Sharp Estimates for Schrödinger Groups on Hardy Spaces for 0<p≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\le 1$$\end{document}

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on L2(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(X)$$\end{document} and suppose that the kernel of e-tL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{-tL}$$\end{document} satisfies a Gaussian upper bound. This paper shows that for 0<p≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p\le 1$$\end{document} and s=n(1/p-1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=n(1/p-1/2)$$\end{document}, ‖(I+L)-seitLf‖HLp(X)≲(1+|t|)s‖f‖HLp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\Vert (I+L)^{-s}e^{itL}f\Vert _{H^p_L(X)} \lesssim (1+|t|)^{s}\Vert f\Vert _{H^p_L(X)} \end{aligned}$$\end{document}for all t∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in {\mathbb {R}}$$\end{document}, where HLp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^p_L(X)$$\end{document} is the Hardy space associated to L. This recovers the classical results in the particular case when L=-Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L=-\Delta $$\end{document} and extends a number of known results.

measure μ(B(x, r )). In this paper we assume that the measure μ satisfies the doubling condition: there exists a constant C > 0 such that for all x ∈ 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 λ ≥ 1, x ∈ X and r > 0; and that for all x, y ∈ 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 −t L whose kernels e −t L satisfy the Gaussian estimate. That is, there exist constants C, c > 0 and m > 1 such that for all x, y ∈ X and t > 0. Through spectral theory we can define the Schrödinger group, for t ∈ R, where E L (λ) 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 = 2, even in the case when L = − is the Laplacian on 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 < ∞ and t ∈ R, Similar results can be found in [2,5,7,11,20,23] and the references therein.
In the classical case when L = − , we also have the following sharp estimate: for all 1 < p < ∞ and t > 0 one has see [22]. Also for p ≤ 1, it was proved by Miyachi [21] that for each 0 < p ≤ 1 and t ∈ R we have where H p (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 1 2 − 1 p . 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 −t L whose kernels satisfy the Gaussian upper bound G. Then for each 0 < p ≤ 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 < ∞, we have See [8]. Thus, Theorem 1.1 completes the scale of sharp estimates for the Schrödinger group for all 0 < p < ∞.
For s > 0, consider the operator defined by and I s,t (L) =Ī 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.
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.

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 B means A ≤ C B, and A ∼ B means that both A B and B A hold.
The space of Schwarz functions on R n is denoted by S (R n ) and given ψ ∈ S (R), λ ∈ R and j ∈ Z, we use the notation ψ j (λ) := ψ(2 − j λ). For f ∈ S (R n ) we denote by 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 μ(E) for any measurable subset E ⊂ X . Also given λ > 0, we will write λB for the B(x B , λr B ).
For each ball B ⊂ X we set Let w ∈ A ∞ and 0 < r < ∞. The Hardy-Littlewood maximal function 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 < ∞ one has whenever p > r . The following elementary estimates will be used frequently. See for example [2].
(a) For any p ∈ [1, ∞] we have for all x ∈ X and s > 0.

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 ≤ 1. Then the Hardy space H p L (X ) is defined as the completion of Next we have a notion of molecules from [16,19].
The molecular property (ii) in particular can be thought of as a mild locality condition on the operator L. We note that if L = − then H p L (R n ) coincides with the standard Hardy space H p (R n ) on R n for p ∈ (0, 1]. In general, depending on the choice of the operator L, the space H p L (R n ) may be quite different to H p (R n ). See for example [12].

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 ψ ∈ S (R) such that where ψ j (λ) := ψ(2 − j λ) for each j ∈ Z. Now let ψ be a partition of unity and define the discrete square function S L,ψ 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 ψ be a partition of unity. Then for each
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.
. Then the kernel K ϕ(t L) of ϕ(t L) satisfies the following: for any N > 0 there exists C such that Next we introduce and give estimates for certain 'Peetre-type' maximal functions.
Obviously, we have Similarly, for s, λ > 0 we set Proposition 2.7 Let ψ ∈ S (R) with supp ψ ⊂ [1/2, 2] and ϕ ∈ S (R) be a partition of unity. Then for any λ > 0 and j ∈ Z we have for all f ∈ L 2 (X ) and x ∈ 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.

Proposition 2.8
Let ψ be a partition of unity. Then for any λ, s > 0 and r ∈ (0, 1) we have: Proof The proof can be done in the same way as [2, Proposition 2.17] and we omit the details.
We next prove the following result.

Proposition 2.9
Let ψ be a partition of unity. Then for 0 < p ≤ 1 and λ > n/ p we have: Choose r < p so that λ > 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.
We now ready to prove Theorem 2.5.

Proof of Theorem 2.5:
for all λ > 0 and d(x, y) < t 1/m . Therefore, where ψ is a partition of unity. By the spectral theory, Hence it follows that for every t > 0, Now let λ > 0, t ∈ [2 −ν−1 , 2 −ν ] for some ν ∈ Z and M > λ. For convenience we may assume c ψ = 1. We then have where in the last line we used ϕ(t L) = (t L)e −t L . We now set ψ M (x) = x −M ψ(x) and ψ(x) = xψ(x). Then the above can be written as M+1 e −t L satisfies the Gaussian upper bound (see [13]), we have where N > n. It follows that Since ψ ∈ S m (R) and supp ψ ⊂ [1/2, 2], x −2m ψ(x) ∈ S (R). Using Lemma 2.6 and an argument similar to the above, we obtain, for j < ν, t ∈ [2 −ν−1 , 2 −ν ] and The above two estimates imply that This, along with Proposition 2.7, implies that for all t ∈ [2 −ν−1 , 2 −ν ] and M > λ. By Young's inequality, Hence, (17) follows from this and Proposition 2.9. The proof of Theorem 2.5 is thus complete.

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 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 ≥ 0, there exists C > 0 such that for every j ∈ N ∪ {0}, 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]. =: where a k = a.1 S k (B) and b k = b.1 S k (B) . Therefore, it suffices to prove that there exists > 0 such that for some > 0.
For each k ≥ 0, setting B t,k = (1 + t)2 k B, we have Using Hölder's inequality and the L 2 -boundedness of S L,ϕ we obtain where in the last inequality we used (2). It remains to estimate the second term E k 12 . To do this, setting we then write where 0 is the largest integer such that 2 0 (m−1)/m ≤ 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 F ,r B ∞ min{1, (2 r m B ) M }2 − n(1/ p−1/2) , implies that On the other hand, since 2 0 (m−1)/m ∼ 2 k r B , we have, for ≥ 0 , We thus deduce that We now take care of F k 11 . For ≥ 0 and j This, in combination with the doubling property (2), yields that By Lemma 3.1, for α = n(1/ p − 1/2) + θ with θ ∈ (0, ), we have We claim that for α > 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 ∈ N, As a consequence, which proves (25). Substituting (25) into (24) we then obtain This, in combination with (23), implies that for α = n(1/ p −1/2)+θ with θ ∈ (0, ), For the first sum, we have as long as M > α.
For the contribution of the second sum we have where we used the fact that 2 0 (m−1)/m ∼ 2 k r B in the second inequality. Therefore, it holds that Collecting the estimates of F k 11 and F k 12 , we arrive at for some > 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 > α.
For the second sum, we have where in the second inequality we used the fact that Arguing similarly to (25), we see that 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.

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.

Laplacian-Like Operators
Let us here consider two additional conditions on the operator L: Hölder regularity: there exists δ 0 ∈ (0, 1] so that whenever d(x,x) < t 1/m we have Conservation: for all y ∈ X and t > 0 we have X e −t L (x, y) dμ (x) = 1.
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].
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