Abstract
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 operator e −tL satisfies the generalized Gaussian (p 0 , p ′ 0 )-estimates of order m for some 1 ≤ p 0 < 2. In this paper we prove sharp endpoint L p -Sobolev bound for the Schrödinger group e itL that for every p ∈ (p 0 , p
As a consequence, the above estimate holds for all 1 < p < ∞ when the heat kernel of L satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces R n . We also give an application to obtain an endpoint estimate for L p -boundedness of the Riesz means of the solutions of the Schrödinger equations.

′ 0 )-estimates of order m for some 1 ≤ p 0 < 2. In this paper we prove sharp endpoint L p -Sobolev bound for the Schrödinger group e itL that for every p ∈ (p 0 , p ′ 0 ), there exists a constant C = C(n, p) > 0 independent of t such that (I + L)
−s e itL f p ≤ C(1 + |t|) s f p , t ∈ R, s = n 1 2 − 1 p .
As a consequence, the above estimate holds for all 1 < p < ∞ when the heat kernel of L satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces R n . We also give an application to obtain an endpoint estimate for L p -boundedness of the Riesz means of the solutions of the Schrödinger equations. Here f denotes the Fourier transform of f . It is well-known that the operator e it∆ acts boundedly on L p (R n ) only if p = 2; see Hörmander [22] . For p 2, it was shown (see for example, [6, 26, 39] )) that for s > n|1/2−1/p|, the operator e it∆ sends the Sobolev space L p
Equivalently, this means that (I + ∆)
−s e it∆ is bounded on L p (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 ([32, 33] ), which states that for every p ∈ (1, ∞),
for some positive constant C = C(n, p) independent of t. The estimate (1.1) is sharp in another way: the factor (1 + |t|) s can not be improved (see [32, p. 169-170] ). See also Feffermann and Stein's work [19] . These results and their generalizations were in fact results on multipliers and relied heavily on Fourier analysis. See, for example, Ouhabaz's monograph [34, Chapter 7] for historical background and more study on the Schrödinger groups.
The purpose of this paper is to establish such sharp endpoint L p estimate (1.1) for the operators e itL t∈R for a large class of non-negative self-adjoint operators acting on L 2 (X) on a metric measure space X. Such an operator L admits a spectral resolution
where E L (λ) is the projection-valued measure supported on the spectrum of L. The operator e itL is defined by
for f ∈ L 2 (X), and forms the Schrödinger group. By the spectral theorem ( [31] ), the operator e itL is continuous on L 2 (X). It is interesting to investigate L p -mapping properties for the Schrödinger group e itL on L p (X) for some p, 1 ≤ p ≤ ∞. As an application of our sharp endpoint L p estimate for the Schrödinger group e itL , we also aim to obtain an endpoint estimate for L p -boundedness of the Riesz means of the solutions of the Schrödinger equations.
Assumptions and main results.
Throughout the paper we assume that X is a metric space, with distance function d, and µ is a nonnegative, Borel, doubling measure on X. We say that (X, d, µ) satisfies the doubling property (see Chapter 3, [10] ) if there exists a constant C > 0 such that V(x, 2r) ≤ CV(x, r) ∀ r > 0, x ∈ X. (1.4) Note that the doubling property implies the following strong homogeneity property, (1.5) V(x, λr) ≤ Cλ n V(x, r) for some C, n > 0 uniformly for all λ ≥ 1 and x ∈ X. In Euclidean space 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 and D, 0 ≤ D ≤ n so that uniformly for all x, y ∈ X and r > 0. Indeed, the property (1.6) with D = n is a direct consequence of triangle inequality of the metric d and the strong homogeneity property. In the cases of Euclidean spaces R n and Lie groups of polynomial growth, D can be chosen to be 0. Consider a non-negative self-adjoint operator L and numbers m ≥ 2 and 1 ≤ p 0 ≤ 2. We say that the semigroup e −tL generated by L, satisfies the generalized Gaussian (p 0 , p −(
Note that condition (GGE p 0 ,p ′ 0 ,m ) for the special case p 0 = 1 is equivalent to m-th order Gaussian estimates (see for example, [5] ). This means that the semigroup e −tL has integral kernels p t (x, y) satisfying the following Gaussian upper estimate:
for every t > 0, x, y ∈ X, where c, C are two positive constants and m ≥ 2. Such estimate (GE m ) is typical for elliptic or sub-elliptic differential operators of order m (see for example, [1, 2, 8, 12, 16, 17, 20, 23, 24, 34, 38, 39, 42] and the references therein). However, there are numbers of operators which satisfy generalized Gaussian estimates and, among them, there exist many for which classical Gaussian estimates (GE m ) fail. This happens, e.g., for Schrödinger operators with rough potentials [36] , second order elliptic operators with rough lower order terms [28] , or higher order elliptic operators with bounded measurable coefficients [13] . See also [3, 4, 5, 9, 25, 37] .
Our main result is that under the generalized Gaussian estimate (GGE p 0 ,p ′ 0 ,m ) for some 1 ≤ p 0 < 2, it is sufficient to ensure that such estimate (1.1) holds for the operator e itL t∈R for p ∈ (p 0 , p ′ 0 ). Our result can be stated as follows. 
As a consequence, this estimate (1.7) holds for all 1 < p < ∞ when the heat kernel of L satisfies a Gaussian upper bound (GE m ).
As a consequence of Theorem 1.1, we have the following result.
We now use the result of Theorem 1.1 to study the property of the solution to the Schrödinger equation
From this it is seen that e itL is bounded on L p only for p = 2. As mentioned by Sjöstrand [39] in the case that L is the Laplacian on the Euclidean space R n , a possible substitute for this operator on L p is its Riesz means, defined by
for t > 0, and I s (t)(L) = I s (−t)(L) for t < 0. Then we have the following result. 
As a consequence, this estimate (1.10) holds for all 1 < p < ∞ when the heat kernel of L satisfies a Gaussian upper bound (GE m ).
The proof of Theorem 1.1 and Corollary 1.2 will be given in Section 3. The proof of Theorem 1.3 will be given in Section 4.
1.3. Comments on the results and methods of the proof. On Lie groups with polynomial growth and manifolds with non-negative Ricci curvature, similar results as in (1.1) for s > n |1/2 − 1/p| have been first announced by Lohoué in [29] , then Alexopoulos obtained them in [1] . There, the method is to replace Fourier analysis by the finite propagation speed of the associated wave equation [41] . In the abstract setting of operators on metric measure spaces, Carron, Coulhon and Ouhabaz [8] showed L p -boundedness of suitable regularizations of the Schrödinger group e itL provided L satisfies Gaussian estimate (GE m ). They proposed a different approach to use some techniques introduced by Davies [12] : the Gaussian semigroup estimates can be extended from real times t > 0 to complex times z ∈ C + = {z ∈ C : Re z > 0} such that
where Γ is the Euler Gamma function. From (1.11), we see that for s > n|1/2 − 1/p|, 
uniformly for φ in bounded subsets of C ∞ 0 (R), by writing
where φ e (λ) = e λ φ(λ) and then applying (1.13) to e −(2 −k −it)L and [4, Theorem 1.1] to φ e (2 −k L), respectively (for more details, see Proposition 3.1 below). As a consequence of (1.14), it follows by a standard scaling argument ( [23, p. 193] ) that for every p ∈ (p 0 , p ′ 0 ) and for every ǫ > 0,
See also previous related results [7, 11, 23, 24] .
Our main result, Theorem 1.1, gives the sharp endpoint estimate (1.15) for the Schrödinger group e itL with ǫ = 0, namely with the optimal number of derivatives and the optimal time growth for the factor (1+|t|) s in (1.15). The proof of Theorem 1.1 is different from that of Fefferman and Stein [19] and Miyachi [32, 33] where their results rely 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 ( [40] ) are not applicable. The lack of smoothness of the kernel will be overcome in Proposition 2.3 below by using some off-diagonal estimates on heat semigroup of non-negative self-adjoint operators, and some techniques in the theory of singular integrals with rough kernels, which lies beyond the scope of the standard Calderón-Zygmund theory (see for example, [2, 4, 5, 9, 16, 17, 18, 25, 34, 37] and the references therein). More specifically, by duality we are reduced to prove the estimate for 2 < p < p ′ 0 , which will follow by the Littlewood-Paley inequality and a variant of the FeffermanStein sharp function (see [2, 18, 19, 30, 37] ),
We then use a variant of an argument in [27, 35] to decompose the function M # T ϕ ,L,K f into several components so that we can employ the off-diagonal estimates (1.20) below. Then we show that the
p by using estimate (1.14) for the Schrödinger group e itL . We note that in the case that L is the Laplace operator ∆ on R n , the kernel estimate relies heavily on Fourier analysis since the operator e it∆ ϕ(2 −k ∆) has convolution kernel
Integration by parts yields that for every M > 0, 
, and this new estimate is crucial for the proof of Theorem 1.1.
The paper is organized as follows. In Section 2 we provide some preliminary results on offdiagonal estimates of the operator e itL ϕ(2 −k L) and spectral multipliers and Littlewood-Paley theory, which we need later, mainly to prove (1.20) in Proposition 2.3. The proof of Theorem 1.1 will be given in Section 3. In Section 4 we will apply Theorem 1.1 to obtain L p -boundedness of the Riesz means of the solutions to the Schrödinger equation.
List of notation.
• (X, d, µ) denotes a metric measure space with a distance d and a measure µ.
• L is a non-negative self-adjoint operator acting on the space L 2 (X).
• For x ∈ X and r > 0, B(x, r) = {y ∈ X : d(x, y) < r} and V(x, r) = µ B(x, r) .
• [t] denotes the integer part of t for any positive real number t.
• N is the set of positive integers.
• For every B ⊂ X, we write B f dµ(y) = µ(B)
• For 1 ≤ r < ∞, M r denote the uncentered r-th maximal operator over balls in X, that is
For simplicity we denote by M the Hardy-Littlewood maximal function M 1 .
2. Off-diagonal estimates and spectral multipliers
In this section we assume that (X, d, µ) is a space of homogeneous type with a dimension n in (1.5) and that L is a self-adjoint non-negative operator in L 2 (X) satisfying the generalized Gaussian estimate (GGE p 0 ,p ′ 0 ,m ) for some 1 ≤ p 0 < 2.
2.1. Off-diagonal estimates. We start by collecting some properties of the generalized Gaussian estimates obtained by Blunck and Kunstmann, see for example, [3, 4, 5, 25] and the references therein. For every j ≥ 1, we recall that 
Let r z = (Re z)
. (i) There exist two positive constants C
′ and c ′ such that for all r > 0, x ∈ X, and z ∈ C with Re z > 0,
(ii) There exist two positive constants C ′′ and c ′′ such that for all r > 0, x ∈ X, k ∈ N and z ∈ C with Re z > 0,
Proof. For the detailed proof we refer readers to [25] . Here we only mention that the proof of Lemma 2.1 relies on the Phragmén-Lindelöf theorem.
Next suppose that m ≥ 2. We say that the semigroup e −tL generated by non-negative self-adjoint operator L satisfies m-th order Davies-Gaffney estimates, if there exist constants C, c > 0 such that for all t > 0, and all x, y ∈ X,
Note that if condition (GGE p 0 ,p ′ 0 ,m ) holds for some 1 ≤ p 0 ≤ 2 with p 0 < 2, then the semigroup e 
P B F(L)P
for all balls B ⊆ X, and all Borel functions F such that supp
In virtue of the Fourier inversion formula
By (ii) of Lemma 2.1 (with
This ends the proof of Lemma 1.5.
The proof of Theorem 1.1 relies on the following off-diagonal estimates for e itL φ k (L), where
Proposition 2.3. Let m ≥ 2 and L satisfies the Davies-Gaffney estimates (DG m ).
For every M > 0, K ∈ N + , s > 0, t ∈ R and k ≥ 1, there exists a constant C = C(M, n, K) independent of t, s, and k such that for j = 2, 3, · · ·
To prove Proposition 2.3, we need the following Lemmas 2.4 and 2.5.
Lemma 2.4. Let m ≥ 2 and L satisfies the Davies-Gaffney estimates (DG m ).
Then for every M > 0, k ∈ N + and t ∈ R, there exists a constant C = C(M, m, n) independent of t and k such that for every j = 2, 3, . . .
for all balls B ⊂ X with r B ≥ c2 (m−1)k/m (1 + |t|) for some c ≥ 1/4. As a consequence, we have
for all balls B ⊂ X with r B ≥ c2 (m−1)k/m (1 + |t|) for some c > 1/4 and for every x ∈ B.
Proof. Note that
. It is clear that Re z = 2 −k > 0, and so r z = (Re z)
for every M > 0. Hence, (2.3) holds. This, in combination with the fact that for every x ∈ B,
as long as we choose M > 3n/2 in (2.4). This proves Lemma 2.4. , 4]), we write φ e (λ) = e λ φ(λ). Then for every M > 0, k ∈ N + and s > 0, there exists a constant C = C(m, n, M) independent of k and s such that for every j = 2, 3, . . .
for all B ⊂ X with r B ≥ c2 (m−1)k/m for some c ≥ 1/4. As a consequence, we have
Proof. We write
Note that the function (
]. We apply Lemma 2.2 with R = 2 k+2 to obtain that for every M > 0 and j ≥ 2,
This, in combination with (2.5), yields that for every x ∈ B,
as long as we choose M > n/2 in the first inequality above and notice the fact that 2 k/m r B ≥ 1/4. This proves Lemma 2.5.
Proof of Proposition 2.3. It suffices to show (2.2) when
. By spectral theory, we write
where we write φ e (λ) = e λ φ(λ). 
where ⌊a⌋ denotes the greatest integer that is smaller than a. Then by noting that S k,t (L) is uniformly bounded on L 2 (X) and by Lemma 2.5,
On the other hand, we apply Lemma 2.4 and the fact that T k (L) is uniformly bounded on L 2 (X) to see that for every M > 0,
Therefore, we combine the estimates (2.7) and (2.8) to obtain that for every M > 0,
which shows that (2.2) holds. The proof of Proposition 2.3 is complete.
In order to prove Theorem 1.3, we also need the following estimate for the operator e itL φ k (tL), t > 0. Recall that φ ∈ C 
Proof. The proof of Proposition 2.6 can be obtained by making minor modifications with the proof of Proposition 2.3, we leave the detail to the reader.
Spectral multipliers.
The following result is a standard known result in the theory of spectral multipliers of non-negative selfadjoint operators. 
In addition, we assume that k≥0 ϕ(λ/2 k ) = 1 for all λ > 0,
Proof. Assertion Let us introduce the Rademacher function, which is defined as follows: i) The function r 0 (t) is defined by r 0 (t) = 1 on [0, 1/2] and r 0 (t) = −1 on (1/2, 1), and then extended to R by periodicity; ii) For k ∈ N\{0}, r k (t) = r 0 (2 k t). Define
A straightforward computation shows that for every integer β > n/2 + 1, sup R>0 ηF(t, Rλ) C β ≤ C β uniformly in t ∈ [0, 1]. Then we apply (2.9) to see that for all p ∈ (p 0 , p
with C > 0 uniformly in t ∈ [0, 1]. This, in combination with the standard inequality for Rademacher functions:
This proves (2.10). By the spectral theory [31] , we have that
From it, we obtain (2.11) by using (2.10) and the standard duality argument (see for example, [40, Chapter IV] 
uniformly for t ∈ R and for φ in bounded subsets of C ∞
(R).
Proof. To Proposition 3.1, we apply (3.1) with z = 2 −k − it to get that for every φ ∈ C ∞ 0 (R),
where φ e (λ) = e λ φ(λ). In the last inequality we used Proposition 2.7 to know that the operator
. This ends the proof of Proposition 3.1.
To prove Theorem 1.1, let us introduce some tools needed in the proof. Let T be a sublinear operator which is bounded on L 2 (X) and {A r } r>0 be a family of linear operators acting on L 2 (X). For f ∈ L 2 (X), we follow [2] to define
, where the supremum is taken over all balls B in X containing x, and r B is the radius of B. 
for all f ∈ L 2 (X), all x ∈ X and all balls B ∋ x, r B being the radius of B.
Then for 0 < p < q, there exists C p such that
for every f ∈ L 2 (X) for which the left-hand side is finite (if µ(X) = ∞, the term C p f p can be omitted in the right-hand side of (3.4) ).
Proof of Theorem 1.1. Let us show Theorem 1.1 for 2 < p < p ′ 0 and s = n|1/2 − 1/p|. We fix a non-zero C ∞ bump function ϕ on R such that
and set ϕ 0 (λ) = ℓ≤0 ϕ(λ/2 ℓ ) and ϕ ℓ (λ) = ϕ(λ/2 ℓ ) for ℓ = 1, 2, . . ..
For this fixed bump function ϕ, we consider an operator T ϕ , given by
where the supremum is taken over all balls B in X containing x, and r B is the radius of B. In order to prove Theorem 1.1, it suffices to show the following two arguments:
(a 1 ) the operator T ϕ satisfies condition (3.3) for every 2 < p < q < p 
Before we prove the above two arguments (a 1 ) and (a 2 ), let us show that Theorem 1.1 is a straightforward consequence of them. Indeed, when (a 1 ) holds for T ϕ , it follows from (b) of Proposition 2.7 and Proposition 3.2 that for 2
This, together with (3.8), yields that
where in the fifth inequality we have used the embedding ℓ 2 ֒→ ℓ p for p ≥ 2, in the sixth inequality the function φ k (λ) = ϕ(2 −k λ)(2 −k λ) −s , and in the last inequality we used (b) of Proposition 2.7 for the Littlewood-Paley result for functions in L p (X). This proves Theorem 1.1.
We now first prove the argument (a 1 ). Indeed, in virtue of the formula
and the commutativity property
Let us prove (3.11) . From hypothesis (GGE p 0 ,p ′ 0 ,m ), it is seen that condition (GGE 2,q,m ) holds for 2 < p < q < p ′ 0 , i.e, there exist constants C, c > 0 such that for every u > 0 and x, y ∈ X,
By Minkowski's inequality, (3.12) and (ii) of Lemma 2.1, conditions (1.5) and (2.1) for every τ = 1, 2, . . . , K and every ball B containing x, the left hand side of (3.11) is less than
The above estimate yields (3.11). Thus, we obtain that the argument (a 1 ) holds. We now show the argument (a 2 ). In the sequel we let φ ∈ C ∞ 0 (R) supported in (1/4, 4) and φ(x) = 1 if x ∈ (1/2, 2), and set
Hence, the proof of (3.8) reduces to show that
where
Here, we use the notation in the above decomposition that the ball B is centered at x B and its radius r B is in [2
Estimate of the term I(x). From this, we see that
For the term I 1 (x), from the arguments in (3.10) and (3.11), it is direct to see that for every x ∈ B,
Then from Proposition 3.1,
For the term I 2 (x), since the function (1 − e
also for u ≥ 2, we use Lemma 2.2 to obtain that for every M > 0,
Those, in combination with k + j ≤ 0 and the fact that for all u ≥ 0,
where M > n/2 and K is large enough so that K > (M +n)/m. We then use the embedding ℓ p ֒→ ℓ ∞ , the Minkowski inequality, L p/2 -boundedness of M and Proposition 3.1 to see that
as desired, as long as K is chosen large enough so that K > (M + n)/m. Combining the estimates of I 1 and I 2 we get that
Estimate of the term II(x). Note that
Similar to the estimate of I 1 (x) above, we see that
It follows from (1.6) that for every B(
and so
Then we have
In this case, since j ≥ (m−1)k+mlog 2 (2+2|t|) and so r B ≥ c2 (m−1)k/m (1+|t|) with c = 2 (m−1)/m ≥ 1/4, we apply Proposition 2.3 to see that for every B(x i , r B ) ∈ J ℓ ,
for every M > 0. This, in combination with the fact that for every x ∈ B,
as long as M in (3.17) is chosen large enough so that M > D + 2n. As a consequence, we have that
Combining the estimates of II 1 and II 2 we obtain the estimate of II as desired.
Estimate of the term III(x). As to be seen later, the term III(x) is the major one.
Similar to the estimates for II and I above, we write
Again, it is clear that 
We will use
to replace the mean value B j in the term III 2 (x). It is seen that for every non-negative function
loc (X) and B j containing x,
and so III 2 (x) ≤ C III 2 (x), where
Now for every k ≥ 1, we choose a sequence (x
τ by the formula
. Hence, one writes
where 
To continue, we claim that the functions {A r B j χ B (k), * τ )} τ have bounded overlap, uniformly in k. Assume this at the moment. Then by setting ℓ = k + j > 0, applying Minkowski's inequality, and the above claim, we obtain that
where
We now show the claim. Note that for B
By this it means that if
Next we will show that
Once (3.20) is proven, we see that
Let us prove estimate (3.20) . First, we observe that for every g ∈ L 1 (X) and p/2 > 1,
From this, we see that the term E ℓ is dominated by a constant times
, we see by the Hölder inequality that the term E ℓ is controlled by a constant multiple of
which yields
This finishes the proof of (3.20) and concludes the desired estimate (3.21) for the term III 21 .
Concerning the term III 22 , we use the embedding ℓ p → ℓ ∞ and the Minkowski inequality to see that the term III 22 p is controlled by
The proof of Theorem 1.1 will be done if we can show that
In this case, we see that • A simple calculation shows that Proof. We prove this theorem by following the approach in the proof of Theorem 1.1 by using Proposition 2.6 instead of Proposition 2.3. For the details, we leave to the reader.
Proof of Theorem 1.3. The proof of Theorem 1.3 is inspired by the idea of [39] . Take a function Φ ∈ C ∞ (R) such that Φ(t) = 0 if t < 1/2 and Φ(t) = 1 if t > 1. Define function F by
where C s is defined by Since I s (t)(L) = I s (−t)(L) for t < 0, we have that I s (t)(L) p→p ≤ C for t < 0. The proof of Theorem 1.3 is complete.

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