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 \(s=n(1/p-1/2)\),
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.
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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
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
for all \(\lambda \ge 1, x\in X\) and \(r>0\); and that
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
for all \(x,y\in X\) and \(t>0\).
Through spectral theory we can define the Schrödinger group, for \(t\in {\mathbb {R}}\),
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 \(t\in {\mathbb {R}}\),
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
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
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]:
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
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
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
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
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,
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
Let \(w\in A_\infty \) and \(0<r<\infty \). The Hardy–Littlewood maximal function \({\mathcal {M}}_{r}\) is defined by
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
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 \(\{f_\nu \}\),
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
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
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
with the norm given by
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
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
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
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
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
Obviously, we have
Similarly, for \(s, \lambda >0\) we set
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
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:
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:
Proof
Since \(|\psi _{j}(\sqrt{L})f|\lesssim \psi ^*_{j,\lambda }({L})f\), it suffices to prove that
Choose \(r<p\) so that \(\lambda >n/r\). Then applying Proposition 2.8 and Lemma 2.1 we have
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
for all \(\lambda >0\) and \(d(x,y)<t^{1/m}\). Therefore,
Since
it suffices to prove that
where \(\psi \) is a partition of unity.
By the spectral theory,
where \(c_\psi =\Big [\int _0^\infty \psi (s )\frac{ds}{s}\Big ]^{-1}\). Hence it follows that for every \(t>0\),
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
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
Since \((tL)^{M}\varphi (t L)=(tL)^{M+1}e^{-tL}\) satisfies the Gaussian upper bound (see [13]), we have
where \(N>n\).
It follows that
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
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 \(s\in [2^{-j-1},2^{-j}]\),
The above two estimates imply that
This, along with Proposition 2.7, implies that
for all \(t\in [2^{-\nu -1},2^{-\nu }]\) and \(M>\lambda \).
By Young’s inequality,
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
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 \(j\in {\mathbb {N}}\cup \{0\}\),
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
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
we can write
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
for each \(k\in {\mathbb {N}}\cup \{0\}.\)
Estimate for \({ E^k_1:}\) We now show that
for some \(\epsilon '>0\).
For each \(k\ge 0\), setting \(B_{t,k}=(1+t)2^kB\), we have
Using Hölder’s inequality and the \(L^2\)-boundedness of \(S_{L,\varphi }\) we obtain
where in the last inequality we used (2).
It remains to estimate the second term \(E_{12}^k\). To do this, setting
we then write
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
By Hölder’s inequality and property (ii) of Definition 2.2 we obtain
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
On the other hand, since \(2^{\ell _0(m-1)/m}\sim 2^{k}r_B\), we have, for \(\ell \ge \ell _0\),
We thus deduce that
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
This, in combination with the doubling property (2), yields that
By Lemma 3.1, for \(\alpha =n(1/p-1/2)+\theta \) with \(\theta \in (0,\epsilon )\), we have
We claim that for \(\alpha >0\),
To show this, as in [8], we write
It is easy to see that
On the other hand,
where \(s=n(1/p-1/2)\). Next, from integration by parts, we have, for each \(N\in {\mathbb {N}}\),
As a consequence,
which proves (25).
Substituting (25) into (24) we then obtain
This, in combination with (23), implies that for \(\alpha =n(1/p-1/2)+\theta \) with \(\theta \in (0,\epsilon )\),
For the first sum, we have
as long as \(M>\alpha \).
For the contribution of the second sum we have
where we used the fact that \(2^{\ell _0(m-1)/m}\sim 2^kr_B\) in the second inequality. Therefore, it holds that
for some \(\epsilon '>0\).
Collecting the estimates of \(F^k_{11}\) and \(F^k_{12}\), we arrive at
for some \(\epsilon '>0\).
It remains to handle the term \(F^k_2\). Indeed, we have
Arguing similarly to the estimate of \(F_{11}\), we have
It is clear that
as long as \(M>\alpha \).
For the second sum, we have
where in the second inequality we used the fact that
along with \(2^{-\ell /2}r_B\ge 2^{-\ell _0/2}r_B\ge 1\). Therefore we may conclude
for some \(\epsilon '>0\). and this, along with the estimate of \(F_1^k\) and (22), implies that
completing the proof of (21).
Estimate for\({ E^k_2:}\) We now show that
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
Arguing similarly to (25), we see that
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
Conservation: for all \(y \in X\) and \(t>0\) we have
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
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
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
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
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
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 \(p_t(x,y)\):
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
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
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
On the other hand, it is well-known that the spectrum of \({\mathcal {H}}\) is contained in \([1,\infty )\) (see [26]). It follows that
It is also well-known that
which implies that the flow \(e^{it{\mathcal {H}}}\) is time-periodic with the period \(T=2\pi \). Therefore,
which completes our proof. \(\square \)
Note that in [25], Thangavelu proved that for each \(t>0\),
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
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
It is well-known (see [2]) that
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
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The authors thank the referees for their valuable input and suggestions. T.A. Bui was supported by the Australian Research Council through the Grant DP220100285.
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Bui, T.A., Ly, F.K. Sharp Estimates for Schrödinger Groups on Hardy Spaces for \(0<p\le 1\). J Fourier Anal Appl 28, 70 (2022). https://doi.org/10.1007/s00041-022-09964-0
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DOI: https://doi.org/10.1007/s00041-022-09964-0