Global Heat Kernel Estimates for $\Delta+\Delta^{\alpha/2}$ in Half-space-like domains

Suppose that $d\ge 1$ and $\alpha\in (0, 2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]\}$ on half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$ for all time $t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a \in (0, 1]$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$. Thus it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{\Delta+ a^\alpha \Delta^{\alpha/2}; \ a\in (0, 1]\}$ in half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$.


Introduction and Setup
This paper is a natural continuation of [5] where small time sharp two-sided estimates for the Dirichlet heat kernel of ∆ + ∆ α/2 on any C 1,1 open sets and large time sharp two-sided estimates for bounded C 1,1 open sets are obtained. In this paper we give sharp two-sided estimates for the Dirichlet heat kernel of ∆ + ∆ α/2 on half-space-like C 1,1 domains for all time. The large time Dirichlet heat kernel estimates for half-space-like domains are very different from those for bounded open sets. See below for the definition of half-space-like C 1,1 open sets.
Throughout this paper, we assume that d ≥ 1 is an integer and α ∈ (0, 2). Let X 0 = (X 0 t , t ≥ 0) be a Brownian motion in R d with generator ∆ = d i=1 ∂ 2 ∂x 2 i , and Y = (Y t , t ≥ 0) be an independent (rotationally) symmetric α-stable process in R d whose generator is the fractional Laplacian ∆ α/2 . For u ∈ C ∞ c (R d ), the space of smooth functions with compact support, the fractional Laplacian can be written in the form ∆ α/2 u(x) = lim ε↓0 {y∈R d : |y−x|>ε} (u(y) − u(x)) A(d, α) |x − y| d+α dy, (1.1) where A(d, α) := α2 α−1 π −d/2 Γ( d+α 2 )Γ(1 − α 2 ) −1 . Here Γ is the Gamma function defined by Γ(λ) := ∞ 0 t λ−1 e −t dt for every λ > 0. For any a > 0, we define X a by X a t := X 0 t + aY t . We will call the process X a the independent sum of the Brownian motion X 0 and the symmetric α-stable process Y with weight a > 0. The Lévy process X a is uniquely determined by its characteristic function for every x ∈ R d and ξ ∈ R d and its infinitesimal generator is ∆ + a α ∆ α/2 . Since (1 − cos(ξ · y)) a α A(d, α) |y| d+α dy, X a has Lévy intensity function The function J a (x, y) determines a Lévy system for X a , which describes the jumps of the process X a : for any non-negative measurable function f on R + × R d × R d with f (s, y, y) = 0 for all y ∈ R d , any stopping time T (with respect to the filtration of X a ) and any x ∈ R d , Let p a (t, x, y) be the transition density of X a with respect to the Lebesgue measure on R d . The function p a (t, x, y) is smooth on (0, ∞) × R d × R d . For any λ > 0, (λX a λ −2 t , t ≥ 0) has the same distribution as (X aλ (α−2)/α t , t ≥ 0) (see the second paragraph of [5,Section 2]), so we have p aλ (α−2)/α (t, x, y) = λ −d p a (λ −2 t, λ −1 x, λ −1 y) for t > 0 and x, y ∈ R d .
For a > 0 and C > 0, define h a C (t, x, y) (1. 4) Here and in the sequel, we use ":=" as a way of definition and, for a, b ∈ R, a ∧ b := min{a, b} and a ∨ b := max{a, b}. The following sharp two-sided estimates on p a (t, x, y) follow from (1.3) and the main results in [10,22] that give the sharp estimates on p 1 (t, x, y).
We record a simple but useful observation. Its proof will be given at the end of this section.
For an open set D ⊂ R d and x ∈ D, we will use δ D (x) to denote the Euclidean distance between x and D c . For a domain D ⊂ R d and λ 0 ≥ 1, we say the path distance in D is comparable to the Euclidean distance with characteristic λ 0 if for every x, y ∈ D, there is a rectifiable curve l in D connecting x to y so that the length of l is no larger than λ 0 |x − y|. Clearly, such a property holds for all bounded C 1,1 domains, C 1,1 domains with compact complements and domains above the graphs of bounded C 1,1 functions.
For any open subset D ⊂ R d , we use τ a D to denote the first time the process X a exits D. We define the process X a,D by X a,D t = X a t for t < τ a D and X a,D t = ∂ for t ≥ τ a D , where ∂ is a cemetery point. X a,D is called the subprocess of X a in D. The generator of X a,D is (∆ + a α ∆ α/2 )| D . It follows from [10] that X a,D has a continuous transition density p a D (t, x, y) with respect to the Lebesgue measure.
One can easily see that, when D is bounded, the operator −(∆ + a α ∆ α/2 )| D has discrete spectrum. In this case, we use λ a,D 1 > 0 to denote the smallest eigenvalue of −(∆ + a α ∆ α/2 )| D . The following is a particular case of a more general result proved in [5, Theorem 1.3] (cf. Proposition 1.2 above).
Note that Theorem 1.3 does not give large time estimates for p a D (t, x, y) when D is unbounded. The goal of this paper is to establish two-sided large time estimates on p a D (t, x, y) for a large class of unbounded C 1,1 domains, namely half-space-like C 1,1 domains. A domain D is said to be halfspace-like if, after isometry, there exist two real numbers b 1 ≤ b 2 such that H b 2 ⊂ D ⊂ H b 1 . Here and throughout this paper, H b stands for the set {x = (x 1 , . . . , x d ) ∈ R d : x d > b}. We will denote H 0 by H. Now we are in a position to state the main result of this paper. For a > 0, define φ a (r) := r ∧ (r/a) α/2 . (1.5) Remark 1.5 (i) The Lévy exponent for X a is Φ a (|ξ|) with Φ a (r) := r 2 + a α r α . The function φ a (r) is related to Φ a (r) as follows.
Here for two non-negative functions f and g, the notation f ≍ g means that there is a positive constant c ≥ 1 so that g(x)/c ≤ f (x) ≤ cg(x) in the common domain of definition for f and g. Hence in view of Theorem 1.1, the estimate (1.5) can be restated as follows. For every M > 0, there are constants c 1 , c 2 ≥ 1 so that for every a ∈ (0, M ] and (t, x, y) ∈ (0, ∞) × D × D, We conjecture that the above Dirichlet heat kernel estimates hold for a large class of rotationally symmetric Lévy processes in R d ; see [6,Conjecture].
a α/(α−2) = 1, and consequently Hence in view of Theorem 1.1 and Proposition 1.2, the statement of Theorem 1.4 can be restated as follows. For all a ∈ (0, M ] and (t, x, y) ∈ (0, a 2α/(α−2) ] × D × D, and for all a ∈ (0, M ] and (t, x, y) ∈ [a 2α/(α−2) , ∞) × D × D, In fact, Theorem 1.4 will be proved in this form. 2 Remark 1.6 Unlike [6,11], there are dramatic differences between the behavior of the heat kernel p a D (x, y) on half-space-like C 1,1 domains and disconnected half-space-like C 1,1 open sets even if x and y are in the same connected component. For example, if D is H ∪ B(x 0 , 1) where x 0 = (0, . . . , 0, −2) and x, y ∈ B(x 0 , 1), then, as a → 0, p a D (x, y) converges to p 0 B(x 0 ,1) (x, y), the Dirichlet heat kernel for Brownian motion on B(x 0 , 1). Thus, in this case, the heat kernel estimates for p a D (t, x, y) when t is large cannot be of the form (1.5) even if x and y are in the same connected component. Furthermore, as one can see from [5,Theorem 1.3], when D is a disconnected half-space-like C 1,1 open set (containing bounded connected component), we can not expect that the heat kernel estimates for p a D (x, y) to be written in a simple form as the one in (1.5). To keep our exposition as transparent as possible, we are content with establishing the heat kernel estimates for half-spacelike C 1,1 domains. 2 Integrating the heat kernel estimates in Theorem 1.4 with respect to t, we get sharp two-sided estimates on the Green function G a D (x, y) := ∞ 0 p a D (t, x, y)dt for X a in half-space-like C 1,1 domains D.
(1.11) Remark 1.8 (i) Note that, when d ≥ 3, g a D (x, y) is independent of a and is comparable to the Green function of Brownian motion in a bounded C 1,1 domain or in a domain above the graph of a bounded C 1,1 function. On the other hand, when d ≤ 2, g a D (x, y) depends on a, which is due to recurrent nature of one-and two-dimensional Brownian motion.
(ii) Observe that if (X a,D t , t ≥ 0) is the subprocess in D of the independent sum of a Brownian motion and a symmetric α-stable process in R d with weight a, then (λX a,D λ −2 t , t ≥ 0) is the subprocess in λD of the independent sum of a Brownian motion and a symmetric α-stable process in R d with weight aλ (α−2)/α (see the second paragraph of [5, Section 2]). Consequently for any λ > 0, we have for t > 0 and x, y ∈ λD. (1.12) When D is a half space, we see from (1.12) that Theorems 1.4 and 1.7 hold with M = ∞.
(iii) The estimates in Theorems 1.4 and 1.7 are uniform in a ∈ (0, M ] in the sense that the constants c 1 , c 2 and c in the estimates are independent of a ∈ (0, M ]. Since X a converges weakly to X 0 , by taking a → 0 these estimates yield the following estimates for the heat kernel p 0 D (t, x, y) and Green function G 0 (x, y) of Brownian motion in half-space-like domains D in which the path distance is comparable to the Euclidean distance: (1.14) The estimates (1.13) and (1.14) extend the main results in [20], where the corresponding estimates were established for domains in R d with d ≥ 3 that are above the graphs of bounded C 1,1 functions.
(iv) By Theorem 1.4, the boundary decay rate of the Dirichlet heat kernel of ∆ + ∆ α/2 is given . This indicates that the Dirichlet heat kernel estimates for ∆ + ∆ α/2 in halfspace-like C 1,1 domains cannot be obtained by a "simple" perturbation argument from ∆ nor from ∆ α/2 .
The main difficulty of this paper is to obtain the correct boundary decay rate of the Dirichlet heat kernel of ∆ + ∆ α/2 . In [5], the correct boundary decay rate for small t was established by using some exit distribution estimates obtained in [7]. Unfortunately the estimates in [7] are not suitable for the present case. Thus, in this paper we give some different forms of exit distribution estimates that are suitable for large time estimates. The first step is, similar to [2,12,7], to compute (∆ + ∆ α/2 )h for certain test functions. But unlike [7], we do not use combinations of test functions to serve as subharmonic and superharmonic functions to obtain our desired estimates. Instead, we use a generalization of Dynkin's formula to obtain the desired exit distribution estimates directly. We believe that our approach to obtain the correct boundary decay rate is quite general and may be used for other types of jump processes.
Throughout this paper, the constants C 1 , C 2 , C 3 , R 0 , R 1 , R 2 , R 3 will be fixed. The lower case constants c 1 , c 2 , . . . will denote generic constants whose exact values are not important and can change from one appearance to another. The dependence of the lower case constants on the dimension d will not be mentioned explicitly. We will use ∂ to denote a cemetery point and for every function f , we extend its definition to ∂ by setting f (∂) = 0. We will use dx or m(dx) to denote the Lebesgue measure in R d . For a Borel set A ⊂ R d , we also use |A| to denote its d-dimensional Lebesgue measure. For every function f , let f + := f ∨ 0.
In the remainder of this paper we will always assume that D is a half-space-like C 1,1 domain with C 1,1 characteristic (R 0 , Λ 0 ) and H b ⊂ D ⊂ H for some b > 0 such that the path distance in D is comparable to the Euclidean distance with characteristic λ 0 and that t 0 , x 0 and y 0 are described as below.
Fix t 0 ≥ b 2 and let e d be the unit vector in the direction of the x d -axis. For x and y in D, define the points (1. 16) and |x − x 0 | = |y − y 0 | = 2t 1/2 0 . Note that when D = H, we can take t 0 to be any positive number. Now as a consequence of Theorem 1.3, we have the following result.
Proof. Let C 2 be the constant in Theorem 1.3 (i) with T = t 0 . From Proposition 1.2 and Theorem 1.3 (i), it is easy to see that (1.18) By Theorem 1.3 (i) and (1. 16), we see that Thus in these cases, (1.17) follows from (1. 19).
In the case z ∈ B(x, Proof. By Theorem 1.3 (i), we see that (1.21) By the same argument as in the proof of Lemma 1.9, . The assertion of the lemma follows by considering each cases in (1.21). 2 The following elementary result will play an important role later in this paper. Recall that D, t 0 , x 0 and y 0 are described as above.
Proof. Note that . Thus in this case, the conclusion of the lemma is trivial. From now on, we assume that δ D (x) ≤ t 1/2 0 . In this case, using the fact t ≥ t 0 and a ∈ (0, M ], we have The proof is now complete. 2 Proof of Proposition 1.2. We first deal with the case a = 1. For t ≥ c 1 and r ≥ 0, On the other hand, for r ≥ 1, So for t ∈ (0, c 1 ] and r ≥ 1, provided either λ 2 t ≥ c 1 or λ|x − y| ≥ 1. This proves the proposition. 2

Preliminary estimates
We will focus on the case D = H in Sections 2-4. In this section we will prove some preliminary estimates that will be used to establish our heat kernel estimates in H. We start with some onedimensional results. Let S be the sum of a unit drift and an α/2-stable subordinator and let W be an independent one-dimensional Brownian motion. Define a process Z by Z t = W St . The process Z is simply the process X 1 in the case of dimension 1 defined in the previous section. We will use the fact that S is a complete subordinator, that is, the Lévy measure of S has a completely monotone density (for more details see [17] or [21]). Let Z t := sup{0 ∨ Z s : 0 ≤ s ≤ t} and let L t be a local time of Z − Z at 0. L is also called a local time of the process Z reflected at the supremum. Then the right continuous inverse L −1 t of L is a subordinator and is called the ladder time process of Z. The process Z L −1 t is also a subordinator and is called the ladder height process of Z. (For the basic properties of the ladder time and ladder height processes, we refer our readers to [1, Chapter 6].) Let V (dr) denote the potential measure of the ladder height process Z L −1 t of Z and v(r) its density, which is a decreasing function on [0, ∞). We know by [16, (5.1)] that v(r) ≍ 1 ∧ r α/2−1 for r > 0. (2.1) Let G (0,∞) be the Green function of Z (0,∞) , the subprocess of Z in (0, ∞). By using [1, Theorem 20, p. 176] which was originally proved in [18], the following formula for G (0,∞) was shown in [14, Proposition 2.8]: For any r > 0, let G (0,r) be the Green function of Z (0,r) , the subprocess of Z in (0, r). Then we have the following result.

Now the proposition follows by the symmetry. 2
Now we return to the process X 1 in R d . Recall that C ∞ c (R d ) is contained in the domain of the L 2 -generator ∆ + ∆ α/2 of X 1 and 3) The following estimates on harmonic measures will play a crucial role in Section 3.

(2.20)
On the other hand, by the Lévy system of X 1 , This together with (2.20) establishes the lemma. 2

Upper bound heat kernel estimates on half-space
In this section we will establish the desired large time upper bound for p 1 H (t, x, y).
Lemma 3.1 For any t 0 > 0 and R > 0, there exists c = c(α, t 0 , R) > 1 such that for t ≥ t 0 and Proof. Clearly, we can assume R ≤ t 1/α 0 and we only need to show the theorem for R ≤ δ H (x) < t 1/α . Let u(x) = (x + d ) α/2 + 1 and U (r) := {x ∈ H; x d < r}. By (2.8), for every x ∈ H with δ H (x) ≥ R, Using the same approximation argument as in the proof of Lemma 2.4 with u k (z) := (g k * u)(z) where g k is the function defined in the proof of Lemma 2.4 and letting k → ∞, we see that for Applying this and Proposition 2.1, we get that for R < δ H (x) < t 1/α .
Proof. By the semigroup property and symmetry, Now the lemma follows from Theorem 1.1 and Lemma 3.1. 2 The next lemma and its proof are given in [5] (also see [3, Lemma 2] and [4, Lemma 2.2]).
Suppose ε < 1 3 , then by the parabolic Harnack inequality in [10,22], where the constant c 1 = c 1 (t 0 , α) > 0 is independent of y ∈ R d . Thus The next result holds for any symmetric discontinuous Hunt process that possesses a transition density and whose Lévy system admitting jumping density kernel. Its proof is the same as that of [6,Lemma 3.3] and so it is omitted here.
(iii) Now we consider the case d = 1. We again deal with three cases separately.