Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators

Let L=−Δ+V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup {e−tLα}t>0,0<α<1, associated with L. By the aid of the fundamental solution of the heat equation: ∂tu+Lu=∂tu−Δu+Vu=0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel Kα,tL(·,·), respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space BMOLγ(Rn) via the fractional heat semigroup {e−tLα}t>0.


Introduction
The aim of this paper is to investigate the fractional heat semigroup of Schrödinger operators where −∆ denotes the Laplace operator ∆ = n ∑ i=1 ∂ 2 /∂x 2 i , and V is a non-negative potential belonging to the reverse Hölder class B q . Definition 1. A non-negative locally L q integrable function V on R n is said to belong to B q , 1 < q < ∞, if there exists C > 0 such that the reverse Hölder inequality, holds for every ball B in R n .
This kind of operator was firstly noted in the famous paper by C. Fefferman [1]. For the special case V = 0, L = −∆, the fractional heat semigroup can be defined via the Fourier transform: In the literature, the fractional heat semigroup {e −t(−∆) α } t>0 has been widely used in the study of partial differential equations, harmonic analysis, potential theory, and Compared with −∆, for arbitrary Schrödinger operator L with the non-negative potential V, the fractional heat semigroup {e −tL α } t>0 , α ∈ (0, 1), can not be defined via (2). In addition, it is obvious that the methods in [6] are invalid for the estimation of the integral kernels of {e −tL α } t>0 . In this paper, by the functional calculus, we observe that the integral kernel of the Poisson semigroup associated with L can be defined as: where K L s (·, ·) denotes the integral kernel of e −sL , i.e., e −sL ( f )(x) := R n K L s (x, y) f (y)dy.
Recall that K L t (·, ·) is a positive, symmetric function on R n × R n , and satisfies R n K L t (x, y)dy ≤ 1. Generally, for α > 0, the subordinative formula (cf., [3]) indicates that there exists a continuous function η α t (·) on (0, ∞), such that: The identity (5) enables us to estimate K L α,t (·, ·) via the heat kernel K L t (·, ·). Let ρ(·) be the auxiliary function defined by (12) below. In Propositions 7 and 8, we can obtain the following pointwise estimates of K L α,t (·, ·): for every N > 0, there exists a constant C N , such that: and for every N > 0, 0 < δ < min{1, 2 − n/q}, and all |h| ≤ t 1/α , there exists a constant C N , such that: K L α,t (x + h, y) − K L α,t (x, y) ≤ C N t(|h|/t 1/2α ) δ (t 1/2α + |x − y|) n+2α 1 + t 1/2α ρ(x) Based on the estimates (6) and (7), we consider the regularity properties of K L α,t (·, ·). Let ∇ x,t denote the gradient operator on R n+1 + , that is, ∇ x,t = (∇ x , ∂/∂t), where ∇ x = (∂/∂x 1 , ∂/∂x 2 , . . . , ∂/∂x n ). Generally speaking, for a differential operator L, if the semigroup {e −tL } t>0 is analytic, then the estimate of the derivative in time of integral kernels can be deduced. However, for the derivatives in spatial variables, it is relatively difficult. Specially, let H = −∆ + |x| 2 be the Hermite operators. The heat kernel related to H, denoted by K H t (·, ·), can be expressed precisely. Hence, the derivative ∇ x K H t (·, ·) can be obtained via a direct computation. (cf., [7,8]). For a general Schrödinger operator, obviously, there does not exist an exact expression of K L t (·, ·), and the regularity estimates of K L t (·, ·) cannot be obtained directly as the case of the Hermite operator H. Alternatively, we obtain an energy estimate of the solution to the equation: By the fundamental solution of −∆, we prove that, for any N > 0, there exists a constant C N , such that: in Lemma 8. A direct computation, together with the subordinative formula, gives: see Proposition 11. By a similar method, we obtain the Hölder regularity of ∇ x K L α,t (·, ·), i.e., for |h| < |x − y|/4 and δ = 1 − n/q, |∇ x K L α,t (x + h, y) − ∇ x K L α,t (x, y)| ≤ C N |h| t 1/2α see Proposition 12. In Section 3.3, we focus on the time-fractional derivatives of K L α,t (·, ·). Recently, there has been an increasing interest in fractional calculus, since time-fractional operators are proven to be very useful for modeling purposes. For example, the following fractional heat equations, are used to describe heat propagation in inhomogeneous media. It is known that, as opposed to the classical heat equation, Equation (9) is known to exhibit sub-diffusive behaviour and is related to anomalous diffusions or diffusions in non-homogeneous media, with random fractal structures. Recall that the fractional derivative of K L α,t (·, ·) is defined as: For some recent works in the frame of confromable derivatives and Mittag-Leffler kernels, see [9,10]. In Section 3.1, we first obtain the regularity estimates of t m ∂ m t K L α,t (·, ·) denoted by D L,m α,t (·, ·); see Proposition 9. Then, the desired estimates of ∂ β t K L α,t (·, ·) can be deduced from (10) and Proposition 9; see Propositions 14-16, respectively.
As an application, in Section 4, we characterize the Camapnato-type spaces associated with L, denoted by BMO γ L (R n ), via the fractional heat semigroup {e −tL α } t>0 . In the last decades, the characterizations of function spaces associated with Schrödinger operators via semigroups and Carleson measures have attracted the attention of many authors. Let V ∈ B q , q > n/2. Using the family of operators {t∂ t e −tL } t>0 , the Carleson measure characterization of BMO L (R n ) was obtained by Dziubański-Garrigós-Martínez-Torrea-Zienkiewicz [11]. Replacing the potential V by a general Radon measure µ, in [12], Wu-Yan extended the result of [11] to generalized Schrödinger operators. The analogue in the setting of Heisenberg groups was obtained by Lin-Liu [13]. Ma-Stinga-Torrea-Zhang [14] characterized the Campanato-type spaces associated with L via the fractional derivatives of the Poisson semigroup. For further information on this topic, we refer to [15][16][17][18][19][20][21] and the references therein. Assume that L = −∆ + V, with V ∈ B q , q > n. By the regularity estimates obtained in Section 3, we establish the following equivalent characterizations: for 0 < γ < min{2α, 2αβ}, . See Theorems 3 and 4, respectively.
(ii) The regularity results for ∇ x K L t (·, ·) obtained in Section 3.2 all are pointwise estimations, which is stronger than the norm estimates. As a corollary of Lemma 8, by a trivial computation, we can obtain the estimates appearing in ( [22], Lemma 2.1) in our suitable setting; see Proposition 10.

Remark 3.
(i) In the regularity estimates of K L α,t (·, ·), one of the main tools is the subordinative formula. Due to the analytic property of the heat semigroup {e −tL } t>0 , the estimates of ∂ t K L t (·, ·) can be deduced from the Cauchy integral formula. Then, we can use the subordinative formula to obtain the related estimates of ∂ t K L α,t (·, ·). However, for the derivatives of K L t (·, ·) in the spatial variables, i.e., ∇ x K L t (·, ·), the method of ∂ t K L t (·, ·) is invalid and we need a more technical estimate; see Lemmas 8-11 for details.
(ii) Following the idea of [11], we can apply the regularities of K L α,t (·, ·) obtained in Section 3 to establish the characterizations of the BMO-type space BMO L (R n ). Since the proofs are similar to those in Section 4, we omit the details.
Some notations: • U ∼ V represents that there is a constant c > 0, such that c −1 V ≤ U ≤ cV, whose right inequality is also written as U V. Similarly, one writes V U for V ≥ cU; • For convenience, the positive constants C may change from one line to another and usually depend on the dimension n, α, β, and other fixed parameters; • Let B be a ball with the radius r. In the rest of this paper, for c > 0, we denote by B cr the ball with the same center and radius cr.

The Schrödinger Operator
Let L = −∆ + V be the Schrödinger operator on R n , n ≥ 3. Throughout the paper, we will assume that V is a nonzero, non-negative potential, and that it belongs to the reverse Hölder class B q , q > n/2,, which is defined in Definition 1. By Hölder's inequality, we can obtain B q 1 ⊂ B q 2 for q 1 ≥ q 2 > 1. One remarkable feature about the class B q is that if V ∈ B q for some q > 1, then there exists ε > 0, which depends only on n and the constant C in (1), such that V ∈ B q+ε . It is also well known that if V ∈ B q , q > 1, then V(x)dx is a doubling measure. Namely, for any r > 0, x ∈ R n , The auxiliary function m(x, V) is defined by: Clearly, 0 < m(x, V) < ∞ for every x ∈ R n , and if r = 1/m(x, V), then 1 r n−2 B(x,r) V(y)dy = 1. For simplicity, we sometimes denote 1/m(x, V) by ρ(x) in the proofs. We state some properties of m(x, V) which will be used in the proofs of the main results.

Fractional Heat Kernels Associated with L
In this section, we first state some backgrounds on the fractional heat semigroup and the fractional heat kernel associated with L. For the case V = 0, the fractional heat semigroup associated with L can not be defined using the Fourier multiplier method (2) as the Laplace operator. We strike out on a new path and introduce the fractional heat semigroup via the subordinative formula.
The Schrödinger operator L can be seen as the generator of the semigroup {e −tL } t>0 , i.e., where the limit is in L 2 (R n ). L is a self-adjointed, positive operator. The integral kernels of the semigroups {e −tL } t>0 are denoted by K L t (·, ·). It is easy to verify that the kernel K L t (·, ·) satisfies the following: For α ∈ (0, 1), the fractional power of L, denoted by L α , is defined as: Here, {e −t √ L } t>0 denotes the Poisson semigroup related to L, with the kernel P L t (·, ·) defined as: e −t 2 /4s 2s 3/2 K L s (x, y)ds.

Campanato-Type Spaces Associated with L
The Campanato-type space associated with L is defined as follows: is defined as the set of all locally integrable functions f , satisfying that there exists a constant C, such that: where the supremum is taken over all balls B centered at x B with radius r B , and: .
The norm f BMO γ L is defined as the infimum of the constants C, such that (16), above, holds.
The space BMO γ L (R n ) is equivalent to the following Lipschitz-type space related to L: Proposition 5. ( [14], Proposition 4.6) If 0 < γ ≤ 1, then the spaces BMO γ L (R n ) and C 0,γ L (R n ) are equal, and their norms are equivalent.

Regularities on Fractional Heat Semigroups Associated with L
The aim of this section is to estimate the regularities of the fractional heat kernel K L α,t (·, ·). By the use of (5), we first estimate ∂ m t K L α,t (·, ·), m ≥ 1. Then, via the solution to (8), we investigate the spatial gradient of K L α,t (·, ·). At last, we obtain the estimation of the time-fractional derivatives of K L α,t (·, ·).

Regularities of the Fractional Heat Kernel
We first investigate the regularities of K L α,t (·, ·).

ds.
By changing variables, we have: which gives: On the other hand, using the change of variables again, we obtain: The above estimate implies that: Now, combining (17) and (18), we have: which, together with the arbitrariness of M, N, indicates that: This completes the proof of Proposition 7.
(i) For any N > 0, there exists a constant C N > 0, such that: For any N > 0, there exists a constant C N > 0, such that for all |h| ≤ t 1/2α , For any N > δ, there exists a constant C N > 0, such that: Hence, By (i) of Proposition 3 and the higher-order derivative formula of the composite function, we can obtain: Notice that η α 1 (τ) ≤ C/τ 1+α . By changing the variables, we obtain: On the other hand, Finally, we have proved that, for arbitrary N > 0, For (ii), via the subordinative Formula (15), we can complete the proof by using (ii) of Proposition 3. We omit the details.
For (iii), it is easy to see that It follows from (iii) of Proposition 3 that: Because the function η α 1 (·) is continuous, the integral On the other hand, recalling that η α 1 (τ) ≤ 1/τ 1+α , we obtain: which implies that:

Lemma 8.
Suppose that V ∈ B q for some q > n. For every N > 0, there exist constants C N > 0 and c > 0, such that for all x, y ∈ R n and t > 0, the kernels K L t (·, ·) satisfy the following estimates: where ω(n) denotes the area of the unit sphere in R n . Fix t > 0 and x 0 , y 0 ∈ R n . Assume that u(·, ·) is a weak solution to the equation: Let η ∈ C ∞ 0 (B(x 0 , 2R)), with some R > 0, such that η = 1 on B(x 0 , 3R/2), |∇η| ≤ C/R, and |∇ 2 η| ≤ C/R 2 . Noticing that ∂ t u + Lu = 0, we can obtain: which, together with integration by parts, gives: Then we can obtain: Notice that it follows from Lemma 5 that (cf., [23], (1.7)): Thus, for x ∈ B(x 0 , R), it holds that: . It follows, from Propositions 1 and 3, that for any N > 0 there exists a constant C N , such that: Finally, it can be deduced from (21) that: The rest of the proof is divided into three cases: It is obvious that: Similarly, for the term M 2 , we can also obtain: Similarly, we can see that: Case 3.1: ρ(x 0 ) ≤ |x 0 − y 0 |/8. Since N is arbitrary, we can deduce from (22) that: Finally, we obtain the following estimates: This proves Lemma 8.
Our spatial gradient estimates in this paper all are pointwise estimations, which is stronger than the norm estimates. From the spatial gradient estimates in Lemma 8, we can obtain the estimates appearing in ( [22], Lemma 2.1) in the following.
Proof. We only give the details for the L 1 -estimate, and the estimate for · L 2 can be dealt with similarly. By Lemma 8, we obtain: By the change of variables, we can obtain: For M 1 , applying the change of variables again, Obviously, M 1,1 t −1/2 . For M 1,2 , we have: Noting that we can obtain M 1 t −1/2 .

Lemma 9.
Suppose that V ∈ B q for some q > n. For every N > 0, there exists a constant C N > 0, such that for all x, y ∈ R n and t > 0, the semigroup kernels K L t (·, ·) satisfy the following estimate: Proof. Assume that u(·, ·) is a weak solution of the equation Similar to Lemma 8, we can prove that for all x ∈ B(x 0 , R), |∂ t u(y, t)|.
Take u(x, t) = K L t (x, y 0 ) for fixed y 0 , and let R ∈ (0, min{ρ(x 0 ), √ t}). It can be deduced from Propositions 1 and 3 that: This, together with R < ρ(x 0 ), implies that: , taking the infimum for R on both sides of (23) reaches: ρ(x 0 )). Taking the infimum again, we obtain: This completes the proof of Lemma 9. Now, we give the gradient estimate of K L α,t (·, ·).
Proposition 11. Suppose α > 0 and V ∈ B q for some q > n. For every N > 0, there exists a constant C N > 0, such that for all x, y ∈ R n and t > 0, Proof. The subordinate formula gives: For L 1 , letting s = t 1/α u, we can obtain: Similarly, for the term L 2 , a change of variables yields: The estimates for L 1 and L 2 indicate that: On the other hand, by Lemma 9 and changing variables τ = s/t 1/α , we obtain: Finally, we obtain: The arbitrariness of N indicates that: Below, we estimate the Lipschitz continuity of |∇ x K L t (·, ·)|.
Let u(x 0 , t) = K L t (x 0 , y 0 ). Then, Define a function F(x) = x 1−n/q (x + 1/x), x > 0. Then, we can see that for x > n/(2q − n), F (x) > 0, i.e., F is increasing, which means that the function f := F( √ t/R) is decreasing for R ∈ (0, (2q − n)t/n). Below, we divide the rest of the proof into two cases: We further divide the discussion into two subcases: For this case, the function f (R) has the infimum f (ρ(x 0 )) for R ∈ (0, ρ(x 0 )). Then, taking the infimum for R on both sides of (24), we can use the fact that ρ(x 0 ) < √ t to obtain: Then, we obtain: It is easy to see that: Case 2: √ t < min{ρ(x 0 ), |x 0 − y 0 |/8}. Similar to Case 1, we divide the discussion into two subcases again: Case 2.1: 0 < R < √ t < (2q − n)t/n < min{ρ(x 0 ), |x 0 − y 0 |/8}. It follows from (24) that: Taking the infimum on both sides (25) reaches: Similarly, taking the infimum on both sides of (25), we obtain: If ρ(x 0 ) < |x 0 − y 0 |/8, then: Lemma 11. Suppose that V ∈ B q for some q > n. Let δ = 1 − n/q. For every N > 0, there exists a constant C N > 0, such that for all x, y ∈ R n and t > 0, the semigroup kernels K L t (·, ·) satisfy the following estimate: for |h| < |x − y|/4, Proof. Similar to Lemma 10, we take R ∈ (0, min{ρ(x 0 ), √ t}) and obtain: Then, the following two cases are considered: It is obvious that Case 2.1 comes back to Case 1. For Case 2.2, letting R → √ t on the right-hand side of (26), we have: Proposition 12. Suppose that α > 0 and V ∈ B q for some q > n. Let δ = 1 − n/q. For every N > 0, there exists a constant C N > 0, such that for all x, y ∈ R n and t > 0, the fractional heat kernels K L α,t (·, ·) satisfy the following estimate: for |h| < |x − y|/4, Proof. By the subordinative formula and Lemma 10, we can obtain:

ds.
We first estimate M 6 and apply a change of variables to obtain: Similarly, for M 7 , we have: which gives: On the other hand, we can deduce from Lemma 11 that: Finally, the arbitrariness of N indicates that: which proves Proposition 12.

Estimation on Time-Fractional Derivatives
In this section, we give some gradient estimates for the fractional heat kernel associated with the variable t. Define an operator: 1), and β > 0.
In the next proposition, we give the Lipschitz continuity of D L α,t (·, ·).

Characterization of Campanato-Morrey Spaces Associated with L
Firstly, we deduce a reproducing formula: Lemma 12. Let α ∈ (0, 1) and β > 0. The operator t β ∂ β t e −tL α defines an isometry from L 2 (R n ) into L 2 (R n+1 + , dxdt/t). Moreover, in the sense of L 2 (R n ), it holds that: Proof. Note that for dE(λ), the spectral resolution of the operator L, it follows from that: Then, for f ∈ L 2 (R n ), we have: Below, we only prove that for every pair of sequences n k ↑ ∞ and k ↓ 0 as k → ∞, If (32) holds, there exists a function h ∈ L 2 (R n ), such that: This means h = C α,β f . Now, we verify (32). As k → ∞, we can apply the functional calculus to deduce that: can be dealt with similarly.
The following inequality was established by Harboure-Salinas-Viviani [37]: For any pair of measurable functions F and G on R n+1 + , we have: In Lemma 13, letting we have: On the left-hand side of (33), since we can obtain, via the change of variables, On the right-hand side of (33), using change of variables again, we obtain: For α ∈ (0, 1) and β > 0, define an area function S L α,β as follows:
Finally, we can obtain the following characterization of BMO γ L (R n ) corresponding to the time-fractional derivative: Theorem 3. Let V ∈ B q , q > n/2. Assume that α ∈ (0, 1), β > 0, and 0 < γ < min{1, 2α, 2βα}. Let f be a function, such that: for some > 0. The following statements are equivalent: Proof. For I, we have: We further divide the estimation of I I into the following two cases: Case 1: ρ(x) ≤ t 1/2α . By Proposition 14, Case 2: ρ(x) > t 1/2α . We use Proposition 16 to obtain that there exists δ > γ, such that: Below, we consider the characterization of BMO γ L (R n ) via the the spatial gradient. Define a general gradient as ∇ α := (∇ x , ∂ 1/2α t ).

Conclusions
In this paper, with the aid of the fundamental solution of the heat equation associated with the Schrödinger operators, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K L α,t (·, ·), respectively. Finally, as an application, we establish a Carleson measure characterization of the Campanato-type space BMO γ L (R n ) via the fractional heat semigroup {e −tL α } t>0 .