the best possible in the case that when $L$ is the classical Laplacian on $\mathbb{R}^{n}$. This can be seen by the following result.

Proposition 3.3**.**

For every $t\geq 1$, there exists a constant $C=C(n)$ independent of $t$ such that

[TABLE]

Proof.

Let $\phi\in C^{\infty}(1/2,\infty)$ with $\operatorname{supp}\phi\subset(1/2,\infty)$ and $\phi=1$ on $[1,\infty)$. We define

[TABLE]

with some constant $c>0$ so that $\|f\|_{1}=1$. Then

[TABLE]

Following [30, Lemma 1], there exist constants $C_{1}$ and $C_{2}$ such that for $|x|\geq C_{2}t,$

[TABLE]

Then we have

[TABLE]

Letting $\lambda=C_{1}t^{-n/2}/(1+C_{2})^{n/2+1}$, we have

[TABLE]

This finishes the proof of Proposition 3.3. ∎

Our results are applicable to Schrödinger group for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with non-negative potentials and Laplace operators acting on Lie groups of polynomial growth (see for example, [9, 12, 17, 24] and the references therein). For example, we consider the Schrödinger operator

[TABLE]

where $V:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}},V\in L^{1}_{\rm loc}({\mathbb{R}}^{n})$ and $V\geq 0.$ The operator $L$ is defined by the quadratic form. The Feynman-Kac formula implies that the semigroup kernels $p_{t}(x,y)$, associated to $e^{-tL}$, satisfy the estimates

[TABLE]

See page 195 of [31].

Theorem 3.4**.**

Assume that $L=-\Delta+V$ where $\Delta$ is the standard Laplace operator acting on ${\mathbb{R}}^{n}$ and $V\in L^{1}_{\rm loc}({\mathbb{R}}^{n})$ is a non-negative function. Then we have

• (i)

The operator $(I+L)^{-n/2}e^{itL}$ is of weak type $(1,1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that for $\lambda>0$,

[TABLE]

• (ii)

For every $1<p<\infty$ and $s\geq n|{1/p-1/2}|$, the operator $(I+L)^{s}e^{itL}$ is bounded on $L^{p}(\mathbb{R}^{n})$, and there exists a constant $C$, independent of $t$ and $f$ so that

[TABLE]

Proof.

This result is a consequence of Theorem 1.1 and [9, Theorem 1]. ∎

We note that under suitable additional assumptions this result can be extended by a similar proof to situation of magnetic Schrödinger operators acting on a complete Riemannian manifold with non-negative potentials.

4. Schrödinger groups on measurable

subsets of a space of homogeneous type

We assume in this section that $\Omega$ is a measurable subset of a space of homogeneous type $(X,d,\mu)$. An example of $\Omega$ is an open domain of the Euclidean space ${\mathbb{R}}^{n}$. If $\Omega$ possesses certain smoothness on its boundary, for example Lipschitz boundary, then it is a space of homogeneous type and the results of Sections 2 and 3 are applicable. However, a general measurable set $\Omega$ needs not satisfy the doubling property, hence it is not a space of homogeneous type. Such a measurable set $\Omega$ appears naturally in boundary value problems, for example partial differential equations with Dirichlet boundary conditions.

In this section we are interested in dealing with Schrödinger groups in those contexts. As it is pointed out in [15], one can extend the singular operators defined in $\Omega$ to the space $X.$ Since there is no assumption on the regularity of the kernels in space variables, the extension of the kernel still satisfies similar conditions. Given $T$, a bounded linear operator on $L^{p}(\Omega),1<p<\infty$, the extension of $T$ to $X$ is defined as

[TABLE]

where $\chi_{\Omega}$ is the characterization function on $\Omega$. Then $T$ is bounded on $L^{p}(\Omega)$ if and only if ${\widetilde{T}}$ is bounded on $L^{p}(X).$ Also $T$ is of weak type $(1,1)$ on $\Omega$ if and only if ${\widetilde{T}}$ is of weak type $(1,1)$ on $X.$ If $K$ is the kernel of $T$, then the associated kernel of ${\widetilde{T}}$ is given by

[TABLE]

As it is observed in [15], the assumptions on the kernels do not involve their regularity so they imply similar properties on the kernels of the extended operators.

The following result gives an example of Schrördiner groups on spaces without the doubling condition.

Theorem 4.1**.**

Suppose that $\Delta_{\Omega}$ is the Laplace operator with Dirichlet boundary condition $\Omega\subseteq{\mathbb{R}}^{n}$. Then we have

• (i)

The operator $(I+\Delta_{\Omega})^{-n/2}e^{it\Delta_{\Omega}}$ is of weak type $(1,1)$, that is, there is a constant $C$, independent of $t$ and $f$ so that

[TABLE]

for $\lambda>0$ when $|\Omega|=\infty$ and $\lambda>\|f\|_{1}/|\Omega|$ when $|\Omega|<\infty$.

• (ii)

For every $1<p<\infty$ and $s\geq n|{1/p-1/2}|$, the operator $(I+\Delta_{\Omega})^{s}e^{it\Delta_{\Omega}}$ is bounded on $L^{p}(\Omega)$, and there exists a constant $C$, independent of $t$ and $f$ so that

[TABLE]

Proof.

We point out that

[TABLE]

(see e.g., Example 2.18, [13]). That is the heat kernels corresponding to $\Delta_{\Omega}$ satisfying Gaussian bounds (${\rm GE}_{m}$) with $m=2$.

Denote $T={(I+\Delta_{\Omega})^{-s}e^{it\Delta_{\Omega}}}$ with $s=n/2$ for $p=1$ and $s\geq n|{1/p-1/2}|$ for $1<p<\infty$. And denote by $\widetilde{T}$ the extension of $T$ as in (4.3). Then (i) and (ii) of Theorem 4.1 follow from Theorem 1.1 and [9, Theorem 1.1] by first applying to the extended operator ${\widetilde{T}}$ and then restricting to $T$, respectively. This completes the proof of Theorem 4.1. ∎

Acknowledgements: P. Chen is supported by NNSF of China 11501583, Guangdong Natural Science Foundation 2016A030313351. Duong and Li are supported by the Australian Research Council (ARC) through the research grants DP160100153 and DP190100970 and by Macquarie University Research Seeding Grant. L. Song is supported by NNSF of China (No. 11622113) and NSF for distinguished Young Scholar of Guangdong Province (No. 2016A030306040). Yan is supported by the NNSF of China, Grant No.  11521101 and  11871480, and Guangdong Special Support Program.