Sharp Estimates for Schrödinger Groups on Hardy Spaces for $0<p\le 1$

Abstract

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on $L^2(X)$ and suppose that the kernel of $e^{-tL}$ satisfies a Gaussian upper bound. This paper shows that for $0<p\le 1$ and $s=n(1/p-1/2)$,

$$\begin{aligned}\Vert (I+L)^{-s}e^{itL}f\Vert _{H^p_L(X)} \lesssim (1+|t|)^{s}\Vert f\Vert _{H^p_L(X)} \end{aligned}$$

for all $t\in {\mathbb {R}}$, where $H^p_L(X)$ is the Hardy space associated to L. This recovers the classical results in the particular case when $L=-\Delta$ and extends a number of known results.

1 Introduction

Let $(X,d, \mu )$ be a metric space endowed with a nonnegative Borel measure $\mu$. Denote by B(x, r) the open ball of radius $r > 0$ and center $x \in X$, and by V(x, r) its measure $\mu (B(x, r))$. In this paper we assume that the measure $\mu$ satisfies the doubling condition: there exists a constant $C>0$ such that

$$\begin{aligned} V(x,2r)\le CV(x,r) \end{aligned}$$

(1)

for all $x\in X$, $r>0$ and all balls B(x, r).

We note that the doubling property (1) yields a constant $n>0$ so that

$$\begin{aligned} V(x,\lambda r)\le C\lambda ^nV(x,r), \end{aligned}$$

(2)

for all $\lambda \ge 1, x\in X$ and $r>0$; and that

$$\begin{aligned} V(x,r)\le C\Big (1+\frac{d(x,y)}{r}\Big )^{n}V(y,r), \end{aligned}$$

(3)

for all $x,y\in X$ and $r>0$.

Suppose that L is a non-negative self-adjoint operator on $L^2(X)$. Suppose further that the operator L generates an analytic semigroup $e^{-tL}$ whose kernels $e^{-tL}$ satisfy the Gaussian estimate. That is, there exist constants $C, c > 0$ and $m>1$ such that

$$\begin{aligned} |e^{-tL}(x,y)|\le \frac{C}{V(x,t^{1/m})}\exp \Big (-\frac{d(x,y)^{m/m-1}}{ct^{1/m-1}}\Big ) \end{aligned}$$

(G)

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

Through spectral theory we can define the Schrödinger group, for $t\in {\mathbb {R}}$,

$$\begin{aligned}e^{itL} =\int _0^\infty e^{it\lambda }dE_L(\lambda ), \end{aligned}$$

where $E_L(\lambda )$ is the spectral decomposition of L.

The mapping properties of the Schrödinger group $e^{it L}$ has a wide range of applications spanning fields such as harmonic analysis and nonlinear dispersive equations. The Schrödinger group is bounded on $L^2(X)$ but not bounded in $L^{p}(X)$ for $p\ne 2$, even in the case when $L=-\Delta$ is the Laplacian on ${\mathbb {R}}^n$. Despite this, $(1+L)^{-s}e^{it L}$ is known to be $L^{p}$-bounded for s sufficiently large. It was shown in [7] that for every $1<p<\infty$ and $t\in {\mathbb {R}}$,

$$\begin{aligned} \Vert (1+L)^{-s}e^{it L} f\Vert _{L^p}\lesssim (1+|t|)^{s}\Vert f\Vert _{L^p}, \qquad s>n\biggl |\frac{1}{2}-\frac{1}{p}\biggr |. \end{aligned}$$

(4)

Similar results can be found in [2, 5, 7, 11, 20, 23] and the references therein.

In the classical case when $L=-\Delta$, we also have the following sharp estimate: for all $1<p<\infty$ and $t>0$ one has

$$\begin{aligned} \Vert (1-\Delta )^{-s}e^{it \Delta } f\Vert _{L^p}\lesssim (1+|t|)^{s}\Vert f\Vert _{L^p}, \qquad s=n\biggl |\frac{1}{2}-\frac{1}{p}\biggr |, \end{aligned}$$

(5)

see [22]. Also for $p\le 1$, it was proved by Miyachi [21] that for each $0<p\le 1$ and $t\in {\mathbb {R}}$ we have

$$\begin{aligned} \Vert (1-\Delta )^{-s}e^{it \Delta } f\Vert _{H^p({\mathbb {R}}^n)}\lesssim (1+|t|)^{s}\Vert f\Vert _{H^p({\mathbb {R}}^n)}, \qquad s=n\Big (\frac{1}{p}-\frac{1}{2}\Big ), \end{aligned}$$

(6)

where $H^p({\mathbb {R}}^n)$ is the classical Hardy spaces. See [24].

Let us turn to some more recent results concerning (4)-(6), which also serves to motivate the results in our paper. The first concerns sharpness for $p>1$. In comparison with (5), estimate (4) is not sharp. However this point has recently been addressed in [9]; more precisely, it was proved there that (4) also holds for $s= n\bigl |\frac{1}{2}-\frac{1}{p}\bigr |$.

Secondly, the following endpoint estimates for $p=1$ were obtained in [8]:

$$\begin{aligned} \Vert (1+L)^{-n/2}e^{it L} f\Vert _{L^1}+\Vert (1+L)^{-n/2}e^{it L} f\Vert _{H^1_L}\lesssim (1+|t|)^{n/2}\Vert f\Vert _{H^1_L}, \end{aligned}$$

(7)

under more general assumptions than (G). Here $H^1_L(X)$ is the Hardy space associated to L (see Sect. 2 for the precise definition of $H^1_L(X)$). In this paper we address the sharp extension of (7) to $p<1$ in the sense of (6). Our main result is the following.

Theorem 1.1

Let L be a non-negative self-adjoint operator on $L^2(X)$ generating an analytic semigroup $e^{-tL}$ whose kernels satisfy the Gaussian upper bound (G). Then for each $0<p\le 1$ and $s=n(1/p-1/2)$, we have

$$\begin{aligned} \Vert (I+L)^{-s}e^{itL}f\Vert _{H^p_L(X)} \lesssim (1+|t|)^{s}\Vert f\Vert _{H^p_L(X)}, \qquad t\in {\mathbb {R}}, \end{aligned}$$

(8)

where $H^p_L(X)$ is the Hardy space associated to L (defined in Sect. 2).

Some comments on Theorem 1.1 are in order.

• (i) It is natural to speculate on the relationship between Theorem 1.1 and [8, Theorem 1.1]. While the endpoint $p=1$ is implied by [8, Theorem 1.1], to the best of our knowledge, the result for $p<1$ is new. It is also important to note that the approach in [8] is not immediately applicable to $p<1$; indeed, the inequality (4.7) in [8], which plays a crucial role in the proof of [8, Theorem 1.1], is not true if the $L^1$-norm is replaced by the $L^p$-norm when $p<1$. We believe therefore that any generalization of Theorem 1.1 under the less restrictive assumptions employed in [8] will require new ideas.

• (ii) By using interpolation, estimate (8) implies the following sharp $L^p$ estimate: for $1<p<\infty$, we have

  $$\begin{aligned}\Vert (1+L)^{-s}e^{it L} f\Vert _{L^p}\lesssim (1+|t|)^{s}\Vert f\Vert _{L^p}, \qquad s=n\biggl |\frac{1}{2}-\frac{1}{p}\biggr |. \end{aligned}$$

  See [8]. Thus, Theorem 1.1 completes the scale of sharp estimates for the Schrödinger group for all $0<p<\infty$.

For $s>0$, consider the operator defined by

$$\begin{aligned} I_{s,t}(L)f =st^{-s}\int _0^t (t-\lambda )^{s-1}e^{-i\lambda L}f d\lambda ,\qquad t>0, \end{aligned}$$

and $I_{s,t}(L)={\bar{I}}_{s,-t}(L)$ for $t<0$. These operators are known as the ‘Riesz means’ associated to L. The Riesz means have close connections with the solution to the Schrödinger equation

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u + Lu =0,\\ u(x,0)=f. \end{array}\right. } \end{aligned}$$

See for example [23].

By using Theorem 1.1, the spectral theorem in [14, Theorem 1.1], and a standard argument from [23], we can obtain the following result.

Corollary 1.2

Assume that L satisfies the conditions of Theorem 1.1. Then for each $0<p\le 1$ there exists a constant $C>0$ independent of t such that

$$\begin{aligned} \Vert I_{s,t}(L)f\Vert _{H^p_L(X)}\le C\Vert f\Vert _{H^p_L(X)}, \ \ \ \ \ \ \ s= \Big (\frac{1}{p}-\frac{1}{2}\Big ) \end{aligned}$$

for all $t\ne 0$.

The organization of this paper is as follows. In Sect. 2, we fix some notations that will be employed throughout the article and detail some properties of the Hardy spaces associated to the operator L. The proof of Theorem 1.1 will be given in Sect. 3. Finally, Sect. 4 will discuss some applications of the main result.

2 Preliminaries

2.1 Notations and Elementary Estimates on the Space of Homogeneous Type

As usual we use C and c to denote positive constants that are independent of the main parameters involved but may differ from line to line. The notation $A\lesssim B$ means $A\le CB$, and $A\sim B$ means that both $A\lesssim B$ and $B\lesssim A$ hold.

The space of Schwarz functions on ${\mathbb {R}}^n$ is denoted by ${\mathscr {S}}({\mathbb {R}}^n)$ and given $\psi \in {\mathscr {S}}({\mathbb {R}})$, $\lambda \in {\mathbb {R}}$ and $j\in {\mathbb {Z}}$, we use the notation $\psi _j(\lambda ):=\psi (2^{-j}\lambda )$. For $f\in {\mathscr {S}}({\mathbb {R}}^n)$ we denote by ${\mathcal {F}}f$ the Fourier transform of f. That is,

$$\begin{aligned} {\mathcal {F}}f(\xi )=\frac{1}{(2\pi )^{n/2}}\int f(x)e^{-ix\cdot \xi }\,dx, \qquad \xi \in {\mathbb {R}}^n. \end{aligned}$$

To simplify notation, we will often just use B for $B(x_B, r_B)$ and V(E) for $\mu (E)$ for any measurable subset $E\subset X$. Also given $\lambda >0$, we will write $\lambda B$ for the $B(x_B, \lambda r_B)$. For each ball $B\subset X$ we set

$$\begin{aligned} S_0(B) = 0, \ \ \ S_j(B)=2^{j}B\backslash 2^{j-1}B \ \ \ \text {for} j\in {\mathbb {N}}. \end{aligned}$$

Let $w\in A_\infty$ and $0<r<\infty$. The Hardy–Littlewood maximal function ${\mathcal {M}}_{r}$ is defined by

$$\begin{aligned} {\mathcal {M}}_{r} f(x)=\sup _{x\in B}\Big (\frac{1}{V(B)}\int _B|f(y)|^r d\mu (y)\Big )^{1/r} \end{aligned}$$

where the sup is taken over all balls B containing x. We will drop the subscripts r when $r=1$. It is well-known that for $0<r<\infty$ one has

$$\begin{aligned} \Vert {\mathcal {M}}_{r} f\Vert _{p}\lesssim \Vert f\Vert _{p}, \end{aligned}$$

(9)

whenever $p>r$.

The following elementary estimates will be used frequently. See for example [2].

Lemma 2.1

Let $\epsilon >0$.

• (a) For any $p\in [1,\infty ]$ we have

  $$\begin{aligned} \Big (\int _X\Big [\Big (1+\frac{d(x,y)}{s}\Big )^{-n-\epsilon }\Big ]^pd\mu (y)\Big )^{1/p}\lesssim V(x,s)^{1/p}, \end{aligned}$$

  for all $x\in X$ and $s>0$.

• (b) For any $f\in L^1_{\mathrm{loc}}(X)$ we have

  $$\begin{aligned} \int _X\frac{1}{V(x\wedge y,s)}\Big (1+\frac{d(x,y)}{s}\Big )^{-n-\epsilon }|f(y)|d\mu (y)\lesssim {\mathcal {M}}f(x), \end{aligned}$$

  for all $x\in X$ and $s>0$, where $V(x\wedge y,s)=\min \{V(x,s),V(y,s)\}$.

We also recall the Fefferman-Stein vector-valued maximal inequality in [17]. For $0<p<\infty$, $0<q\le \infty$ and $0<r<\min \{p,q\}$, we have for any sequence of measurable functions $\{f_\nu \}$,

$$\begin{aligned} \Big \Vert \Big (\sum _{\nu }|{\mathcal {M}}_rf_\nu |^q\Big )^{1/q}\Big \Vert _{p}\lesssim \Big \Vert \Big (\sum _{\nu }|f_\nu |^q\Big )^{1/q}\Big \Vert _{p}. \end{aligned}$$

(10)

2.2 Hardy Spaces Associated to the Operator L

We first recall from [16, 19] the definition of the Hardy spaces associated to an operator. Let L be a nonnegative self-adjoint operator on $L^2(X)$ satisfying the Gaussian upper bound (G). Let $0<p\le 1$. Then the Hardy space $H_{L}^p(X)$ is defined as the completion of

$$\begin{aligned} \{f\in L^2(X): {\mathcal {A}}_Lf \in L^p(X)\} \end{aligned}$$

under the norm $\Vert f\Vert _{H_{L}^p(X)}=\Vert {\mathcal {A}}_Lf\Vert _{L^p}$, where the square function ${\mathcal {A}}_{L}$ is defined as

$$\begin{aligned} {\mathcal {A}}_Lf(x)=\Bigl (\int _0^\infty \int _{d(x,y)<t}|t^mLe^{-t^mL}f(y)|^2\frac{d\mu (y) dt}{tV(x,t)}\Bigr )^{1/2}. \end{aligned}$$

(11)

Next we have a notion of molecules from [16, 19].

Definition 2.2

(Molecules for L) Let $\epsilon >0$, $0<p\le 1$ and $M\in {\mathbb {N}}$. A function m(x) is called a $(p,2,M,L,\epsilon )$-molecule associated to a ball $B \subset X$ of radius $r_{B}$ if there exists a function $b\in D(L^{M})$ such that

• (i) $m=L^M b$;

• (i) $\Vert L^{k}b\Vert _{L^2(S_j(B))}\le 2^{-j\epsilon }r_B^{m(M-k)}V(2^jB)^{1/2-1/p}$ for all $k=0,1,\dots ,M$ and $j=0,1,2\ldots$.

The molecular property (ii) in particular can be thought of as a mild locality condition on the operator L.

Definition 2.3

(Hardy spaces associated to L) Given $\epsilon >0$, $0<p\le 1$ and $M\in {\mathbb {N}}$, we say that $f=\sum \lambda _jm_j$ is a molecule $(p,2,M,L,\epsilon )$-representation if $\{\lambda _j\}_{j=0}^\infty \in \ell ^p$, each $m_j$ is a $(p,2,M,L,\epsilon )$-atom, and the sum converges in $L^2(X)$. The space $H^{p}_{L,\mathrm{mol},M,\epsilon }(X)$ is then defined as the completion of

$$\begin{aligned} \left\{ f\in L^2(X):f \ \text {has a molecule} (p,2,M,L,\epsilon )-representation\right\} , \end{aligned}$$

with the norm given by

$$\begin{aligned} \Vert f\Vert ^p_{H^{p}_{L,\mathrm{mol},M,\epsilon }(X)}= & {} \inf \left\{ \sum |\lambda _j|^p : f\right. \\&\left. =\sum \lambda _jm_j \ \text {is a molecule} (p,2,M,L,\epsilon )-representation\right\} . \end{aligned}$$

The following gives a molecular characterization for the Hardy spaces $H^p_L(X)$.

Theorem 2.4

([6, 16, 19]) Let $\epsilon >0$, $p\in (0,1]$ and $M>\frac{n(2-p)}{2mp}$. Then the Hardy spaces $H^{p}_{L,\mathrm{mol},M,\epsilon }(X)$ and $H^{p}_{L}(X)$ coincide and have equivalent norms.

We note that if $L=-\Delta$ then $H^p_L({\mathbb {R}}^n)$ coincides with the standard Hardy space $H^p({\mathbb {R}}^n)$ on ${\mathbb {R}}^n$ for $p\in (0,1]$. In general, depending on the choice of the operator L, the space $H^p_L({\mathbb {R}}^n)$ may be quite different to $H^p({\mathbb {R}}^n)$. See for example [12].

2.3 Discrete Square Functions

In this section we obtain an inequality for certain square functions that will be important in the proof of Theorem 1.1.

In what follows, by a “partition of unity” we shall mean a function $\psi \in {\mathscr {S}}({\mathbb {R}})$ such that ${\text {supp}}\psi \subset [1/2,2]$, $\int \psi (\xi )\,\frac{d\xi }{\xi }\ne 0$ and

$$\begin{aligned}\sum _{j\in {\mathbb {Z}}}\psi _j(\lambda )=1 on (0,\infty ).\end{aligned}$$

where $\psi _j(\lambda ):=\psi (2^{-j}\lambda )$ for each $j\in {\mathbb {Z}}$. Now let $\psi$ be a partition of unity and define the discrete square function $S_{L,\psi }$ by

$$\begin{aligned} S_{L,\psi }f =\Big (\sum _{j\in {\mathbb {Z}}}|\psi _j(L)f|^2\Big )^{1/2}, \end{aligned}$$

which is bounded on $L^2(X)$ by Khintchine’s inequality. We also have the following, which is the main result of this section.

Theorem 2.5

Let $\psi$ be a partition of unity. Then for each $0<p\le 1$, we have

$$\begin{aligned} \Vert f\Vert _{H^p_L}\lesssim \Vert S_{L,\psi }f\Vert _p \end{aligned}$$

for all $f\in H^p_L(X)$.

In order to prove the theorem we follow the ideas in [2]. Before presenting the proof we gather some technical elements which will play a core role in the proof of the theorem.

The first concerns certain kernel estimates.

Lemma 2.6

([18]) Let $\varphi , \psi \in {\mathscr {S}}({\mathbb {R}})$ supported in [1/2, 2]. Then the kernel $K_{\varphi (tL)}$ of $\varphi (tL)$ satisfies the following: for any $N>0$ there exists C such that

$$\begin{aligned} |K_{\varphi (t L)}(x,y)|\le \frac{C}{V(x\vee y,t^{1/m})}\Big (1+\frac{d(x,y)}{t^{1/m}}\Big )^{-N}, \end{aligned}$$

(12)

for all $t>0$ and $x,y\in X$, where $V(x\vee y,t^{1/m})=\max \{V(x,t^{1/m}),V(y,t^{1/m})\}$.

Next we introduce and give estimates for certain ‘Peetre-type’ maximal functions. For $\lambda >0, j\in {\mathbb {Z}}$ and $\varphi \in {\mathscr {S}}({\mathbb {R}})$ the Peetre-type function is defined, for $f\in {\mathcal {L}}^2(X)$, by

$$\begin{aligned} \varphi _{j,\lambda }^*(L)f(x)=\sup _{y\in X}\frac{|\varphi _j(L)f(y)|}{(1+2^{j/m}d(x,y))^\lambda }\, , x\in X. \end{aligned}$$

(13)

Obviously, we have

$$\begin{aligned} \varphi _{j,\lambda }^*(L)f(x)\ge |\varphi _j(L)f(x)|, \ \ \ \ x\in X. \end{aligned}$$

Similarly, for $s, \lambda >0$ we set

$$\begin{aligned} \varphi _{\lambda }^*(sL)f(x)=\sup _{y\in X}\frac{|\varphi (sL)f(y)|}{(1+d(x,y)/s^{1/m})^\lambda }, \ \ \ f\in L^2(X). \end{aligned}$$

(14)

Proposition 2.7

Let $\psi \in {\mathscr {S}}({\mathbb {R}})$ with $\mathrm{supp}\,\psi \subset [1/2,2]$ and $\varphi \in {\mathscr {S}}({\mathbb {R}})$ be a partition of unity. Then for any $\lambda >0$ and $j\in {\mathbb {Z}}$ we have

$$\begin{aligned} \sup _{s\in [2^{-j-1},2^{-j}]}\psi ^*_{\lambda }(s L)f(x) \lesssim \sum _{k=j-2}^{j+3} \varphi ^*_{k,\lambda }( L)f(x) \end{aligned}$$

(15)

for all $f\in L^2(X)$ and $x\in X$.

Proof

The proof can be done in the same way as [2, Proposition 2.16] with $s^{1/m}$ and $2^{j/m}$ in place of s and $2^j$ respectively. We omit the details. $\square$

Proposition 2.8

Let $\psi$ be a partition of unity. Then for any $\lambda , s>0$ and $r\in (0,1)$ we have:

$$\begin{aligned} \psi ^*_\lambda (s L)f(x)\lesssim \Big [\int _X \frac{1}{V(z,s^{1/m})}\frac{|\psi (s L)f(z)|^r}{(1+d(x,z)/s^{1/m})^{\lambda r}}d\mu (z)\Big ]^{1/r} \end{aligned}$$

for all $f\in L^2(X)$ and $x\in X$.

Proof

The proof can be done in the same way as [2, Proposition 2.17] and we omit the details.

$\square$

We next prove the following result.

Proposition 2.9

Let $\psi$ be a partition of unity. Then for $0<p\le 1$ and $\lambda >n/p$ we have:

$$\begin{aligned} \Big \Vert \Big [\sum _{j\in {\mathbb {Z}}} |\psi ^*_{j,\lambda }({L})f|^2\Big ]^{1/2}\Big \Vert _{p}\sim \Vert S_{L,\psi }f\Vert _{p}. \end{aligned}$$

Proof

Since $|\psi _{j}(\sqrt{L})f|\lesssim \psi ^*_{j,\lambda }({L})f$, it suffices to prove that

$$\begin{aligned} \Big \Vert \Big [\sum _{j\in {\mathbb {Z}}} |\psi ^*_{j,\lambda }({L})f|^2\Big ]^{1/2}\Big \Vert _{p}\lesssim \Vert S_{L,\psi }f\Vert _{p}. \end{aligned}$$

(16)

Choose $r<p$ so that $\lambda >n/r$. Then applying Proposition 2.8 and Lemma 2.1 we have

$$\begin{aligned} \begin{aligned} \psi ^*_{j,\lambda }( L)f(x)&\lesssim \Big [\int _X \frac{1}{V(z,2^{-j})}\frac{|\psi _j( L)f(z)|^r}{(1+2^jd(x,z))^{\lambda r}}d\mu (z)\Big ]^{1/r} \lesssim {\mathcal {M}}_r\big (|\psi _j( L)f|\big )(x) \end{aligned} \end{aligned}$$

At this stage, we may apply the weighted Fefferman-Stein maximal inequality (10) to obtain (16) as desired. $\square$

We now ready to prove Theorem 2.5.

Proof of Theorem 2.5:

Setting $\varphi (\lambda )=\lambda e^{-\lambda }$. Observe that

$$\begin{aligned} |\varphi (t L)f(y)|\le \varphi ^*_\lambda (t L)f(x) \end{aligned}$$

for all $\lambda >0$ and $d(x,y)<t^{1/m}$. Therefore,

$$\begin{aligned} \begin{aligned}&\Big (\int _0^\infty \int _{d(x,y)<t^{1/m}}|\varphi (t L)f(y)|^2 \frac{d\mu (y)dt}{tV(x,t^{1/m})}\Big )^{1/2}\\&\le \left[ \int _0^\infty \int _{d(x,y)<t^{1/m}} |\varphi ^*_\lambda (t L)f(x)|^2\frac{d\mu (y) dt}{tV(x,t^{1/m})}\right] ^{1/2}\\&\lesssim \left[ \int _0^\infty |\varphi ^*_\lambda (t L)f(x)|^2\frac{dt}{t}\right] ^{1/2}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \Vert f\Vert _{H^p_L} =\Big \Vert \Big (\int _0^\infty \int _{d(x,y)<t^{1/m}}|\varphi (t L)f(y)|^2 \frac{d\mu (y)dt}{tV(x,t^{1/m})}\Big )^{1/2}\Big \Vert _p, \end{aligned}$$

it suffices to prove that

$$\begin{aligned} \Big \Vert \left[ \int _0^\infty |\varphi ^*_\lambda (t L)f(x)|^2\frac{dt}{t}\right] ^{1/2}\Big \Vert _p\lesssim \Vert S_{L,\psi }f\Vert _p, \end{aligned}$$

(17)

where $\psi$ is a partition of unity.

By the spectral theory,

$$\begin{aligned} f =c_\psi \int _0^\infty \psi (s L)f\frac{ds}{s} \quad \hbox { in}\ L^2(X), \end{aligned}$$

where $c_\psi =\Big [\int _0^\infty \psi (s )\frac{ds}{s}\Big ]^{-1}$. Hence it follows that for every $t>0$,

$$\begin{aligned} \varphi (t L)(f )=c_\psi \int _0^\infty \varphi (t L)\psi (s L)f\frac{ds}{s}. \end{aligned}$$

(18)

Now let $\lambda >0$, $t\in [2^{-\nu -1},2^{-\nu }]$ for some $\nu \in {\mathbb {Z}}$ and $M>\lambda$. For convenience we may assume $c_\psi = 1$. We then have

$$\begin{aligned} \varphi (t L)(f )= & {} \int _0^\infty \psi (s L)\varphi (t L)f\frac{ds}{s}\\= & {} \sum _{j\ge \nu }\int _{2^{-j-1}}^{2^{-j}} \psi (s L)\varphi (t L)f\frac{ds}{s}+\sum _{j< \nu }\int _{2^{-j-1}}^{2^{-j}} \psi (s L)\varphi (t L)f\frac{ds}{s}\\= & {} \sum _{j\ge \nu }\int _{2^{-j-1}}^{2^{-j}}\Big ( \frac{s}{t}\Big )^{M} (sL)^{-M}\psi (s L)(tL)^{M}\varphi (t L)f\frac{ds}{s}\\&+\sum _{j< \nu }\int _{2^{-j-1}}^{2^{-j}} \frac{t}{s} (sL)\psi (s L)e^{-tL}f\frac{ds}{s}, \end{aligned}$$

where in the last line we used $\varphi (t L)= (tL)e^{-tL}$.

We now set $\psi _{M}(x)=x^{-M}\psi (x)$ and ${\widetilde{\psi }}(x)=x \psi (x)$. Then the above can be written as

$$\begin{aligned} \varphi (t L)(f )= & {} \sum _{j\ge \nu }\int _{2^{-j-1}}^{2^{-j}}\Big (\frac{s}{t}\Big )^{M} (tL)^{M}\varphi (t L)\psi _{M}( sL)f\frac{ds}{s}\\&+\sum _{j< \nu }\int _{2^{-j-1}}^{2^{-j}} \frac{t}{s} e^{-tL}{\widetilde{\psi }} (s L)f\frac{ds}{s}. \end{aligned}$$

Since $(tL)^{M}\varphi (t L)=(tL)^{M+1}e^{-tL}$ satisfies the Gaussian upper bound (see [13]), we have

$$\begin{aligned} \begin{aligned} |(tL)^{M}\varphi (t L)\psi _{M}( sL)f(y)|\lesssim \int _X\frac{1}{V(y,t^{1/m})}\Big (1+\frac{d(y,z)}{t^{1/m}}\Big )^{-\lambda -N}|\psi _{M}( sL)f(z)|d\mu (z) \end{aligned} \end{aligned}$$

where $N>n$.

It follows that

$$\begin{aligned} \begin{aligned}&\frac{|(t^2L)^{M}\varphi (t L)\psi _{M}(tL)f(y)|}{(1+d(x,y)/t^{1/m})^\lambda }\lesssim \int _X\frac{1}{V(y,t^{1/m})}\Big (1+\frac{d(y,z)}{t^{1/m}}\Big )^{-N}\\&\quad \frac{|\psi _{M}(s L)f(z)|}{(1+d(x,z)/t^{1/m})^\lambda }d\mu (z) \end{aligned} \end{aligned}$$

for $x,y\in X$.

Hence, for $j\ge \nu$, $t\in [2^{-\nu -1},2^{-\nu }]$ and $s\in [2^{-j-1},2^{-j}]$ we have

$$\begin{aligned} \begin{aligned} \frac{|(t^2L)^{M}\varphi (t L)\psi _{M}( sL)f(y)|}{(1+d(x,y)/t^{1/m})^\lambda }&\lesssim 2^{\lambda (j-\nu )}\psi ^*_{M,\lambda }( sL)f(x)\\&\quad \int _X\frac{1}{V(y,t^{1/m})}\Big (1+\frac{d(y,z)}{t^{1/m}}\Big )^{-N}d\mu (y)\\&\lesssim 2^{\lambda (j-\nu )}\psi ^*_{M,\lambda }( sL)f(x). \end{aligned} \end{aligned}$$

Since $\psi \in {\mathscr {S}}_m({\mathbb {R}})$ and ${\text {supp}}\psi \subset [1/2,2]$, $x^{-2m}\psi (x)\in {\mathscr {S}}({\mathbb {R}})$. Using Lemma 2.6 and an argument similar to the above, we obtain, for $j< \nu$, $t\in [2^{-\nu -1},2^{-\nu }]$ and $s\in [2^{-j-1},2^{-j}]$,

$$\begin{aligned} \frac{|e^{-tL}{\widetilde{\psi }}( sL)f(y)|}{(1+d(x,y)/t^{1/m})^\lambda }\lesssim \widetilde{\psi }^*_{\lambda }(s L)f(x). \end{aligned}$$

The above two estimates imply that

$$\begin{aligned} \begin{aligned} |\varphi ^*_\lambda (t L)(f )|&\le \sum _{j\ge \nu }2^{-(j-\nu )(M-\lambda )}\sup _{s\in (2^{-j-1},2^{-j}]}\psi ^*_{M,\lambda }( sL)f \\&\quad + \sum _{j< \nu }2^{-2m(\nu -j)}\sup _{s\in (2^{-j-1},2^{-j}]}\widetilde{\psi }^*_{\lambda }(s L)f. \end{aligned} \end{aligned}$$

This, along with Proposition 2.7, implies that

$$\begin{aligned} \begin{aligned} |\varphi ^*_\lambda (t L)(f )|&\lesssim \sum _{j\ge \nu -1}2^{-(M-\lambda )(j-\nu )}\psi ^*_{j,\lambda }( L)f +\sum _{j< \nu +3}2^{-2m(\nu -j)}\psi ^*_{j,\lambda }( L)f \\&\lesssim \sum _{j\in {\mathbb {Z}}}2^{-2m|\nu -j|} \psi ^*_{j,\lambda }( L)f \end{aligned} \end{aligned}$$

(19)

for all $t\in [2^{-\nu -1},2^{-\nu }]$ and $M>\lambda$.

By Young’s inequality,

$$\begin{aligned} \begin{aligned} \Big (\int _{0}^{\infty } |\varphi ^*_\lambda (t L)(f)|^2\frac{dt}{t}\Big )^{1/2}&\lesssim \Big (\sum _{\nu \in {\mathbb {Z}}}\Big [\sum _{j\in {\mathbb {Z}}}2^{-(2m-\alpha )|\nu -j|} |\psi ^*_{j,\lambda }( L)f|^2\Big )^{1/2} \\&\lesssim \Big (\sum _{j\in {\mathbb {Z}}}|\psi ^*_{j,\lambda }( L)f|^2\Big )^{1/2}. \end{aligned} \end{aligned}$$

Hence, (17) follows from this and Proposition 2.9. The proof of Theorem 2.5 is thus complete. $\square$

3 Estimates for the Schrödinger Group on Hardy Spaces

This section is devoted to the proof of Theorem 1.1. Before embarking on the proof, we need the following result from [8, Proposition 3.4]. Define

$$\begin{aligned} \Vert f\Vert _{{{\textbf {B}}}^s} =\int _{-\infty }^\infty |{\mathcal {F}} f (\tau )|(1+|\tau |)^s d\tau , \end{aligned}$$

where ${\mathcal {F}} f$ denotes the Fourier transform of f.

Lemma 3.1

([8]) Suppose that L is a non-negative self-adjoint operator on $L^2(X)$ and satisfies the Gaussian upper bound (G). Then for every $s\ge 0$, there exists $C>0$ such that for every $j\in {\mathbb {N}}\cup \{0\}$,

$$\begin{aligned} \Vert 1_B F(L) 1_{S_j(B)}\Vert _{2\rightarrow 2}\le C(\root m \of {R} 2^jr_B)^{-s}\Vert \delta _RF\Vert _{{{\textbf {B}}}^s} \end{aligned}$$

for all balls B, and all Borel functions F such that $\mathrm{supp}\,F\subset [-R, R]$, where $\delta _R F(\cdot )=F(R\cdot )$.

We are now ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1:

To prove the theorem, we will use Theorem 2.5 and the standard argument in, for example, [8, 14, 16, 19].

Set $F(\lambda )=(1+\lambda )^{-s}e^{it\lambda }$ with $t>0$ and $s=n(1/p-1/2)$. Let $\varphi$ be a partition of unity. By Theorem 2.5 it suffices to show that there exists $C>0$ such that

$$\begin{aligned} \Vert S_{L,\varphi } a\Vert _p\le C \end{aligned}$$

for every $(p,2,M,L,\epsilon )$ molecule a with $\epsilon >0$ and $M>n(1/p-1/2)+1$.

Suppose a is a such a molecule that is associated with some ball B, and b be a function satisfying $a=L^M b$ from Definition 2.2. Using the following identity

$$\begin{aligned} \begin{aligned} \mathrm{Id}&= (I-e^{-r_B^mL})^M + \sum _{k=1}^m (-1)^{k+1}C^M_k e^{-kr_B^mL}=:(I-e^{-r_B^mL})^M + P(r_B^mL) \end{aligned} \end{aligned}$$

we can write

$$\begin{aligned} \begin{aligned} S_{L,\varphi }(F(L)a)&= S_{L,\varphi }[(I-e^{-r_B^mL})^MF(L)a] + S_{L,\varphi }[(r_B^mL)^{M}P(r_B^{m}L)F(L)r_B^{-mM}b]\\&\lesssim \sum _{k\ge 0}S_{L,\varphi }[(I-e^{-r_B^mL})^MF(L)a_k]\\&\quad + \sum _{k\ge 0}S_{L,\varphi }[(r_B^mL)^{M}P(r_B^{m}L)F(L)r_B^{-mM}b_k]\\&= :\sum _{k\ge 0}E^k_1 + \sum _{k\ge 0}E_{2}^k, \end{aligned} \end{aligned}$$

where $a_k= a.1_{S_k(B)}$ and $b_k= b.1_{S_k(B)}$.

Therefore, it suffices to prove that there exists $\epsilon '>0$ such that

$$\begin{aligned} \Vert E^k_1\Vert _p +\Vert E^k_2\Vert _p \lesssim 2^{-k\epsilon '}(1+t)^{n(1/p-1/2)} \end{aligned}$$

(20)

for each $k\in {\mathbb {N}}\cup \{0\}.$

Estimate for ${ E^k_1:}$ We now show that

$$\begin{aligned} \Vert E^k_1\Vert _p \lesssim 2^{-k\epsilon '}(1+t)^{n(1/p-1/2)}, \ \ \ \ \ \ k\in {\mathbb {N}}\cup \{0\} \end{aligned}$$

(21)

for some $\epsilon '>0$.

For each $k\ge 0$, setting $B_{t,k}=(1+t)2^kB$, we have

$$\begin{aligned} \begin{aligned} \Vert E_1^k\Vert _p^p&= \Vert S_{L,\varphi }[(I-e^{-r_B^mL})^MF(L)a_k]\Vert ^p_{L^p(4B_{t,k})}\\&\quad +\Vert S_{L,\varphi }[(I-e^{-r_B^mL})^MF(L)a_k]\Vert ^p_{L^p(X\backslash 4B_{t,k})}\\&=E^k_{11} + E^k_{12}. \end{aligned} \end{aligned}$$

Using Hölder’s inequality and the $L^2$-boundedness of $S_{L,\varphi }$ we obtain

$$\begin{aligned} \begin{aligned} \Vert E^k_{11}\Vert _p^p&\lesssim V(2^k(1+t)B)^{\frac{2-p}{2}}\Vert S_{L,\varphi }[(I-e^{-r_B^mL})^MF(L)a_k]\Vert ^p_{L^2(4B_{t,k})}\\&\lesssim V(4(1+t)2^kB)^{1-p/2}\Vert a_k\Vert ^p_{2}\\&\lesssim 2^{-\epsilon kp}V(4(1+t)2^kB)^{1-p/2}V(2^kB)^{p/2-1}\\&\lesssim 2^{-\epsilon kp}(1+t)^{n(1-p/2)}, \end{aligned} \end{aligned}$$

where in the last inequality we used (2).

It remains to estimate the second term $E_{12}^k$. To do this, setting

$$\begin{aligned} F_{\ell , r_B}(\lambda )= \varphi _\ell (\lambda )(I-e^{-r_B^m\lambda })^MF(\lambda ), \end{aligned}$$

we then write

$$\begin{aligned} \begin{aligned} \Vert E^k_{12}\Vert _p^p&= \Big \Vert \Big (\sum _{\ell \in {\mathbb {Z}}}|F_{\ell , r_B}(L)a_k|^2\Big )^{1/2} \Big \Vert _{L^p(X\backslash 4B_{t,k})}^p\\&\lesssim \Big \Vert \sum _{\ell \in {\mathbb {Z}}}|F_{\ell , r_B}(L)a_k| \Big \Vert _{L^p(X\backslash 4B_{t,k})}^p\\&\lesssim \sum _{\ell \in {\mathbb {Z}}}\Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(X\backslash 4B_{t,k})}^p\\&= \sum _{\ell \ge \ell _0}\ldots + \sum _{\ell < \ell _0}\ldots =: F^k_1 + F^k_2, \end{aligned} \end{aligned}$$

(22)

where $\ell _0$ is the largest integer such that $2^{\ell _0(m-1)/m}\le 2^{k}r_B$.

We estimate $F^k_1$ first. To do this, we write

$$\begin{aligned} \begin{aligned} F_1^k&=\sum _{\ell \ge \ell _0}\Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(X\backslash 4B_{t,k})}^p\\&\le \sum _{\ell \ge \ell _0}\sum _{j\ge \frac{\ell -\ell _0}{2}}\Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(S_j(B_{t,k}))}^p +\sum _{\ell \ge \ell _0} \Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(B(x_B, 2^{\ell (m-1)/m}(1+t)))}^p\\&=: F^k_{11} + F^k_{12}. \end{aligned} \end{aligned}$$

By Hölder’s inequality and property (ii) of Definition 2.2 we obtain

$$\begin{aligned} \begin{aligned} F^k_{12}&\lesssim \sum _{\ell \ge \ell _0} V(B(x_B, 2^{\ell (m-1)/m}(1+t)))^{1-p/2}\Vert F_{\ell , r_B}(L)a\Vert _{L^2(B(x_B, 2^{\ell (m-1)/m}(1+t)))}^p\\&\lesssim \sum _{\ell \ge \ell _0} V(B(x_B, 2^{\ell (m-1)/m}(1+t)))^{1-p/2}\Vert F_{\ell , r_B}\Vert ^p_{\infty }\Vert a_k\Vert ^p_2. \end{aligned} \end{aligned}$$

This, along with the fact that $\Vert F_{\ell , r_B}\Vert _{\infty }\lesssim \min \{1, (2^\ell r_B^m)^M\} 2^{-\ell n(1/p-1/2)}$, implies that

$$\begin{aligned} \begin{aligned} F^k_{12}&\lesssim \sum _{\ell \ge \ell _0} 2^{-kp\epsilon }V(B(x_B, 2^{\ell (m-1)/m}(1+t)))^{1-p/2}\\&\quad \min \{1, (2^\ell r_B^m)^{pM}\} 2^{-\ell n(1-p/2)}V(2^kB)^{1-p/2}. \end{aligned} \end{aligned}$$

On the other hand, since $2^{\ell _0(m-1)/m}\sim 2^{k}r_B$, we have, for $\ell \ge \ell _0$,

$$\begin{aligned} \begin{aligned} \frac{V(x_B, 2^{\ell (m-1)/m}(1+t))}{V(2^kB)}&\sim \frac{V(x_B, 2^{(\ell -\ell _0)(m-1)/m}(1+t)2^kr_B)}{V(2^kB)}\\&\lesssim [2^{(\ell -\ell _0)(m-1)/m}(1+t)]^{n}\\&\sim (1+t)^n[2^{\ell (m-1)/m}(2^kr_B)^{-1}]^n\\&\lesssim (1+t)^n[2^{\ell (m-1)/m} r_B ^{-1}]^n \end{aligned} \end{aligned}$$

We thus deduce that

$$\begin{aligned} \begin{aligned} F^k_{12}&\lesssim 2^{-kp\epsilon }(1+t)^{n(1-p/2)}\sum _{\ell \ge \ell _0} \min \{1, (2^\ell r_B^m)^{pM}\} 2^{-\ell n(1-p/2)} [2^{\ell (m-1)/m} r_B^{-1}]^{n(1-p/2)}\\&\lesssim 2^{-kp\epsilon }(1+t)^{n(1-p/2)}\sum _{\ell \ge \ell _0} \min \{1, (2^\ell r_B^m)^{pM}\} [2^{\ell /m}r_B]^{-n(1-p/2)}\\&\lesssim 2^{-kp\epsilon }(1+t)^{n(1-p/2)}. \end{aligned} \end{aligned}$$

We now take care of $F^k_{11}$. For $\ell \ge \ell _0$ and $j\ge \frac{(\ell -\ell _0)(m-1)}{m}$ we have

$$\begin{aligned} \begin{aligned} F^k_{11}&\le \sum _{\ell \ge \ell _0}\sum _{j\ge \frac{\ell -\ell _0}{2}}\Vert F_{\ell ,r_B}a_k \Vert ^p_{L^2(S_j(B_{t,k}))} V(2^jB_{t,k})^{1-p/2}\\&\lesssim \sum _{\ell \in {\mathbb {N}}}\sum _{j\ge \frac{(\ell -\ell _0)(m-1)}{m}}\Vert F_{r_B,\ell }(L)\Vert ^p_{L^2(S_k(B))\rightarrow L^2(S_j(B_{t,k}))} \Vert a_k \Vert ^p_{2}V(2^jB_{t,k})^{1-p/2}\\&\lesssim \sum _{\ell \in {\mathbb {N}}}\sum _{j\ge \frac{(\ell -\ell _0)(m-1)}{m}}2^{-kp\epsilon }\Vert F_{r_B,\ell }(L)\Vert ^p_{L^2(S_k(B))\rightarrow L^2(S_j(B_{t,k}))} V(2^kB)^{p/2-1}V(2^jB_{t,k})^{1-p/2}. \end{aligned} \end{aligned}$$

This, in combination with the doubling property (2), yields that

$$\begin{aligned} \begin{aligned} F^k_{11}&\lesssim \sum _{\ell \ge \ell _0}\sum _{j\ge \frac{(\ell -\ell _0)(m-1)}{m}} 2^{-kp\epsilon }[2^{j}(1+t)]^{n(1-p/2)}\Vert F_{r_B,\ell }(L)\Vert ^p_{L^2(S_k(B))\rightarrow L^2(S_j(B_{t,k}))}. \end{aligned}\nonumber \\ \end{aligned}$$

(23)

By Lemma 3.1, for $\alpha =n(1/p-1/2)+\theta$ with $\theta \in (0,\epsilon )$, we have

$$\begin{aligned} \Vert F_{r_B,\ell }(L)\Vert _{L^2(S_k(B))\rightarrow L^2(S_j(B_{t,k}))}\lesssim (2^{\ell /m} 2^j(1+t)2^kr_B)^{-\alpha } \Vert \delta _{2^\ell }F_{r_B,\ell }\Vert _{{{\textbf {B}}}^\alpha }. \end{aligned}$$

(24)

We claim that for $\alpha >0$,

$$\begin{aligned} \Vert \delta _{2^\ell }F_{r_B,\ell }\Vert _{{{\textbf {B}}}^\alpha }\lesssim \max \{1,2^{(\alpha -n(1/p-1/2))\ell }\} (1+t)^\alpha \min \{1, (2^\ell r_B^m)^M\}. \end{aligned}$$

(25)

To show this, as in [8], we write

$$\begin{aligned} \begin{aligned} \Vert \delta _{2^\ell }F_{r_B,\ell }\Vert _{{{\textbf {B}}}^\alpha }&=\Vert \varphi (\lambda )(I-e^{-2^\ell r_B^m\lambda })^MF(2^\ell \lambda )\Vert _{{{\textbf {B}}}^\alpha } \\&\lesssim \Vert \varphi (\lambda )(I-e^{-2^\ell r_B^m\lambda })^M\Vert _{{{\textbf {B}}}^\alpha }\Vert \varphi (\lambda )F(2^\ell \lambda )\Vert _{{{\textbf {B}}}^\alpha }. \end{aligned} \end{aligned}$$

It is easy to see that

$$\begin{aligned} \Vert \varphi (\lambda )(I-e^{-2^\ell r_B^m\lambda })^M\Vert _{{{\textbf {B}}}^\alpha }\lesssim \min \{1, (2^\ell r_B^m)^M\}. \end{aligned}$$

On the other hand,

$$\begin{aligned} {\mathcal {F}}(\varphi (\lambda )F(2^\ell \lambda ))(\tau ) =\int _{-\infty }^\infty \varphi (\lambda )\frac{e^{i(2^\ell t -\tau )\lambda }}{(1+2^\ell \lambda )^s}d\lambda , \end{aligned}$$

where $s=n(1/p-1/2)$. Next, from integration by parts, we have, for each $N\in {\mathbb {N}}$,

$$\begin{aligned} {\mathcal {F}}(\varphi (\lambda )F(2^\ell \lambda ))(\tau )\le C_N \min \{1, 2^{-\ell s}\}(1+|2^\ell t -\tau |)^{-N}. \end{aligned}$$

As a consequence,

$$\begin{aligned} \begin{aligned} \Vert \varphi (\lambda )F(2^\ell \lambda )\Vert _{{{\textbf {B}}}^\alpha }&\lesssim \min \{1, 2^{-\ell s}\}\int _{{\mathbb {R}}}(1+|2^\ell t -\tau |)^{-N}(1+|\tau |)^\alpha d\tau \\&\lesssim \max \{1, 2^{(\alpha -s)\ell }\}(1+t)^\alpha , \end{aligned} \end{aligned}$$

which proves (25).

Substituting (25) into (24) we then obtain

$$\begin{aligned}&\Vert F_{r_B,\ell }(L)\Vert _{L^2(S_k(B))\rightarrow L^2(S_j(B_{t,k}))}\lesssim (2^{\ell /m} 2^j2^kr_B)^{-\alpha }\\&\quad \max \{1,2^{(\alpha -n(1/p-1/2))\ell }\} \min \{1, (2^\ell r_B^m)^M\}. \end{aligned}$$

This, in combination with (23), implies that for $\alpha =n(1/p-1/2)+\theta$ with $\theta \in (0,\epsilon )$,

$$\begin{aligned} \begin{aligned} F_{11}^k&\lesssim \sum _{\ell \ge \ell _0}\sum _{j\ge \frac{(\ell -\ell _0)(m-1)}{m}} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} 2^{-jp\theta }(2^{\ell /m}2^k r_B)^{-\alpha p} \\&\quad \max \{1,2^{ \ell \theta p}\} \min \{1, (2^\ell r_B^m)^{pM}\}\\&=: \sum _{\ell \ge \ell _0: \;\ell < 0 }\ldots + \sum _{\ell \ge \ell _0: \;\ell \ge 0 }\ldots . \end{aligned} \end{aligned}$$

For the first sum, we have

$$\begin{aligned} \begin{aligned} \sum _{\ell \ge \ell _0: \;\ell < 0 }\ldots&\lesssim \sum _{\ell \ge \ell _0} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} (2^{\ell /m}2^k r_B)^{-\alpha p} \min \{1, (2^\ell r_B^m)^{pM}\}\\ {}&\lesssim \sum _{\ell \ge \ell _0} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} (2^{\ell /m}r_B)^{-\alpha p} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim 2^{-kp\epsilon }(1+t)^{n(1-p/2)}, \end{aligned} \end{aligned}$$

as long as $M>\alpha$.

For the contribution of the second sum we have

$$\begin{aligned} \begin{aligned} \sum _{\ell \ge \ell _0: \;\ell \ge 0 }\ldots&\lesssim \sum _{\ell \ge \ell _0: \;\ell \ge 0 }2^{-kp\epsilon }(1+t)^{n(1-p/2)} 2^{-\frac{p\theta (\ell -\ell _0)(m-1)}{m}}(2^{\ell /m}2^k r_B)^{-\alpha p} 2^{ \ell \theta p} \\&\quad \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{\ell \ge \ell _0} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} [2^{\ell (m-1)/m}(2^kr_B)^{-1}]^{-\theta p}(2^{\ell /m} 2^kr_B)^{-\alpha p}2^{\theta p\ell }\\&\quad \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{\ell \ge \ell _0\vee 0}2^{-kp(\epsilon -\theta )}(1+t)^{n(1-p/2)} (2^{\ell /m} r_B)^{-p(\alpha -\theta ) } \\&\quad \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim 2^{-kp(\epsilon -\theta )}(1+t)^{n(1-p/2)}, \end{aligned} \end{aligned}$$

where we used the fact that $2^{\ell _0(m-1)/m}\sim 2^kr_B$ in the second inequality. Therefore, it holds that

$$\begin{aligned} F^k_{11}\lesssim 2^{-k\epsilon '}(1+t)^{n(1-p/2)} \end{aligned}$$

for some $\epsilon '>0$.

Collecting the estimates of $F^k_{11}$ and $F^k_{12}$, we arrive at

$$\begin{aligned} F_1^k \lesssim 2^{-k\epsilon '}(1+t)^{n(1-p/2)} \end{aligned}$$

for some $\epsilon '>0$.

It remains to handle the term $F^k_2$. Indeed, we have

$$\begin{aligned} \begin{aligned} F^k_2=\sum _{\ell<\ell _0}\Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(X\backslash 4B_{t,k})}^p =\sum _{\ell <\ell _0}\sum _{j\ge 3}\Vert F_{\ell , r_B}(L)a_k\Vert _{L^p(S_j(B_{t,k}))}^p. \end{aligned} \end{aligned}$$

Arguing similarly to the estimate of $F_{11}$, we have

$$\begin{aligned} \begin{aligned} F^k_{2}&\lesssim \sum _{\ell< \ell _0}\sum _{j\ge 3} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} 2^{-jp\theta }(2^{\ell /m}2^k r_B)^{-\alpha p}\max \{1,2^{\theta \ell p}\} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{\ell< \ell _0}2^{-kp\epsilon } (1+t)^{n(1-p/2)} (2^{\ell /m} 2^kr_B)^{-\alpha p}\max \{1,2^{\theta \ell p}\} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{\ell< 0}2^{-kp\epsilon } (1+t)^{n(1-p/2)} (2^{\ell /m} r_B)^{-\alpha p} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\ \ \ + \sum _{0<\ell < \ell _0}2^{-kp\epsilon } (1+t)^{n(1-p/2)} (2^{\ell /m} 2^kr_B)^{-\alpha p} 2^{\theta \ell p} \min \{1, (2^\ell r_B^m)^{pM}\}. \end{aligned} \end{aligned}$$

It is clear that

$$\begin{aligned} \sum _{\ell < 0}2^{-kp\epsilon } (1+t)^{n(1-p/2)} (2^{\ell /m} r_B)^{-\alpha p} \min \{1, (2^\ell r_B^m)^{pM}\}\lesssim 2^{-kp\epsilon } (1+t)^{n(1-p/2)}, \end{aligned}$$

as long as $M>\alpha$.

For the second sum, we have

$$\begin{aligned} \begin{aligned} \sum _{0<\ell< \ell _0}2^{-kp\epsilon }&(1+t)^{n(1-p/2)} (2^{\ell /m} 2^kr_B)^{-\alpha p} 2^{\theta \ell p} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{0<\ell< \ell _0}2^{-kp\epsilon } (1+t)^{n(1-p/2)} (2^{\ell /m} 2^kr_B)^{-\alpha p} 2^{\theta \ell p} \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{0<\ell < \ell _0} 2^{-kp\epsilon }(1+t)^{n(1-p/2)} [2^{\ell (m-1)/m}(2^kr_B)^{-1}]^{-\theta p}(2^{\ell /m} 2^kr_B)^{-\alpha p}2^{\theta p\ell } \\&\quad \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim \sum _{\ell > 0}2^{-kp(\epsilon -\theta )}(1+t)^{n(1-p/2)} (2^{\ell /m} r_B)^{-p(\alpha -\theta ) } \min \{1, (2^\ell r_B^m)^{pM}\}\\&\lesssim 2^{-kp(\epsilon -\theta )}(1+t)^{n(1-p/2)} \end{aligned} \end{aligned}$$

where in the second inequality we used the fact that

$$\begin{aligned} 2^{\ell (m-1)/m}(2^kr_B)^{-1}\le 2^{\ell _0(m-1)/m}(2^kr_B)^{-1} \le 1, \end{aligned}$$

along with $2^{-\ell /2}r_B\ge 2^{-\ell _0/2}r_B\ge 1$. Therefore we may conclude

$$\begin{aligned} F^k_2\lesssim 2^{-kp\epsilon '}(1+t)^{n(1-p/2)} \end{aligned}$$

for some $\epsilon '>0$. and this, along with the estimate of $F_1^k$ and (22), implies that

$$\begin{aligned} E^k_{12}\lesssim 2^{-kp\epsilon '}(1+t)^{n(1-p/2)}, \end{aligned}$$

completing the proof of (21).

Estimate for${ E^k_2:}$ We now show that

$$\begin{aligned} \Vert E^k_2\Vert _p \lesssim 2^{-k\epsilon '}(1+t)^{n(1/p-1/2)}, \ \ \ \ \ \ k=0,1,2,\ldots \end{aligned}$$

(26)

for some $\epsilon '>0$.

Set $G_{\ell , r_B}(\lambda )= \varphi _\ell (\lambda )(r_B^m\lambda )^M P(r_B^m\lambda )F(\lambda )$. Then we have

$$\begin{aligned} \Vert G_{\ell , r_B}\Vert _{\infty } \lesssim \min \{(2^\ell r_B^m)^{-M}, (2^\ell r_B^m)^M\} 2^{-\ell n(1/p-1/2)}. \end{aligned}$$

Arguing similarly to (25), we see that

$$\begin{aligned} \Vert \delta _{2^\ell }G_{r_B,\ell }\Vert _{{{\textbf {B}}}^\alpha }\lesssim \max \{1,2^{(\alpha -n(1/p-1/2))\ell }\} (1+t)^\alpha \min \{(2^\ell r_B^m)^{-M}, (2^\ell r_B^m)^M\}, \end{aligned}$$

as along as $M>n(1/p-1/2)+1>\alpha =n(1/p-1/2)+\theta$ with $\theta \in (0,\epsilon )$.

At this stage, proceed along the same lines as in the proof of (21) to obtain (26). This completes the proof of (20), and thus of Theorem 1.1. $\square$

4 Some Applications

Our framework is sufficiently general to include a large variety of applications; in this section we survey a few of the more interesting cases.

4.1 Laplacian-Like Operators

Let us here consider two additional conditions on the operator L: Hölder regularity: there exists $\delta _{0} \in (0,1]$ so that whenever $d(x,{{\bar{x}}})< t^{1/m}$ we have

$$\begin{aligned} | e^{-tL}(x,y) - e^{-tL}({{\bar{x}}},y) | \lesssim \Big (\frac{d(x,\bar{x})}{t^{1/m}}\Big )^{\delta _0}\frac{1}{{V\left( {x,\sqrt{t} } \right) }}\exp \Big (-\frac{d(x,y)^{m/(m-1)}}{ct^{1/(m-1)}}\Big ), \end{aligned}$$

(H)

Conservation: for all $y \in X$ and $t>0$ we have

$$\begin{aligned} \int _X {e^{-tL}\left( {x,y} \right) d \mu \left( x \right) } = 1. \end{aligned}$$

(C)

Examples of typical operators satisfying (G), (H) and (C) include the 2k-higher order elliptic operator in divergence form with smooth coefficients, the homogeneous sub-Laplacian on a homogeneous group and the Laplace-Beltrami operator on a doubling manifold admits the Poincaré’s inequality as in [1].

We recall the definition of the Hardy spaces $H^p (X)$ for $\frac{n}{n+1}<p\le 1$ from [10]. For $0<p\le 1$, we say that a function a is a (2, p) atom if there exists a ball B such that

• (i) supp $a\subset B$;

• (ii) $\Vert a\Vert _{L^2}\le V(B)^{1/2-1/p}$;

• (iii) $\int a(x)\mu (x)=0$.

For $p=1$ the atomic Hardy space $H^1$ is defined as follows. We say that a function $f\in H^1 (X)$, if $f\in L^1$ and there exist a sequence $(\lambda _j)_{j\in {\mathbb {N}}}\in l^1$ and a sequence of (2, 1)-atoms $(a_j)_{j\in {\mathbb {N}}}$ such that $f=\sum _{j}\lambda _ja_j$. We set

$$\begin{aligned} \Vert f\Vert _{H^1 }=\inf \{\sum _{j}|\lambda _j|: f=\sum _{j}\lambda _ja_j\}. \end{aligned}$$

For $0<p<1$, as in [10], we need to introduce the Lipschitz space ${\mathfrak {L}}_\alpha$. We say that the function $f\in {\mathfrak {L}}_\alpha$ if there exists a constant $c>0$, such that

$$\begin{aligned} |f(x)-f(y)|\le c|B|^{\alpha } \end{aligned}$$

for all ball B and $x, y\in B$. The best constant c above can be taken to be the norm of f and is denoted by $\Vert f\Vert _{{\mathfrak {L}}_\alpha }$.

Now let $0<p<1$ and $\alpha =1/p-1$. We say that a function $f\in H^p (X)$, if $f\in ({\mathfrak {L}}_{\alpha })^*$ and there is a sequence $(\lambda _j)_{j\in {\mathbb {N}}}\in l^p$ and a sequence of (2, p)-atoms $(a_j)_{j\in {\mathbb {N}}}$ such that $f=\sum _{j}\lambda _ja_j$. Furthermore, we set

$$\begin{aligned} \Vert f\Vert _{H^p }=\inf \Big \{\Big (\sum _{j}|\lambda _j|^p\Big )^{1/p}: f=\sum _{j}\lambda _ja_j\Big \}. \end{aligned}$$

Note that when $0<p<1$, the quantity $\Vert \cdot \Vert _{H^p }$ is not the norm but $d(f,g):=\Vert f-g\Vert _{H^p }$ forms a metric.

Lemma 4.1

Let L be a nonnegative self-adjoint operator satisfying (G), (H) and (C). Then $H^p_{L}(X)\equiv H^p (X)$ for $\frac{n}{n+\delta _0}<p\le 1$.

Proof

The proof of this lemma is fairly standard but we could not find in the existing literature. Thus for the reader’s benefit, we will provide a sketch of its proof. Firstly, arguing similarly to Lemma 9.1 in [16], we have that every $(p,2,M,L,\epsilon )$ molecule m satisfies

$$\begin{aligned} \int _X m(x)d\mu (x) = 0. \end{aligned}$$

Therefore, by the argument as in the proof of [3, Proposition 4.16] we can show that $\Vert m\Vert _{H^p (X)}\lesssim ~1$ uniformly for every $(p,2,M,L,\epsilon )$ molecule m with $M>\frac{n(2-p)}{2mp}$, $\epsilon >n$ and $\frac{n}{n+1}<p\le 1$. It follows immediately that $H^p_{L}(X)\subset H^p (X)$.

Conversely, if a is a (2, p) atom with $\frac{n}{n+\delta _0}<p\le 1$, then by a standard argument we can show that $\Vert {\mathcal {A}} a\Vert _p\lesssim 1$, where ${\mathcal {A}}$ is the square function defined by (11). It follows $\Vert a\Vert _{H^p_L}\lesssim 1$ and this gives $H^p(X)\subset H^p_{L}(X)$. The proof is thus complete. $\square$

From Lemma 4.1 and Theorem 1.1 we deduce the following.

Theorem 4.2

Let L be a nonnegative self-adjoint operator satisfying (G), (H) and (C). Then for each $\frac{n}{n+\delta _0}<p\le 1$ and $s=n(1/p-1/2)$ we have

$$\begin{aligned} \Vert (I+L)^{-s}e^{itL}f\Vert _{H^p(X)} \lesssim (1+|t|)^{s}\Vert f\Vert _{H^p(X)}, \qquad t\in {\mathbb {R}}. \end{aligned}$$

4.2 Hermite Operators

Let ${\mathcal {H}}=-\Delta +|x|^2$ be the Hermite operator on ${\mathbb {R}}^n$ with $n\ge 1$. Let $p_t(x,y)$ denote the kernel of the semigroup $e^{-t{\mathcal {H}}}$. It is clear that $p_t(x,y)$ enjoys the Gaussian upper bound (G). Moreover we have an explicit representation for the kernel $p_t(x,y)$:

$$\begin{aligned} p_t(x,y)=\frac{1}{\pi ^{n/2}}\Big (\frac{e^{-2t}}{1-e^{-4t}}\Big )^{n/2}\exp \Big (-\frac{1}{4}\frac{1+e^{-2t}}{1-e^{-2t}}|x-y|^2-\frac{1}{4}\frac{1-e^{-2t}}{1+e^{-2t}}|x+y|^2\Big ) \end{aligned}$$

for all $t>0$ and $x,y\in {\mathbb {R}}^n$. This representation is well known – see for example [26].

Let $\rho (x)=\min \{1, |x|^{-1}\}$ for $x\in {\mathbb {R}}^n$. Let $p\in (0,1]$. A function a is called a $(p, \infty , \rho )$-atom associated to the ball $B(x_0,r)$ if

• (i) $\mathrm{supp}\, a\subset B(x_0,r)$;

• (ii) $\Vert a\Vert _{L^\infty }\le |B(x_0,r)|^{-1/p}$;

• (iii) $\displaystyle \int x^\alpha a(x)dx =0$ for all $|\alpha |\le \lfloor n(1/p-1)\rfloor$ if $r<\rho (x_0)/4$.

The Hardy space $H^p_{at, \rho }({\mathbb {R}}^n)$ is then defined to be the set of all functions f which can be expressed in the form $f=\sum _j \lambda _j a_j$ where $(\lambda _j)_j\in \ell ^p$ and $a_j$ are $(p,\infty , \rho )$-atoms. Its norm is given by

$$\begin{aligned} \Vert f\Vert _{H^p_{at, \rho }({\mathbb {R}}^n)}:= \inf \Bigl \{\Big (\sum _j|\lambda _j|^p\Big )^{1/p}: f=\sum _j \lambda _j a_j\Bigr \}, \end{aligned}$$

where the infimum is taken over all possible atomic decompositions of f. From the definition, it is obvious that $H^p({\mathbb {R}}^n)\subsetneqq H^p_{at, \rho }({\mathbb {R}}^n)$ for all $p\in (0,1]$; more importantly, we have $H^p_{at, \rho }({\mathbb {R}}^n)\equiv H^p_{{\mathcal {H}}}({\mathbb {R}}^n)$ for all $0<p\le 1$ (see for instance [4, 15]), thus the Hardy space associated to the Hermite operator contains the standard Hardy spaces $H^p({\mathbb {R}}^n)$.

Theorem 4.3

Let ${\mathcal {H}}=-\Delta +|x|^2$ be the Hermite operator on ${\mathbb {R}}^n$ with $n\ge 1$. Then for each $0<p\le 1$ and $s=n(1/p-1/2)$ we have

$$\begin{aligned} \Vert {\mathcal {H}}^{-s}e^{it{\mathcal {H}}}f\Vert _{H^p_{at,\rho }({\mathbb {R}}^n)} \lesssim \Vert f\Vert _{H^p_{at, \rho }({\mathbb {R}}^n)}, \qquad t\in {\mathbb {R}}. \end{aligned}$$

Proof

Since ${\mathcal {H}}$ is a nonnegative self-adjoint operator and satisfies the Gaussian upper bound (G) with $m=2$, then by Theorem 1.1 and the coincidence $H^p_{at, \rho }({\mathbb {R}}^n)\equiv H^p_{{\mathcal {H}}}({\mathbb {R}}^n)$ for every $0<p\le 1$, we have

$$\begin{aligned} \Vert (I+{\mathcal {H}})^{-s}e^{it{\mathcal {H}}}f\Vert _{H^p_{at,\rho }(\mathbb R^n)} \lesssim (1+|t|)^s\Vert f\Vert _{H^p_{at, \rho }({\mathbb {R}}^n)}. \end{aligned}$$

On the other hand, it is well-known that the spectrum of ${\mathcal {H}}$ is contained in $[1,\infty )$ (see [26]). It follows that

$$\begin{aligned} \Vert {\mathcal {H}}^{-s}e^{it{\mathcal {H}}}f\Vert _{H^p_{at,\rho }({\mathbb {R}}^n)} \lesssim (1+|t|)^s\Vert f\Vert _{H^p_{at, \rho }({\mathbb {R}}^n)}. \end{aligned}$$

It is also well-known that

$$\begin{aligned} e^{it{\mathcal {H}}}f =\int _{{\mathbb {R}}^n} \exp \bigg (2\frac{(|x|^2+|y|^2)\cos 2t -2\langle x,y\rangle }{i\sin 2t}\bigg )f(y)dy, \end{aligned}$$

which implies that the flow $e^{it{\mathcal {H}}}$ is time-periodic with the period $T=2\pi$. Therefore,

$$\begin{aligned} \Vert {\mathcal {H}}^{-s}e^{it{\mathcal {H}}}f\Vert _{H^p_{at,\rho }({\mathbb {R}}^n)} \lesssim \Vert f\Vert _{H^p_{at, \rho }({\mathbb {R}}^n)}, \end{aligned}$$

which completes our proof. $\square$

Note that in [25], Thangavelu proved that for each $t>0$,

$$\begin{aligned} \Vert {\mathcal {H}}^{-s}e^{it{\mathcal {H}}}f\Vert _{L^1 ({\mathbb {R}}^n)} \le C_t \Vert f\Vert _{H^1({\mathbb {R}}^n)}, \end{aligned}$$

where $C_t$ is a constant dependent on t. In comparison, our result in Theorem 4.3 improves upon the result in [25] significantly, even in the case $p=1$ since $H^1_{at, \rho }({\mathbb {R}}^n)\subset H^1({\mathbb {R}}^n) \subset L^1({\mathbb {R}}^n)$. Moreover, the constant in Theorem 4.3 is independent of t.

We now consider an application of Theorem 4.3 to the Schödinger equation

$$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u + {\mathcal {H}}{\mathcal {u}} =0,\\ u(x,0)=f. \end{array}\right. } \end{aligned}$$

(27)

For each $0<p\le 1$ and $s>0$ we define the Hardy-Sobolev space $\dot{H}^{p,s}_{{\mathcal {H}}}({\mathbb {R}}^n)$ associated to ${\mathcal {H}}$ by

$$\begin{aligned} \Vert f\Vert _{\dot{H}^{p,s}_{{\mathcal {H}}}({\mathbb {R}}^n)} =\Vert \mathcal H^{s/2}f\Vert _{H^p_{at,\rho }({\mathbb {R}}^n)}. \end{aligned}$$

It is well-known (see [2]) that

$$\begin{aligned} \Vert f\Vert _{\dot{H}^{p,s}_{{\mathcal {H}}}({\mathbb {R}}^n)}= \Big \Vert \Big [\sum _j (2^{-js}|\psi _j(\sqrt{\mathcal {H}}))^{2}f|^2\Big ]^{1/2}\Big \Vert _p. \end{aligned}$$

This means that similar to the classical setting, the Hardy-Sobolev space $\dot{H}^{p,s}_{{\mathcal {H}}}({\mathbb {R}}^n)$ can be viewed as a Triebel–Lizorkin type space $\dot{F}^{s}_{p,2}({\mathbb {R}}^n)$ that is associated to ${\mathcal {H}}$.

Returning to the equation (27), we note that its solution can be formally written as $u=e^{it{\mathcal {H}}}f$. From Theorem 4.3 we can then deduce the following result.

Corollary 4.4

Suppose u is a solution to (27) and let $0<p\le 1$. If the initial data $f\in \dot{H}^{p,s}_{\mathcal H}({\mathbb {R}}^n)$ with $s=n(1/p-1/2)$, then we have

$$\begin{aligned} \Vert u\Vert _{H^p_{at,\rho }({\mathbb {R}}^n)} \lesssim \Vert f\Vert _{H^{p,2s}_{\mathcal H}({\mathbb {R}}^n)}. \end{aligned}$$