Estimates of Dirichlet heat kernels for subordinate Brownian motions

In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and Laplace exponent of the corresponding subordinator only.


Introduction
Transition densities of Lévy processes killed upon leaving an open set D are Dirichlet heat kernels of the generators of such processes on D. For example, the classical Dirichlet heat kernel, which is the fundamental solution of the heat equation in D with zero boundary values, is the transition density of Brownian motion killed upon leaving D. Since, except in some special cases, explicit forms of the Dirichlet heat kernels are impossible to obtain, obtaining sharp estimates of the Dirichlet heat kernels has been a fundamental problem both in probability theory and in analysis.
After the fundamental work in [11], sharp two-sided estimates for the Dirichlet heat kernel p D (t, x, y) of non-local operators in open sets have been studied a lot (see [2,3,5,6,7,12,13,15,16,17,14,18,19,20,25,35,36,37]). In particular, very recently in [6,19], sharp two-sided estimates of p D (t, x, y) were obtained for a large class of rotationally symmetric Lévy processes when the radial parts of their characteristic exponents satisfy weak scaling conditions whose upper scaling exponent is strictly less than 2. A still remaining open question in this direction is that, when the upper scaling exponent is not strictly less than 2, for how general discontinuous Lévy processes one can prove sharp two-sided estimates for their Dirichlet heat kernels. In this paper we investigate this question for subordinate Brownian motions, which form a very large class of Lévy processes.
A subordinate Brownian motion in R d is a Lévy process which can be obtained by replacing the time of Brownian motion in R d by an independent subordinator (i.e., an increasing Lévy process starting from 0). The subordinator used to define the subordinate Brownian motion X can be interpreted as "operational" time or "intrinsic" time. For this reason, subordinate Brownian motions have been used in applied fields a lot.
To obtain the sharp Dirichlet heat kernel estimates, it is necessary to know the sharp heat kernel estimates in R d . Recently heat kernel estimates for discontinuous Markov processes have been a very active research area and, for a large class of purely discontinuous Markov processes, the sharp heat kernel estimates were obtained in [4,8,10,21,22,23,32,49,50]. But except [45,50], for the estimates of the heat kernel, a common assumption on the purely discontinuous Markov processes in R d considered so far is that their weak scaling orders were always strictly between 0 and 2. Very recently in [45], the second-named author considered a large class of purely discontinuous subordinate Brownian motions whose weak scaling order is between 0 and 2 including 2, and succeeded in obtaining sharp heat kernel estimates of such processes. In this sense, the results in [45] extend earlier works in [4].
Motivated by [45], the main purpose of this paper is to establish sharp two-sided estimates of p D (t, x, y) for a large class of subordinate Brownian motions in C 1,1 open set whose weak scaling order is not necessarily strictly below 2. Our estimates are explicit and written in terms of the dimension d, the Euclidian distance |x − y| for x, y ∈ D, the distance to the boundary of D for x, y ∈ D and the Laplace exponent of the corresponding subordinator only. See Section 8 for examples, in particular, (8.2)-(8.3) for estimates of the Dirichlet heat kernels.
This paper is also motivated by [5,7], and, several results and ideas in [7,45] will be used here. It is shown in [5] that, when weak scaling orders of characteristic exponents of unimodal Lévy processes in R d are strictly below 2, sharp estimates on the survival probabilities for the unimodal Lévy processes can be obtained without the information on sharp two-sided estimates for the Dirichlet heat kernels. Such estimates in [5] can not be used in the setting of this paper.
We will use the symbol ":=," which is read as "is defined to be." In this paper, for a, b ∈ R we denote a ∧ b := min{a, b} and a ∨ b := max{a, b}. By B(x, r) = {y ∈ R d : |x − y| < r} we denote the open ball around x ∈ R d with radius r > 0 . We also use convention 0 −1 = +∞. For any open set V , we denote by δ V (x) the distance of a point x to V c . We sometimes write point z = (z 1 , . . . , z d ) ∈ R d as ( z, z d ) with z ∈ R d−1 .
(1) We say that f satisfies the lower scaling condition L a (γ, C L ) if there exist a ≥ 0, γ > 0 and C L ∈ (0, 1] such that f (λt) f (λ) ≥ C L t γ for all λ > a and t ≥ 1 . (1.5) We say that f satisfies the lower scaling condition near infinity if the above constant a is strictly positive and we say f satisfies the lower scaling condition globally if a = 0. (2) We say f satisfies the upper scaling condition U a (δ, C U ) if there exist a ≥ 0, δ > 0 and C U ∈ [1, ∞) such that f (λt) f (λ) ≤ C U t δ for all λ > a and t ≥ 1 . For any open set D ⊂ R d , the first exit time of D by the process X is defined by the formula τ D := inf{t > 0 : X t / ∈ D} and we use X D to denote the process obtained by killing the process X upon exiting D. By the strong Markov property, it can easily be verified that p D (t, x, y) := p(t, x, y) − E x [p(t − τ D , X τ D , y) : τ D < t], t > 0, x, y ∈ D, (1.7) is the transition density of X D . Note that from (1.4) we see that sup |x|≥β,t>0 p(t, x) < ∞ for all β > 0. Using this estimate and the continuity of p, it is routine to show that p D (t, x, y) is symmetric and continuous (see [27]).
We say that D ⊂ R d (when d ≥ 2) is a C 1,1 open set with C 1,1 characteristics (R 0 , Λ) if there exist a localization radius R 0 > 0 and a constant Λ > 0 such that for every z ∈ ∂D there exist a C 1,1 -function ϕ = ϕ z : R d−1 → R satisfying ϕ(0) = 0, ∇ϕ(0) = (0, . . . , 0), ∇ϕ ∞ ≤ Λ, |∇ϕ(x) − ∇ϕ(w)| ≤ Λ|x − w| and an orthonormal coordinate system CS z of z = (z 1 , · · · , z d−1 , z d ) := ( z, z d ) with origin at z such that D ∩ B(z, R 0 ) = {y = (ỹ, y d ) ∈ B(0, R 0 ) in CS z : y d > ϕ( y)}. The pair (R 0 , Λ) will be called the C 1,1 characteristics of the open set D. Note that a C 1,1 open set D with characteristics (R 0 , Λ) can be unbounded and disconnected, and the distance between two distinct components of D is at least R 0 . By a C 1,1 open set in R with a characteristic R 0 > 0, we mean an open set that can be written as the union of disjoint intervals so that the infimum of the lengths of all these intervals is at least R 0 and the infimum of the distances between these intervals is at least R 0 .
It is well-known that C 1,1 open set D with the characteristic (R 0 , Λ) satisfies the interior and exterior ball conditions with the characteristic R 1 > 0, that is, there exists R 1 > 0 such that the following holds: for all x ∈ D with δ D (x) ≤ R 1 there exist balls B 1 ⊂ D and B 2 ⊂ D c whose radii are R 1 such that x ∈ B 1 and δ B 1 (x) = δ D (x) = δ B 2 c (x). Without loss of generality whenever we consider a C 1,1 open set D with the characteristic (R 0 , Λ), we will take R 0 as the characteristic of the interior and exterior ball conditions of D, that is, We say that the path distance in a connected open set U is comparable to the Euclidean distance with characteristic λ 1 if for every x and y in U there is a rectifiable curve l in U which connects x to y such that the length of l is less than or equal to λ 1 |x − y|. Clearly, such a property holds for all bounded C 1,1 domains (connected open sets), C 1,1 domains with compact complements, and a domain consisting of all the points above the graph of a bounded globally C 1,1 function.
In this paper, for the Laplace exponent φ of a subordinator, we define the function H : (0, ∞) → [0, ∞) by H(λ) := φ(λ) − λφ ′ (λ). The function H, which appeared earlier in the work of Jain and Pruitt [31], took a central role in [45] in obtaining the sharp heat kernel estimates of the transition density of the corresponding subordinate Brownian motion X in R d .
Obviously, this function H will also naturally appear in this paper in the estimates of the transition density of X in open subsets. Under the weak scaling assumptions on H we will obtain the sharp two-sided estimates of p D (t, x, y). Recall that δ D (x) is the distance between x and the boundary of D.
In the main results of this paper, we will impose the following assumption: there exists a positive constant c > 0 such that j(r) ≤ cj(r + 1), r > 1.
We are now ready to state the main result of this paper.
be a subordinator with zero drift whose Laplace exponent is φ and let X = (X t ) t≥0 be the corresponding subordinate Brownian motion in R d . Assume that (1.8) holds and that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 and γ > 2 −1 1 δ≥1 (a) For every T > 0, there exist constants c 1 , C 0 and a U > 0 such that for every (t, x, y) When D is an unbounded, we further assume that H satisfies L 0 (γ 0 , C L ) and U 0 (δ, C U ) with δ < 2 and that the path distance in each connected component of D is comparable to the Euclidean distance with characteristic λ 1 . Then for every T > 0 there exist constants c 2 , a L > 0 such that for every (t, x, y) ∈ (0, T ] × D × D, where −λ D < 0 is the largest eigenvalue of the generator of X D . We emphasize that we put the assumption γ > 2 −1 1 δ≥1 on lower scaling condition near infinity, not globally, i.e., we don't assume that γ 0 > 2 −1 1 δ≥1 in Theorem 1.3(b).
When D is a half space-like domain, we have the global estimates for all t > 0 on the Dirichlet heat kernel. Theorem 1.4. Let S = (S t ) t≥0 be a subordinator with zero drift whose Laplace exponent is φ and let X = (X t ) t≥0 be the corresponding subordinate Brownian motion in R d . Suppose that D is a domain consisting of all the points above the graph of a bounded globally C 1,1 function and H satisfies L 0 (γ, C L ) and U 0 (δ, C U ) with δ < 2. Then there exist c ≥ 1 and a L , a U > 0 such that both (1.11) and (1.12) hold for all (t, x, y) ∈ (0, ∞) × D × D.
The assumption that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 in Theorems 1.3 and 1.4 allows us to cover several interesting cases where the scaling order of the characteristic exponent Ψ(ξ) = φ(|ξ| 2 ) of X is 2.
The rest of the paper is organized as follows. In Section 2, we revisit [45] and improve one of the main results of [45] in Theorem 2.9. This result will be used in Sections 5-7 to show the sharp two-sided estimates of the Dirichlet heat kernel when φ satisfies the lower scaling condition near infinity or H(λ) = φ(λ) − λφ ′ (λ) satisfies the lower and upper scaling conditions near infinity. In Section 3 we first show that the scale-invariant parabolic Harnack inequality holds with explicit scaling in terms of Laplace exponent. Then using this we give some preliminary interior lower bound of the Dirichlet heat kernel. Using such lower bound of the Dirichlet heat kernel, Theorem 2.9, (4.1), and the estimates on exit probabilities in Section 4 we prove the estimates of the survival probabilities and the sharp two-sided estimates of the transition density p D (t, x, y) for the killed process X D . This is done in Sections 5-6. As an application of Theorem 1.3, in Section 7 we establish the estimates on the Green functions in bounded C 1,1 domain. Section 8 contains some examples of subordinate Brownian motions and the sharp two-sided estimates of transition density and Green function of them.
In this paper, we use the following notations. For a Borel set W in R d , ∂W , W and |W | denote the boundary, the closure and the Lebesgue measure of W in R d , respectively. For s ∈ R, s + := s ∨ 0 Throughout the rest of this paper, the positive constants a 0 , a 1 , T 1 , M 0 , M 1 , R, R * , R 0 , R 1 , C, C i , i = 0, 1, 2, . . . , can be regarded as fixed, while the constants c i = c i (a, b, c, . . .), i = 0, 1, 2, . . . , denote generic constants depending on a, b, c, . . ., whose exact values are unimportant. They start anew in each statement and each proof. The dependence of the constants on φ, γ, δ, C L , C U and the dimension d ≥ 1, may not be mentioned explicitly.

Preliminary Heat kernel estimates in R d
Throughout this paper we assume that φ is the Laplace exponent of a subordinator S. Without loss of generality we assume that φ(1) = 1. In this section we revisit [45] and improve the main result of [45] for the case that φ satisfies the lower scaling condition near infinity.
In fact, suppose a ≤ λ < b and x ≥ 1.
In fact, suppose a ≤ λ < b and x ≥ 1.
(2.5) By Remark 2.2 we also have Lemma 2.4. If φ satisfies L a (γ, C L ) for some a > 0, then for every b ∈ (0, a], Throughout this paper, the process X = (X t : t ≥ 0) is a subordinate Brownian motion whose characteristic exponent is φ(|x| 2 ). Recall that x → j(|x|) is the Lévy density of the subordinate Brownian motion X defined in (1.2), which gives rise to a Lévy system for X describing the jumps of X; For any x ∈ R d , stopping time τ (with respect to the filtration of X), and nonnegative measurable function f on R + × R d × R d with f (s, y, y) = 0 for all y ∈ R d and s ≥ 0 we have (e.g., see [22, Appendix A]). The next lemma holds for every symmetric Lévy process and it follows from [46, (3.2)] and [29,Corollary 1]. Recall that τ D is the first exit time of D by the process X.
Recall that X has a transition density p(t, x, y) = p(t, y−x) = p(t, |y−x|) of the form (1.4). We first consider the estimates of p(t, x) under the assumption that φ satisfies L a (γ, C L ) for some a > 0. Note that L a (γ, C L ) implies lim λ→∞ φ(λ) = ∞. Proposition 2.6. If φ satisfies L a (γ, C L ) for some a > 0, then for every T > 0 there exists c = c(T ) > 0 such that for all t ≤ T and x ∈ R d , Proposition 2.7. If φ satisfies L a (γ, C L ) for some a > 0, then for every T > 0 there exist c 1 , c 2 > 0 such that for all t ≤ T and x ∈ R d satisfying tφ(|x| −2 ) ≤ 1, Proposition 2.8. If φ satisfies L a (γ, C L ) for some a > 0, then for every T > 0 there exists c = c(T ) > 0 such that for all t ≤ T and x ∈ R d , In particular, if additionally tφ(M|x| −2 ) ≥ 1 holds for some M > 0, then we have (2.7) Proof. We closely follow the proof of [45,Proposition 3.5]. Let ρ ∈ (0, 1) be the constant in Proposition 2.1 and, without loss of generality, we assume T ≥ ρφ −1 (a). Using (1.4) we get Note that, by Lemma 2.4, we have that for 0 < t < T = ρφ(b) −1 , ✷ We now revisit [45].
Theorem 2.9. Let S = (S t ) t≥0 be a subordinator with zero drift whose Laplace exponent is φ and let X = (X t ) t≥0 be the corresponding subordinate Brownian motion in R d and p(t, x, y) = p(t, y − x) be the transition density of X.
If φ satisfies L a (γ, C L ) for some a > 0, then for every T > 0 there exist

11)
and Throughout this section, we assume that φ has no drift and satisfies L a (γ, C L ) for some a ≥ 0. Recall that p D (t, x, y) defined in (1.7) is the transition density for X D , the subprocess of X killed upon leaving D. Let The law of the time-space process s → Z s starting from (t, x) will be denoted as P (t,x) .
Recall that Φ(r) = 1 φ(1/r 2 ) . In this section, we will first prove that X satisfies the scaleinvariant parabolic Harnack inequality with explicit scaling in terms of Φ. That is, Theorem 3.2. Suppose that φ has no drift and satisfies L a (γ, C L ) for some a ≥ 0. For every M > 0, there exist c > 0 and c 1 , c 2 ∈ (0, 1) depending on d, γ and C L (also depending on M and a if a > 0) such that for every Theorem 3.2 clearly implies the elliptic Harnack inequality. Thus this extends the main result of [29].
To prove Theorem 3.2, we first observe that for each c 1 , b > 0 and every r, t > 0 satisfying Using this and the fact that φ ≥ H, we see that for each Thus by [45] (for a = 0) and Proposition 2.8 and (2.10) (for a > 0) we have the following bounds: and where the above constant C > 1 depends on T if a > 0. Now, using (3.2) and (3.3) we get the following lower bound.
Suppose that φ has no drift and satisfies L a (γ, C L ) for some a ≥ 0. For every M > 0, there exist constants c > 0 and ε ∈ (0, 1/2) such that for every Proof. Since the proof for the case a = 0 is almost identical to the proof for the case a > 0, we will prove the proposition for the case a > 0 only. Fix x 0 ∈ R d and let B r := B(x 0 , r). The constant ε ∈ (0, 1/2) will be chosen later. For Now combining (1.7), (3.2), (3.3) and (3.5) we have that for x, y ∈ B εΦ −1 (t) and t ∈ (0, Φ(εr)], Observe that and so Consequently, we have from (3.6) and (3.7), Integrating (3.2) and (3.4), we obtain that there exist constants c 1 , c 2 > 0 such that We say (UJS) holds for J if there exists a positive constant c such that for every y ∈ R d , Proof of Theorem 3.2. Note that (UJS) always holds for our Lévy density x → j(|x|) since j is non-increasing. (see [9, page 1070]). Thus, using Proposition 3.
Proof. Using Theorem 3.2, the proof for the case that φ satisfies L 0 (γ, C L ) is identical to that of [7,Proposition 3.4]. Even through the proof is similar, for reader's convenience we provide the proof for the case that φ satisfies L a (γ, C L ) for a > 0.
Without loss of generality we assume a = 1. We fix b, T > 0 and (t, Thus, by the symmetry of p D , Theorem 3.2 and Lemma 2.3(a), there exists This together with Lemma 2.5 yields that there exist c 2 , c 3 > 0 such that Proof. Again, using Proposition 3.4, the proof for the case that φ satisfies L 0 (γ, C L ) is the same as that of [7,Proposition 3.5], and for reader's convenience we provide the proof for the case that φ satisfies L a (γ, C L ) for a > 0. Without loss of generality we assume a = 1. Throughout the proof we assume that t ∈ (0, T ). By Lemma 2.5, starting at z ∈ B(y, (12) −1 bΦ −1 (t)), with probability at least c 1 = c 1 (b, T ) > 0 the process X does not move more than (18) −1 bΦ −1 (t) by time t. Thus, using the strong Markov property and the Lévy system in (2.6), we obtain Using the (UJS) property of j (see [9, page 1070]), we obtain Since, for t/2 < s < t and w ∈ B(x, (72) −1 bΦ −1 (t/2)), we have by Proposition 3.4 that for t/2 < s < t and w ∈ B(x, (72) −1 bΦ −1 (t/2)), Combining (3.11), (3.12) with (3.13) and applying (UJS) again, we get In the last inequality we have used Lemma 2.3(a). Since by the semigroup property of p D and Proposition 3.4, , the proposition now follows from this and (3.14). ✷ For a C 1,1 open set D in R d with characteristics (R 0 , Λ), consider a z ∈ ∂D and a C 1,1 - (3.16) where ( x, x d ) are the coordinates of x in CS z . We also define It is easy to see that for every z ∈ ∂D and r ≤ κR 0 , r). In this paper, given a C 1,1 open set D, V z (r) always refers to the C 1,1 domain above.
Proposition 3.6. Suppose that φ has no drift and satisfies L a (γ, Moreover, there exist c 3 , c 4 > 0 such that for all z ∈ ∂D, r ≤ κR and (t, Using this and (3.21) we get Now, using (2.9) and Proposition 2.1, we conclude from (3.22) that We have proved (3.19).
(b) Suppose that D is a domain consisting of all the points above the graph of a bounded globally C 1,1 function. Then by [48], (3.21) holds for all (s, z, w) ∈ (0, ∞) × D × D. Using this fact and the assumption L 0 (γ, C L ), one can follow the arguments in (a) line by line and prove (b). We skip the details. ✷

Key estimates
In this section we prove key estimates on exit distribution for X in C 1,1 open set with explicit decay rate.
Throughout this section we assume that H satisfies L a (γ, C L ) and U a (δ, C U ) for some a > 0 with δ < 2 and the drift of the subordinator is zero.
Thus, for all x ∈ B(0, M), Proof. The proof is just a combination of Proposition 3.5 and the proof of [45,Proposition 3.6]. We spell out the details for completeness. By [45,Proposition 2.8] there exist L 1 , L 2 > 1 and c 1 > 0 such that for |x| ≤ (aL 1 ) −1/2 and tφ(|x| −2 ) ≤ 1 it holds that Without loss of generality, we assume that M > (aL 1 ) −1/2 and consider the following two cases separately.
Recall from the paragraph before Proposition 3.6 that, for z ∈ ∂D and r ≤ κR 0 , V 0 (r) is a C 1,1 domain with characteristics (rR 0 /L 0 , ΛL 0 /r) such that D 0 (3r/2, r/2) ⊂ V 0 (r) ⊂ D 0 (2r, r). Note that for w ∈ D 0 (2 −3 r, 2 −4 r), we have δ V 0 (r) (w) = δ D (w). Using this, (3.20) and (4.3), we have that for w ∈ D 0 (2 −3 r, 2 −4 r), We define, for i ≥ 1, and s 0 = s 1 . Note that r/(10) < s i < r/8. For i ≥ 1, set Repeating the argument leading to [42, (6.29)], we get that for z ∈ J i and i ≥ 2, We first claim that for all w ∈ J i , P w (X σ i,1 / ∈ J i ) is bounded below by a strictly positive constant. We prove the claim for w ∈ We choose ε ∈ (0, 2 −4 /Λ) small so that If y ∈ V ∩ J i and y d < w d , then clearly . Thus using (4.8), we have that for y ∈ V ∩ J i , which implies that On the other hand, for y ∈ 1 Since we assume that γ > 1/2, we can find a large M so that . Thus, by Proposition 3.4, for such y, z and s, using this and a chaining argument through the semigroup property, we have By (4.3) and (4.8)-(4.11), we have that for all w ∈ J i \ D 0 (2 −i−3 r, s i−1 ), which is a positive constant independent of i. We have proved the claim.
Thus, we have that there exists k 1 ∈ (0, 1) such that For the purpose of further estimates, we now choose a positive integer l ≥ 1 such that k l 1 ≤ 4 −1 . Next we choose i 0 ≥ 2 large enough so that 2 −i < 1/(200li 3 ) for all i ≥ i 0 . Now we assume i ≥ i 0 . Using (4.12) and the strong Markov property we have that for z ∈ J i , which implies for some 1 ≤ k ′ ≤ k ≤ li, Thus, using the strong Markov property and then using (4.4) (noting that 4 by (4.5), (4.13), (4.14) and Lemma 2.
By this and (4.7), for z ∈ J i and i ≥ i 0 , Thus the claim above is valid, since D 0 (2 −3 r, 2 −4 r) ⊂ ∪ ∞ k=1 J k . The proof is now complete. ✷ The next two results should be well-known but we could not find any reference. We provide the full details.
Proof. Using the change of variables u = t − s in the first integral and u = s − t in the second integral, we get that for ε ∈ (0, s/2), Letting ε → 0, we also have proved the second claim of the lemma. ✷ Proof. By Lemma 4.4, for all small ε ∈ (0, x d /2), Thus by the monotone convergence theorem, (4.16) is equal to Since x d (2u − x d ) ≤ u 2 , wee also have the upper bound as ✷ Let ψ(r) = 1/H(r −2 ). We first note that Φ(r) ≤ ψ(r) and for every 0 < r < R < 1.
Φ and ψ are also related as (4.18) Using (4.1), (4.17) and (4.18), we get that for R < 1, and  For any function f : R d → R and x ∈ R d , we define an operator as follows: Lf (x) := P.V.  Recall that C 2 0 (R d ) is the collection of C 2 functions in R d vanishing at infinity. It is well known that C 2 0 (R d ) ⊂ D(L) and that, by the rotational symmetry of X, where A is the infinitesimal generator of X. We also recall that δ D (x) is the distance of the point x to D c .
Proof. Since the case of d = 1 is easier, we give the proof only for d ≥ 2. Without loss of generality we assume that Λ > 1. For x ∈ D ∩ B(z, r/M 0 ), choose z x ∈ ∂D be a point satisfying δ D (x) = |x − z x |. Let ϕ be a C 1,1 function and CS = CS zx be an orthonormal coordinate system with z x chosen as the origin so that We fix the function ϕ and the coordinate system CS, and consider the truncated square function and we define ϕ : B( 0, r) → R by ϕ( y) := 2Λ| y| 2 . Since ∇ϕ( 0) = 0, by the mean value theorem we have − ϕ( y) ≤ ϕ( y) ≤ ϕ( y) for any y ∈ D ∩ B(x, r/2) and so that where we have used y 2 d + δ D (y) 2 ≤ 2(2 ϕ( y)) 2 = 2(4Λ| y|) 2 for y ∈ A. We will show that the above is less than c 1 r 3 /Φ(r). Second, when y ∈ E, we have that |y d − δ D (y)| ≤ (1 ∧ R 0 ) −1 ϕ( y). Indeed, if 0 < y d ≤ δ D (y) and y ∈ E, δ D (y) ≤ y d + |ϕ( y)| ≤ y d + ϕ( y). Since we assume that Λ > 1, we have Thus, if y d ≥ δ D (y) and y ∈ E, using the interior ball condition, we have Thus, Since E ⊂ {( y, y d ) : | y| < r, ϕ( y) < y d < ϕ( y) + 2r}, using the polar coordinates for | y| = v and the change of the variable s := y d − ϕ(v), we have by (2.11) and Lemma 2.3, .
We have proved the lemma. ✷ Since (4.23) holds, we have Dynkin's formula for L: for each g ∈ C 2 c (R d ) and any bounded open subset U of R d we have (4.34) Note that, since H may not be comparable to φ, the next result can not be obtained using Lévy system and (4.1).
Proof. Fix z ∈ ∂D, r ≤ R and an open set U ⊂ D ∩ B(z, r/M 0 ). Define f (y) = (δ D (y)) 2 1 D∩B(z,2r) (y). Then by Lemma 4.6, there exists c 1 = c 1 (φ, Λ, d) ∈ (0, 1) such that for all r ≤ R and y ∈ D ∩ B(z, r/M 0 ), c −1 Let v ≥ 0 be a smooth radial function such that v(y) = 0 for |y| > 1 and R d v(y)dy = 1. For k ≥ 1, define v k (y) := 2 kd v(2 k y) and f (k) r By letting ε ↓ 0 and using the dominated convergence theorem, it follows that for w ∈ B k and all large k, Therefore, by the Dynkin's formula in (4.34) we have that for x ∈ B k and all large k, By letting k → ∞, for any x ∈ U, we conclude that ✷ Let X d be the last coordinate of X and let L t be the local time at 0 for (sup s≤t X d s ) − X d t . Using its right-continuous inverse L −1 s , define the ascending ladder-height process as H s = X d L −1 s . We define V , the renewal function of the ascending ladder-height process H, as It is well-known that V is subadditive (see [1, p.74]). Note that, since the resolvent measure of X d t is absolutely continuous, by [47,Theorem 2], V is absolutely continuous and V and V ′ are harmonic for the process X d t on (0, ∞). Thus, by the strong Markov property, V ((x d ) + ) and V ′ ((x d ) + ) are harmonic in the upper half space R d with respect to X. Furthermore, the function V (r) is comparable to Φ(r) 1/2 (see [4,Corollary 3]): there exists c > 1 such that  We observe that, by a direct calculation using (4.18), Thus, using this and the fact lim s→0 sΦ(s) −1/2 = 0 which also can be seen from (4.18), we have (4.36) Lemma 4.9. Assume that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 and γ > 2 −1 1 δ≥1 for some a ≥ 0. Let γ 1 := γ1 δ<1 + (2γ − 1)1 δ≥1 > 0. There exist c 1 , c 2 , c 3 > 0 such that for all positive constants R ≤ 1 and λ > 1, Proof. If δ < 1 then ψ and Φ are comparable near 0, thus, by (2.5) for t ≤ R ≤ 1, By (4.17) and Lemma 2.3(a), if δ ≥ 1 then for t ≤ R ≤ 1, Thus, for t ≤ R ≤ 1, Using (4.39) we have that for all R ≤ 1 and λ > 1, The second inequality in (4.37) also follows from (4.39) (with R = 1 and t = R). ✷ Proposition 4.10. Let D ⊂ R d be a C 1,1 open set with characteristics (R 0 , Λ). Assume that (1.8) holds and that H satisfies L a (γ, C L ) and U a (δ, C U ) with a > 0, δ < 2 and γ > 2 −1 1 δ≥1 . For any z ∈ ∂D and r ≤ 1 ∧ R 0 , we define h r (y) = h r,z (y) := V (δ D (y))1 D∩B(z,r) (y).
Then, there exists C * = C * (φ, Λ, d) > 0 independent of z such that Lh r is well-defined in D ∩ B(z, r/4) and for all x ∈ D ∩ B(z, r/4).
Using the same approximation argument in the proof of Proposition 4.7 and the Dynkin's formula, we have that, for every λ ≥ 4, open set U ⊂ D ∩ B(0, λ −1 R) and x ∈ U, where C * > 0 is the constant in Proposition 4.10.

Upper bound estimates
In this section we discuss the upper bound of the Dirichlet heat kernels on C 1,1 open sets. Throughout the remainder of this paper, we always assume that (1.8) holds, that φ has no drift and that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 and γ > 2 −1 1 δ≥1 for some a > 0.
We first establish sharp estimates on the survival probability. Lemma 5.1 is proved in [5] when weak scaling order of characteristic exponent is strictly below 2. We emphasize here that results in [5] can not be used here.

Lower bound estimates
Recall that we always assume that (1.8) holds, that φ has no drift and that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 and γ > 2 −1 1 δ≥1 for some a > 0.
Using Lemma 5.1 from Section 5, in this section we will prove Theorem 1.3(b). The main ideas in this section come from [7]. We first observe the following simple lemma.
Proof. By [7, Theorem 2.2] there exists a constant c 1 = c 1 (d) > 0 such that The condition L a (γ, C L ) and Remark 2.2 imply that for all λ ≥ 1, which goes to zero as λ → ∞. ✷ We now discuss some lower bound estimates of p D (t, x, y). We first note that by Lemma 5.1, there exist C 3 ≥ 1 and T 1 ∈ (0, 1 ∧ Φ(R 0 )] such that For x ∈ D we use z x to denote a point on ∂D such that |z x − x| = δ D (x) and n(z x ) := (x − z x )/|z x − x|. By a simple geometric argument, one can easily see that (6.2) Lemma 6.2. There exist a 1 > 0 and M 1 > 1 ∨ 4a 1 such that for all a ∈ (0, a 1 ], x ∈ D and t ∈ (0, T 1 ], we have that where ξ a x (t) := x + aΦ −1 (t)n(z x ) and C 3 and T 1 are the constants in (6.1).
Proof. By (1.10) and a change of variable, for every a > 0, t ∈ (0, T 1 ] and x ∈ D, For the rest of the proof, we assume that x ∈ D, a ∈ (0, a 1 ] and t ∈ (0, Thus using this, (1.10) and the monotonicity of r → p(t, r), we have that for every λ ≥ 2a 1 , By Lemma 6.1, we can choose M 1 > 1 ∨ 4a 1 large so that C 0 6 d H(6 −1 M 1 ) < (4C 3 ) −1 . Then by (6.1)-(6.4) and our choice of a 1 and M 1 , we conclude that The next result is easy to check (see the proof of [20, Lemma 2.5] for a similar computation). We skip the proof. Lemma 6.3. For any given positive constants c 1 , r 1 , T and r 2 > r 1 , there is a positive constant c 2 = c 2 (r 1 , r 2 , T, c 1 , φ) so that for every r 1 ≤ r < r 2 (a∧1) −1 and t ∈ (0, T ]. , for all x, y ∈ D (6.5) Recall that a 1 > 0 and M 1 > 1 ∨ 4a 1 are the constants in Lemma 6.2 and C 3 and T 1 are the constants in (6.1). We also recall that for x ∈ D, z x ∈ ∂D such that |z x − x| = δ D (x) and n(z x ) = (x − z x )/|z x − x|. Without loss of the generality we assume that T > 3T 1 .

Green function estimates
In next two sections we use the notation f (x) ≍ g(x), x ∈ I, which means that there exist constants c 1 , c 2 > 0 such that c 1 f (x) ≤ g(x) ≤ c 2 g(x) for x ∈ I.
Recall that Φ(r) = (φ(1/r 2 )) −1 where φ is the Laplace exponent φ of the subordinator S. When φ satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 for some a > 0, Green function estimates for the corresponding subordinate Brownian motion were already discussed in [19]. In this section we discuss Green function estimates when φ has no drift and that H satisfies L a (γ, C L ) and U a (δ, C U ) with δ < 2 and γ > 2 −1 1 δ≥1 for some a > 0.
By the exactly same proof as the one of [19, Lemma 7.1], we have the following.  Note that, by Lemma 2.4, for every T > 0, there exists C T > 1 such that for 0 < r ≤ R ≤ T.
The proof of the next lemma is very similar (and simpler) to the one of Lemma 7.2 so we skip the proof. Lemma 7.4. Suppose that (1.8) holds, that φ has no drift and that H satisfies L 0 (γ, C L )