On sharp estimates for Schrödinger groups of fractional powers of nonnegative self-adjoint operators

Let L be a non negative, selfadjoint operator on L 2 (X) , where X is a metric space endowed with a doubling measure. Consider the Schrödinger group for fractional powers of L . If the heat ﬂow e − tL satisﬁes suitable conditions of Davies–Gaffney type, we obtain the following estimate in Hardy spaces associated to L :


Introduction
The Schrödinger flow e iτ (− ) γ /2 with γ > 0 is a group of isometries on L 2 (R n ) but is unbounded on every other L p space with p = 2.It is well known, however, that boundedness in L p can be recovered at the price of a loss of derivatives.More precisely, for 0 < γ = 1, 1 < p < ∞ and τ > 0 one has while for 0 < p ≤ 1, τ ∈ R one has where H p (R n ) denotes the classical Hardy spaces (see [37]).A similar situation holds for the half-wave flow e it (− ) 1/2 corresponding to γ = 1, but with a loss (n − 1)s p , see for example [27,28,26,32].These results can be regarded as instances of general L p estimates for Fourier integral operators ( [34,29,30]) and are strongly connected to the Schrödinger equation with a fractional Laplacian ⎧ ⎨ ⎩ i ∂u ∂τ + (− ) γ /2 u = 0, u(x, 0) = f (x).
(3) Indeed, estimate (1) implies that any solution u of equation (3) satisfies Since the flows e itL γ /2 are well defined for arbitrary non negative selfadjoint operators via spectral calculus, it is natural to investigate possible extensions of the previous results beyond the case of the Laplacian.The study of Schrödinger flows beyond the Laplacian case is an interesting topic and has attracted a great deal of attention, see [29,30,6,17,16,7,13,14,9].
To introduce our results, we give a brief overview of related research.In the case γ = 2, Lohoué [24] (see also [1]) proved a similar result to (1) for β > 2ns p on Lie groups with polynomial growth and manifolds with nonnegative curvature.In [12], Carron, Coulhon and Ouhabaz prove the following inequality (I + L) −s− e iτ L f p (1 for s = ns p and > 0, where L is a nonnegative self-adjoint operator satisfying the Gaussian upper bound on spaces of homogeneous type space.It is easy to see that in comparison with the classical case (1) the above estimate is not sharp.
In [16], given a selfadjoint, non-negative operator L on L 2 (R n ) whose heat kernel satisfies a mild smoothness effect and a mild off-diagonal decay, the following estimate is proved for k ∈ Z, τ > 0 and a suitable p ∈ (1, ∞): where φ ∈ C ∞ c (R n ) is a cut-off function.This result includes the case of Schrödinger operators with Kato class electromagnetic potentials.Note that (5) implies the estimate (4).The results in [16] were extended in [7] to the very general setting of metric measure spaces with a doubling measure (homogeneous spaces).This goes far beyond the Euclidean case and includes Riemannian manifolds, homogeneous groups, sublaplacians on Heisenberg groups, and operators with singular potentials.Hence, it is natural to raise a question on the validity of a sharp estimate for s = ns p (I + L) −s e iτ L f p (1 Recently, the sharp estimate (6) was proved in [13] (see also [19]) under the assumption of Gaussian upper bounds of order m ≥ 2 on the heat kernel of L. The estimate was later extended to Hardy type spaces, see for example [14,4].However, to the best of our knowledge, only partial results are known for the general flows e itL γ /2 .This leads to our purpose to establish sharp estimates for the flows e itL γ /2 with 0 < γ = 1 generated by arbitrary fractional powers of the operator L.
We now introduce the setting of our results.Let (X, d, μ) be a metric space with distance d, endowed with a nonnegative Borel measure μ.Denote by B(x, r) the open ball of radius r > 0 and center x ∈ X, and by V (x, r) = μ(B(x, r)) its volume.We say that (X, d, μ) is a space of homogeneous type (in the sense of Coifman and Weiss [15]) if it satisfies the doubling property, i.e. there exists a constant C > 0 such that V (x, 2r) ≤ CV (x, r) (7) for all x ∈ X and r > 0. Notice that the doubling property (7) implies the following properties: and for all x, y ∈ X and r > 0. A direct consequence of ( 9) is that V (x, r) ≈ V (y, r) when d(x, y) ≤ r.
Let L be a non-negative, self-adjoint operator L on L 2 (X) which generates an analytic semigroup {e −tL } t>0 .If E L (λ) is the spectral decomposition of L and F : [0, ∞) → C is any bounded Borel function, we denote by F ( √ λ) the bounded operator on L 2 (X) defined as We shall consider two different assumptions on the semigroup.We say that e −tL satisfies the Davies-Gaffney estimates if there exist constants C, c > 0 such that for any open subsets for every We also consider the following condition: there exist p 0 ∈ [1, 2) and constants C and c > 0 such that for all balls B ⊂ X and t > 0, Note that assumption (11) is implied by the following generalized Gaussian condition GGE(p 0 ): for all t > 0 and x, y ∈ X.It is important to note that the generalized Gaussian GGE(1) is equivalent to the Gaussian upper bound, i.e., there exist constants C, c > 0 such that for all x, y ∈ X and t > 0. Recalling the notation the main results of this paper are the following two theorems.
The Hardy space H p L (X) mentioned in the next statement is defined in Section 2: Theorem 1.2.Let L be a nonnegative self-adjoint operator on L 2 (X).Assume that L satisfies (10).For 0 < γ = 1 and β = γ ns p , we have for all τ ∈ R and p ∈ (0, 1], where H p L (X) is the Hardy space associated to the operator L.Moreover, if L satisfies (11) additionally, then the estimate (14) holds true by the interpolation.
Note that Theorem 1.2 gives only a weak type estimate at the endpoint p 0 .Hence, Theorem 1.1 is not a consequence of Theorem 1.2.In the special case L = − , the Hardy space H p L (R n ) turns out to be the classical Hardy space H p (R n ) (see [11]), and hence our estimate recovers the classical estimate (2).
We emphasize that Theorems 1.1 and 1.2 are non-trivial extensions of the classical results.Indeed, the Fourier transform is not available in the setting of metric spaces.Moreover, while the proof of the classical cases relies heavily on the Calderón-Zygmund theory, it seems that this theory might not be applicable in our setting due to the mild assumption on the main operator L, hence new ideas and techniques are required.As a consequence, our paper not only extends, but also provides new proofs for the classical results.Further comments on Theorems 1.1 and 1.2 will be given after Corollary 1.4 below.
As it is well-known, Theorem 1.1 is closely connected with the Schrödinger equation where 0 < γ = 1.Indeed, from Theorem 1.1 we have: Let L be a nonnegative self-adjoint operator on L 2 (X).Assume that L satisfies (10) and (11).If u(x, τ ) is a solution to (16) with 0 < γ = 1, then for p ∈ (p 0 , p 0 ) and β = γ ns p , Another application of Theorems 1.1 Theorem 1.2 concerns the Riesz means associated to L, defined via the following operator: dλ, s, t > 0, while I s,t (L) = Īs,−t (L) for t < 0. See [27,36] for the study of these operators in the case L is the standard Laplacian on R n and [24,1] for extensions to more general contexts.From Theorems 1.1 and 1.2, by standard arguments we obtain the following: Corollary 1.4.Let L be a nonnegative self-adjoint operator on L 2 (X).Assume that L satisfies (10).Then for p ∈ (0, 1], 0 < γ = 1 and s = γ ns p , If in addition L satisfies (11), then for p ∈ (p 0 , p 0 ) we have Some comments regarding Theorems 1.1 and 1.2 are in order: (i) By a careful examination of our proofs, the results in Theorem 1.1, Theorem 1.2, Corollary 1.3 and Corollary 1.4 hold true also for β ≥ γ ns p .Our approach could be modified to study the case γ = 1, but the resulting estimate is not sharp in comparison with the classical cases (1) and ( 2).Hence, we do not pursue the case γ = 1 here.(ii) As mentioned above, boundedness of the Schrödinger group, corresponding to the case γ = 2, has been studied extensively.L p -boundedness of the Schrödinger groups when γ = 2 under the assumption of Gaussian upper bounds of order m ≥ 2 was proved in [13].
Boundedness on the Hardy spaces H p L (X) was obtained in [14,4].Boundedness under the generalized Gaussian estimates of order m ≥ 2 was obtained in [19].(iii) Much less was known for γ = 2.In the special case of the Hermite operator, boundedness on L p (X) and on Hardy spaces associated to the operator (with the exception of the weak type estimate) was obtained in [8].(iv) Theorem 1.2 is new even when γ = 2. Boundedness on the Hardy space H 1 L (X) was obtained in [14], but the approach there does not work for the case 0 < p < 1.Although Theorem 1.2 also implies the boundedness of (I + L) −β/2 e iτ L γ /2 on L p for p 0 < p < p 0 , the weak type boundedness (p 0 , p 0 ) in Theorem 1.1 is unique.(v) We emphasize that the techniques in [13] do not work in our setting.The approach in [13] relies heavily on estimates for operators of the form e −tL e iτ L , leading to Besov norm estimates of the function e −(t−τ )• .This approach fails completely if we replace the flow e iτ L by the general flow e iτ L γ /2 .To overcome this problem, we need to establish new operator estimates in Lemma 2.10, which play a crucial role in the proofs of our main results.Our approach can be used to obtain sharp estimates for imaginary powers of L, and this will be the topic of an upcoming paper.
Our theory is highly comprehensive, encompassing a broad range of significant operators in harmonic analysis and partial differential equations (PDEs).Notable examples include Schrödinger operators with inverse-square potentials [5,25], the Kohn-Laplacian on pseudoconvex manifolds of finite type, as studied by Nagel-Stein [31], and the Laplace-Beltrami operators on doubling manifolds [2].Additional operators of interest can be found in references such as [7,10] and the references therein.To demonstrate the practical applications of our theory, we present two compelling instances in the realm of PDEs.
Schrödinger operators with inverse-square potentials.Consider the following Schrödinger operators with inverse square potential on R n , n ≥ 3: Set The Schrödinger operator L a is understood as the Friedrichs extension of guarantees that L a is nonnegative.It is well-known that L a is self-adjoint and the extension may not be unique as . For further details, we refer the readers to [20,33,38].Set n σ = n/σ if σ > 0 and for any n σ < p ≤ q < n σ there exist C, c > 0 such that for every t > 0, any measurable subsets E, F ⊂ R n , and all f ∈ L p (E), we have: Hence, the operator L a satisfies ( 10) and ( 11) with p 0 = n σ .Therefore, from Theorem 1.1, for 0 < γ = 1 and β = γ ns p , we have

Dirichlet Laplacians on open domains.
Let be an open subset of R n .Note that may not satisfy the doubling condition.Let − D be Dirichlet Laplacian on the domain .It is well known that the semigroup kernel e t D (x, y) of e t D satisfies the Gaussian upper bound for all t > 0 and all x, y ∈ .By the extension argument as in [18], we can obtain the estimates for the Schrödinger group associated to the fractional Laplacian (− D ) γ /2 .More precisely, we have for 0 < γ = 1 and β = γ ns p , we have To the best of our knowledge, this result is new.
In the next Section, we recall the properties of Hardy spaces associated to the operator L, and some estimates for functions of the operator.The proof of Theorems 1.1 and 1.2 is given in Section 3.

Notation.
Throughout this paper, we use C to denote positive constants, which are independent of the main parameters involved and whose values may vary at every occurrence.By writing f g, we mean that f ≤ Cg.We also use f ∼ g to denote that C −1 g ≤ f ≤ Cg.
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

Hardy spaces associated to the operator L
In this section, we assume that the operator L is a nonnegative self-adjoint operator on L 2 (X) satisfying the Davies-Gaffney estimates (10).We first recall from [22,23] 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 (A2).Let 0 < p ≤ 2. Then the Hardy space , where the square function S L is defined as . Definition 2.1 ([22,23]).Let 0 < p ≤ 1 and Definition 2.2 (Atomic Hardy spaces for L).Given 0 < p ≤ 1 and M ∈ N, we say that f = λ j a j is an atomic (p, 2, M, L)-representation if {λ j } ∞ j =0 ∈ p , each a j is a (p, 2, M, L)-atom, and the sum converges in L 2 (X).The space H p L,at,M (X) is then defined as the completion of f ∈ L 2 (X) : f has an atomic (p, 2, M, L)-representation , with the norm given by We note that if L = − on L 2 (R n ), then H p L (R n ) reduces to 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, it may happen that either See for example [11].
By a careful examination of the proofs in the papers [22,23], it can be verified that the Hardy space H p L (X) can be defined by using S ψ,L instead of S L .

Some estimates on the functional calculus
Assume that the operator L is a nonnegative self-adjoint operator on L 2 (X) satisfying the Davies-Gaffney estimates (10).It is well-known that the kernel K cos(t See for example [10].We first recall the following result in [15, Lemma 1]. Lemma 2.7.Let L be a nonnegative self-adjoint operator on L 2 (X) satisfying the Davies-Gaffney estimates (10).If F is an even bounded Borel function with supp F ⊂ [−r, r] for some r > 0, then Lemma 2.8.Let L be a nonnegative self-adjoint operator on L 2 (X) satisfying the Davies-Gaffney estimates (10).Assume that the operator L satisfies (11) for some p 0 ∈ [1, 2) additionally.Then for p ∈ [p 0 , 2] and N > ns p /2, we have for any ball B ⊂ X.
Proof.For any N ∈ N, Note that under the Davies-Gaffney estimates (10) and (11), by the interpolation we have, for any p ∈ [p 0 , 2], Therefore, This, together with (8), as long as N > ns p /2.Similarly, as long as N > ns p /2.Hence, this completes our proof.
We have the following useful lemma.
For the inequality ( 22), we fix p ∈ [p 0 , 2] and N ∈ N, N > ns p , and then we write It follows that Similarly, This completes our proof.Lemma 2.10.Let L be a nonnegative self-adjoint operator on L 2 (X) satisfying the Davies-Gaffney estimates (10) and (11) for some p 0 ∈ [1,2).Let be the function in Lemma 2.9 and let ψ ∈ C ∞ c (R) be an even function such that supp ψ ⊂ {ξ : 1/4 ≤ |ξ | ≤ 4} and ψ = 1 on {ξ : where C is a constant independent of s, τ, and the sets E, F .
The inequalities (23) and (24) still hold true if we replace (I − (s √ L)) M by (I − e −s 2 L ) M .In the case p = 2, the condition (11) can be omitted.
Proof.We need only to give the proof for the case p = 2 since the case p = 2 can be done similarly.
Proof of (23). Consequently, From Lemma 2.8, we have for a fixed N > n/2, It reduces to estimate the L ∞ norm of the function Since supp H ,s ⊂ {t : 2 −2 ≤ |t| ≤ 2 +2 }, it suffices to estimate H ,s ∞ .To do this, we borrow some ideas in [35].Recall that which follows that for any k 0 ∈ N, It follows that Recall that Since is an even Schwartz function with (0) = 0, 1 − (t) t 2 as t → 0. Hence, it is easy to verify that This, along with (26), implies (23).This completes our proof of (23).
From ( 25), we have Similarly, it can be verified that the inequalities ( 23) and ( 24) still hold true if we replace (I − (s √ L)) M by (I − e −s 2 L ) M .This completes our proof.

Sharp weak type estimates for the Schrödinger flows e itL γ /2
This section is dedicated to proving Theorem 1.1.
To do this, let f ∈ L p (X) and λ > 0. Then by the Calderón-Zygmund decomposition [15] we and where M is an positive integer which will be fixed later.We have Using Chebyshev's inequality and the L 2 -boundedness of For the bad part, let be the function in Lemma 2.9.Setting θ,r B k (t) = (θ r B k t), then we have We will take care of the first term E 1 .Note that where c k are coefficients.From Lemma 2.9, and By the Chebyshev inequality and the L 2 -boundedness of This, along with (iv), ( 31), ( 32) and (ii), implies that We now apply (8) to further obtain where in the last inequality we used (iii).It remains to estimate E 2 .To do this, we set Then, where By ( 8) and (iii), Hence, it suffices to estimate the term Then we have We now take care of each term E 221 , E 222 and E 223 individually.

Estimate of the term E 221
By Chebyshev's inequality, For each k and < 0, by Hölder's inequality, the doubling property ( 8) and (ii), Using ( 23) in Lemma 2.10, for a fixed k 0 > n/2 we have where in the second inequality we used (8).Consequently, we have (36) This, together with the fact that max{1, and by a simple calculation, we come up Therefore,

Estimate of the term E 222
This term can be done similarly to the term E 221 with some modifications.Indeed, similarly to the term E 221 , we also obtain Since ≥ 0 and 2 Hence, similarly to (36) we obtain that Hence,

Estimate of the term E 223
This term is quite complicated and can be estimated by the duality argument.By Chebyshev's inequality and L 2 -boundedness of where where u j = u.1 S j (B k ) .By Hölder's inequality, we have We will claim that Indeed, for j = 0, 1, 2, using Lemma 2.11 we obtain This, in combination with (8), implies that On the other hand, for j = 0, 1, 2, , where M is the Hardy-Littlewood maximal function defined by Therefore, for j = 0, 1, 2, dμ(z). (42) For j ≥ 3, also using Lemma 2.11 and arguing similarly to (35), we have, for It follows that In addition, These two last estimates give us that This and (42) prove the claim (41).We now insert (41) into (40) and raise both side to the power of 2 to obtain further Using Kolmogorov's inequality, the weak type (1,1) of the maximal function M and (iii), We now estimate E 12 .To do this, setting we then write Therefore, where 0 is the smallest integer such that 2 0 ≥ r B , which implies 2 0 r B .For the first term E 121 , we can see that =: E 1211 + E 1212 .
By Hölder's inequality and ( 8), On the other hand, we have which, together with the fact that Inserting this into (45) we arrive at where in the second inequality we used (8) and in the third inequality we used the fact 2 0 r B .
(48) At this stage, arguing similarly to the estimate of the term E 121 we come up with For the details we would like to leave to the interested reader.
This completes the proof of (15).In order to reduce to the sharp estimate (14).We note that by interpolation, the Davies-Gaffney estimates (10) and (11) implies that the operator L satisfies GGE(p) for all p 0 < p < p 0 .This, together with Proposition 2.5, implies that H p L (X) = L p (X) for all p 0 < p < p 0 .
At this stage, by using the standard argument (see for example [27]) and Proposition 2.4 the estimate (14) follows immediately from (15).This completes our proof.