Two-sided Green function estimates for killed subordinate Brownian motions
aa r X i v : . [ m a t h . P R ] F e b Two-sided Green function estimates for killed subordinateBrownian motions
Panki Kim ∗ Renming Song and
Zoran Vondraˇcek † February 8, 2011
Abstract
A subordinate Brownian motion is a L´evy process which can be obtained by replacing the time of theBrownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownianmotion is − φ ( − ∆), where φ is the Laplace exponent of the subordinator. In this paper, we consider a largeclass of subordinate Brownian motions without diffusion component and with φ comparable to a regularlyvarying function at infinity. This class of processes includes symmetric stable processes, relativisticstable processes, sums of independent symmetric stable processes, sums of independent relativistic stableprocesses, and much more. We give sharp two-sided estimates on the Green functions of these subordinateBrownian motions in any bounded κ -fat open set D . When D is a bounded C , open set, we establishan explicit form of the estimates in terms of the distance to the boundary. As a consequence of suchsharp Green function estimates, we obtain a boundary Harnack principle in C , open sets with explicitrate of decay. AMS 2010 Mathematics Subject Classification : Primary 60J45, Secondary 60J75, 60G51.
Keywords and phrases:
Green function, Poisson kernel, subordinate Brownian motion, κ -fat open set, C , open set, symmetric stable process, relativistic stable process, harmonic functions, Harnack inequality,boundary Harnack principle, fluctuation theory, regularly varying function. The investigation of fine potential-theoretic properties of discontinuous Markov processes in the Euclideanspace began in the late 1990’s with the study of symmetric stable processes. One of the first results obtainedin this area was sharp Green function estimates of symmetric α -stable processes in bounded C , domainsin R d , 0 < α < d ≥
2. Recall that if X is a symmetric Markov process in R d and D is an open subset of R d , then the Green function G D ( x, y ) of X in D (if it exists) is the density of the mean occupation time for X before exiting D , that is, the density of the measure U E x Z τ D U ( X t ) dt, U ⊂ D, where τ D is the first time the process X exits D . Analytically speaking, if L is the infinitesimal generatorof X and L| D is the restriction of L to D with zero exterior condition, then G D ( · , y ) is the solution of( L| D ) u = − δ y .A process X = ( X t : t ≥
0) is called a (rotationally) symmetric α -stable (L´evy) process, 0 < α <
2, ifit is a L´evy process whose characteristic exponent Φ, defined by E [exp { iθ · X t } ] = exp {− t Φ( θ ) } , is given ∗ Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MEST)(2010-0001984). † Supported in part by the MZOS grant 037-0372790-2801.
1y Φ( θ ) = | θ | α . The infinitesimal generator of a symmetric α -stable process is − ( − ∆) α/ . The paths ofthe symmetric α -stable process X are purely discontinuous, as opposed to the case α = 2 correspondingto Brownian motion which has continuous paths. It was independently shown in [11] and [22] that if D isa bounded C , domain in R d , G D ( x, y ) the Green function of the symmetric α -stable process in D , and δ D ( x ) the distance between the point x and the complement D c of D , then there exists a constant c > D and α ) such that c − (cid:18) ∧ ( δ D ( x ) δ D ( y )) α/ | x − y | α (cid:19) | x − y | d − α ≤ G D ( x, y ) ≤ c (cid:18) ∧ ( δ D ( x ) δ D ( y )) α/ | x − y | α (cid:19) | x − y | d − α , (1.1)for all x, y ∈ D . Here and in the sequel, for a, b ∈ R , a ∧ b := min { a, b } and a ∨ b := max { a, b } . The sameform of the estimates in the case α = 2 (and d ≥
3) were obtained much earlier in [33] and [35] for theBrownian motion case.The proofs of (1.1) for symmetric α -stable processes relied heavily on the explicit formulae for the Greenfunctions and the Poisson kernels of the ball. Moving away from stable processes, such formulae were notavailable and new methods had to be developed. [25] studied the relativistic α -stable process (with relativisticmass m >
0) whose characteristic exponent is given by Φ( θ ) = ( | θ | + m /α ) α/ − m and infinitesimal generatoris given by m − ( − ∆ + m /α ) α/ , and showed that the Green function of this process in any bounded C , domain D satisfies the same sharp estimates (1.1). Soon after, [12], using a perturbation method, establisheda general result which includes the main result of [25] as a special case. For different generalizations of themain result of [25], see the recent papers [14, 19].Quite recently, [8] studied the L´evy process which is the sum of independent symmetric β -stable and α -stable processes, 0 < β < α <
2. The characteristic exponent of this process is given by Φ( θ ) = | θ | α + | θ | β and the infinitesimal generator by − ( − ∆) α/ − ( − ∆) β/ . Sharp two-sided estimates on the heat kernel ofthis process in C , open sets were established in [8]. As a by-product of the heat kernel estimates, sharpGreen function estimates of the process in any bounded C , open set were obtained in [8]. Estimates havethe form (1.1). In contrast with the relativistic stable processes, these Green function estimates cannot beobtained using the methods of [12, 14, 19, 25]. The case α = 2 (i.e., one of the processes is a Brownianmotion) was covered in [10] with analogous estimates.The common feature of these Green function estimates is that both the distance between the points, | x − y | , and distances to D c , δ D ( x ) , δ D ( y ), appear as arguments of power functions . However, it follows from[6, Chapter 5] that the asymptotic behavior of the free Green function G ( x, y ) of many transient symmetricL´evy processes is of the form G ( x, y ) ∼ | x − y | d − α ℓ ( | x − y | − ) as | x − y | → α ∈ (0 ,
2) and ℓ is a nontrivial slowly varying function at infinity. (See, also Theorem 2.9 below.)Therefore, Green function estimates of the form (1.1) cannot be true for these general symmetric L´evyprocesses. The purpose of this paper is to establish sharp two-sided Green function estimates for these moregeneral L´evy processes in open sets of R d . In our estimates, δ D ( x ) , δ D ( y ) and | x − y | appear as argumentsof regularly varying functions, not necessarily power functions. In order to explain our setting and results,let us first note that stable processes, relativistic stable processes and sums of independent stable processescan be obtained as subordinate Brownian motions. Indeed, let W = ( W t = ( W t , . . . , W dt ) : t ≥
0) be a d -dimensional Brownian motion, and let S = ( S t : t ≥
0) be an independent subordinator. Recall thata subordinator is an increasing L´evy process on [0 , ∞ ), which can be characterized through its Laplaceexponent φ : E [exp {− λS t } ] = exp {− tφ ( λ ) } , λ >
0. The process X = ( X t : t ≥
0) defined by X t := W S t iscalled a subordinate Brownian motion. The infinitesimal generator of X is − φ ( − ∆). By choosing the Laplaceexponent φ ( λ ) as λ α/ , ( λ + m /α ) α/ − m and λ α/ + λ β/ respectively, the resulting subordinate Brownian2otion turns out to be a symmetric α -stable process, a relativistic stable process and an independent sumof β and α -stable processes respectively. The Laplace exponent of a subordinator is a Bernstein functionand hence has the representation φ ( λ ) = bλ + Z (0 , ∞ ) (1 − e − λt ) µ ( dt ) , where b ≥ µ is a measure (called the L´evy measure of φ ) such that R (0 , ∞ ) (1 ∧ t ) µ ( dt ) < ∞ . Ifthe measure µ has a completely monotone density, the Laplace exponent φ is called a complete Bernsteinfunction. The common feature of the Laplace exponents φ ( λ ) = λ α/ , φ ( λ ) = ( λ + m /α ) α/ − m and φ ( λ ) = λ β/ + λ α/ is that all three of them are complete Bernstein functions whose behavior at infinity isgiven by lim λ →∞ φ ( λ ) /λ α/ = 1. We will see that those two properties (the latter slightly weakened) are thedetermining factors for the Green function estimates (1.1).Recall that an open set D in R d ( d ≥
2) is said to be a C , open set if there exist a localization radius R > > z ∈ ∂D , there exist a C , -function ψ = ψ z : R d − → R satisfying ψ (0) = 0, ∇ ψ (0) = (0 , . . . , k∇ ψ k ∞ ≤ Λ, |∇ ψ ( x ) − ∇ ψ ( z ) | ≤ Λ | x − z | , and an orthonormalcoordinate system CS z : y = ( y , · · · , y d − , y d ) := ( e y, y d ) with origin at z such that B ( z, R ) ∩ D = { y = ( e y, y d ) ∈ B (0 , R ) in CS z : y d > ψ ( e y ) } . The pair ( R, Λ) is called the characteristics of the C , open set D . We remark that in some literature, the C , open set defined above is called a uniform C , open set since ( R, Λ) is universal for all z ∈ ∂D . By a C , open set in R we mean an open set which can be written as the union of disjoint intervals so that theminimum of the lengths of all these intervals is positive and the minimum of the distances between theseintervals is positive. Note that a bounded C , open set can be disconnected.The main result of this paper is the following sharp Green function estimates. In the statement andthroughout the paper we use notation f ( t ) ≍ g ( t ) as t → ∞ (resp. t → f ( t ) /g ( t ) staysbounded between two positive constants as t → ∞ (resp. t → Theorem 1.1
Suppose that X = ( X t : t ≥ is a L´evy process whose characteristic exponent is given by Φ( θ ) = φ ( | θ | ) , θ ∈ R d , where φ : (0 , ∞ ) → [0 , ∞ ) is a complete Bernstein function such that φ ( λ ) ≍ λ α/ ℓ ( λ ) , λ → ∞ , (1.2) α ∈ (0 , ∧ d ) and ℓ : (0 , ∞ ) → (0 , ∞ ) is a measurable, locally bounded function which is slowly varying atinfinity. When d ≤ , we assume an additional assumption, see (2.14) . Then for every bounded C , openset D in R d with characteristics ( R, Λ) , there exists C = C ( diam ( D ) , R, Λ , α, ℓ, d ) > such that the Greenfunction G D ( x, y ) of X in D satisfies the following estimates: C − ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! ℓ ( | x − y | − ) | x − y | d − α ≤ G D ( x, y ) ≤ C ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! ℓ ( | x − y | − ) | x − y | d − α . (1.3)Since the subordinate Brownian motion X is completely determined by the Laplace exponent φ , onewould expect that the above estimates can be expressed in terms of the function φ only. And indeed, analternative form of (1.3) reads as follows: There exists c > c − ∧ φ ( | x − y | − ) p φ ( δ D ( x ) − ) φ ( δ D ( x ) − ) ! | x − y | d φ ( | x − y | − )3 G D ( x, y ) ≤ c ∧ φ ( | x − y | − ) p φ ( δ D ( x ) − ) φ ( δ D ( x ) − ) ! | x − y | d φ ( | x − y | − ) (1.4)(see (1.6) below for yet another alternative form of these estimates). To the best of our knowledge, the aboveGreen function estimates include all known Green function estimates of pure-jump transient subordinateBrownian motions in bounded C , open set D in R d as special cases. The estimates above include a lotmore processes: In the case d ≥
3, our estimates are valid for all subordinate Brownian motions satisfying(1.2). For more concrete examples, see Example 2.16.Let us give the main ingredients of the proof of Theorem 1.1. The groundwork has been laid down inthe recent paper [20] where a similar class of subordinate Brownian motions was studied. One differenceto the current setting was that in [20] the Laplace exponent was assumed to be precisely regularly varyingat infinity and not just comparable to a regularly varying function. Another difference is that [20] containssome additional assumptions that we showed to be redundant. The results of [20] are reproved in [21] underconditions valid in this paper (with the redundant assumptions removed). When referring to those resultswe will quote both sources. The main result of [20, 21] is the boundary Harnack principle for nonnegativeharmonic functions of the subordinate Brownian motion X in bounded κ -fat open sets. Based on theboundary Harnack principle and using the well-established methodology of [4, 16], we will first obtain Greenfunction estimates of the form (1.5) (in the spirit of [4, 16]) in bounded κ -fat open sets.Recall from [32] that an open set D in R d is κ -fat if there exists R > Q ∈ ∂D and r ∈ (0 , R ), D ∩ B ( Q, r ) contains a ball B ( A r ( Q ) , κr ). The pair ( R , κ ) is called the characteristics of the κ -fat open set D . All Lipschitz domains, non-tangentially accessible domains and John domains are κ -fat(cf. [18, 32] and the references therein). In general, the boundary of a κ -fat open set can be nonrectifiable. Theorem 1.2
Suppose that X = ( X t : t ≥ is a L´evy process satisfying the same conditions as inTheorem 1.1 and that D is a bounded κ -fat open set with characteristics ( R , κ ) . Then there exists C = C ( diam ( D ) , R , κ, α, ℓ, d ) > such that for every x, y ∈ D , C − g ( x ) g ( y ) g ( A ) | x − y | d − α ℓ ( | x − y | − ) ≤ G D ( x, y ) ≤ C g ( x ) g ( y ) g ( A ) | x − y | d − α ℓ ( | x − y | − ) , A ∈ B ( x, y ) , (1.5) where g and B ( x, y ) are defined in (3.11) and (3.7) respectively. In the case 0 < c ≤ ℓ ( λ ) ≤ c < ∞ for large λ , using the Harnack inequality and the boundary Harnackprinciple, the above form of Green function estimates has been established by several authors in specialcases. See [10, Theorem 1.1], [16, Theorem 2.4] and [17, Theorem 1].To obtain the interior estimates in Theorem 1.2 (i.e. for points x, y away from the boundary), we use theasymptotic behavior of the Green function of X in R d proved in [21, Theorem 3.2] (see also [20, Theorem3.1]). Using the interior estimates and following the method developed in [4, 16], we obtain the full estimatesin a bounded κ -fat open set using the boundary Harnack principle from [20, 21].Even though a lot of complications occur due to the appearance of the slowly varying function ℓ , theproof of Theorem 1.2 is still routine. But the precise estimates (1.3) in bounded C , open sets are verydelicate. One of the ingredients comes from the fluctuation theory of one-dimensional L´evy processes. Let Z = ( Z t : t ≥
0) be the one-dimensional subordinate Brownian motion defined by Z t := W dS t , and let V bethe renewal function of the ladder height process of Z . The function V is harmonic for the process Z killedupon exiting (0 , ∞ ), and the function w ( x ) := V ( x d ) { x d > } , x ∈ R d , is harmonic for the process X killedupon exiting the half space R d + := { x = ( x , . . . , x d − , x d ) ∈ R d : x d > } (Theorem 4.1). Therefore, w givesthe correct rate of decay of harmonic functions near the boundary of R d + . This shows the importance of thefluctuation theory (of one-dimensional L´evy processes) in our approach.4he second ingredient is the “test function” method applied to the operator A defined by A f ( x ) = lim ε → Z { y ∈ R d : | y − x | >ε } ( f ( y ) − f ( x )) J ( y − x ) dy , with the domain consisting of functions f for which the limit exists and is finite. Here J denotes the density ofthe L´evy measure of X . On the space of smooth functions with compact support, this operator coincides withthe infinitesimal generator of X . We emphasize that, compared to the test function methods of [5, 9, 15],there are several differences in our approach. In [5, 9, 15], appropriate subharmonic and superharmonicfunctions of X (or the truncated version of X ) are chosen as test functions, first in the case of half spacesand then for C , open sets, and the values of the generator acting on these test functions are computedin detail. Then suitable combinations of the test functions are used to find the correct exit distributionestimates. In [5, 9, 15], the test functions are power functions of the form x → ( x d ) p and the densities of theL´evy measures of the processes have closed forms. However, the density J of the L´evy measure of our processdoes not have a simple form. We do not even know the asymptotic behavior of J near infinity in general.Furthermore, in general, power functions of the form x → ( x d ) p are neither subharmonic nor superharmonicfunctions for our processes, and it is not clear what are the appropriate choices for the test functions.Due to the above differences and difficulties, obtaining the correct boundary decay rate of the Greenfunction in C , open set D requires new ideas and approaches. In this paper, we will use the function w which is smooth and harmonic on the half space, as our only test function. Using this and the characterizationof harmonic functions recently established in [7], we show that A w ≡ D is a C , open set with characteristics ( R, Λ), Q ∈ ∂D and h ( y ) = V ( δ D ( y )) D ∩ B ( Q,R ) , then A h ( y ) is welldefined and bounded for y ∈ D close enough to the boundary point Q . Using this lemma, we give certainexit distribution estimates in Lemma 4.5, which provide the correct rate of decay of Green functions nearthe boundary of D . Unlike [5, 9, 15], in Lemma 4.5 we do not construct subharmonic and superharmonicfunctions on C , open set D . Instead we use Dynkin’s formula on h to obtain the desired exit distributionestimates directly. In fact, our approach is simpler than the previous approaches and may be used for othertypes of jump processes. We hope our approach will shed new light on the understanding of the boundarybehavior of nonnegative harmonic functions of general Markov processes.The estimates (1.3) are best understood in terms of the renewal function V which provides the exact rateof decay of G D near the boundary. Let G be the Green function of X in the whole space R d . An equivalentform of (1.3) is given by c − (cid:18) ∧ V ( δ D ( x )) V ( δ D ( y )) V ( | x − y | ) (cid:19) G ( x, y ) ≤ G D ( x, y ) ≤ c (cid:18) ∧ V ( δ D ( x )) V ( δ D ( y )) V ( | x − y | ) (cid:19) G ( x, y ) . (1.6)By combining the sharp estimates of the Green function in a bounded C , open set with the boundaryHarnack principle proved in [20, 21] (see Theorem 2.15 below), we obtain a boundary Harnack principle withexplicit decay rate. In the next theorem we give an extension to unbounded C , open sets. Recall that,given Q ∈ ∂D , a function u : R d → R is said to vanish continuously on D c ∩ B ( Q, r ) if u = 0 on D c ∩ B ( Q, r )and u is continuous at every point of ∂D ∩ B ( Q, r ). Theorem 1.3
Suppose that X = ( X t : t ≥ is a L´evy process satisfying the same conditions as in Theorem1.1 and that D is a (possibly unbounded) C , open set in R d with characteristics ( R, Λ) . Then there exists C = C ( R, Λ , α, ℓ, d ) > such that for r ∈ (0 , ( R ∧ / , Q ∈ ∂D and any nonnegative function u in R d that is harmonic in D ∩ B ( Q, r ) with respect to X and vanishes continuously on D c ∩ B ( Q, r ) , we have u ( x ) δ D ( x ) α/ p ℓ (( δ D ( y )) − ) ≤ C u ( y ) δ D ( y ) α/ p ℓ (( δ D ( x )) − ) for every x, y ∈ D ∩ B ( Q, r/ . (1.7)5n alternative form of (1.7) reads as follows: There exists a constant c > u ( x ) V ( δ D ( x )) ≤ c u ( y ) V ( δ D ( y )) for every x, y ∈ D ∩ B ( Q, r/ . (1.8)Note that unlike the usual form of the boundary Harnack principle where one considers the ratio of twoharmonic functions, functions in the denominator of (1.7) and (1.8) are not harmonic. Instead, they providethe correct boundary decay of non-negative harmonic functions. Indeed, an equivalent form of Theorem 1.3says that there exists a constant c > u in R d that is harmonic in D ∩ B ( Q, r ) with respect to X and vanishes continuously on D c ∩ B ( Q, r ) it holds that c − V ( δ D ( x )) ≤ u ( x ) ≤ cV ( δ D ( x )) for every x ∈ D ∩ B ( Q, r/ . This paper is organized as follows: In the next section we establish the setting and notation, proveseveral new results for complete Bernstein functions, and describe some of the known results from [20, 21].In Section 3 we prove the Green function estimates in bounded κ -fat open sets. Section 4 is devoted to theGreen function estimates in bounded C , open sets.We will use the following conventions in this paper. The values of the constants C , C , . . . , M , ε and R, R , R , · · · will remain the same throughout this paper, while c, c , c , c , · · · and r, r , r , r , . . . standfor constants whose values are unimportant and which may change from one appearance to another. Allconstants are positive finite numbers. The labeling of the constants c , c , c , · · · starts anew in the statementof each result. The dependence of the constants on dimension d may not be mentioned explicitly. We willuse “:=” to denote a definition, which is read as “is defined to be”. Further, f ( t ) ∼ g ( t ), t → f ( t ) ∼ g ( t ), t → ∞ , respectively) means lim t → f ( t ) /g ( t ) = 1 (lim t →∞ f ( t ) /g ( t ) = 1, respectively). For any open set U ,we denote by δ U ( x ) the distance between x and the complement of U , i.e., δ U ( x ) = dist( x, U c ). We will use ∂ to denote the cemetery point and for every function f , we extend its definition to ∂ by setting f ( ∂ ) = 0.For every function f , let f + := f ∨
0. We will use dx to denote the Lebesgue measure in R d . For a Borel set A ⊂ R d , we also use | A | to denote its Lebesgue measure and diam( A ) to denote the diameter of the set A . In this section we collect and explain preliminary results necessary for further development in Sections 3and 4. Most of these results originate from [20] where they were proved under somewhat stronger conditionsthan in this paper. Their extensions to the current setting, in particular Theorems 2.9, 2.11 and 2.15, aregiven with full proofs in [21]. Here we prove only results that have not appeared in [20]. Lemma 2.1 andPropositions 2.4 and 2.6 about complete Bernstein functions may be of independent interest. The differencebetween the assumptions in [20] and this paper is discussed in Remark 2.2.A C ∞ function φ : (0 , ∞ ) → [0 , ∞ ) is called a Bernstein function if ( − n D n φ ≤ n . Every Bernstein function has a representation φ ( λ ) = a + bλ + R (0 , ∞ ) (1 − e − λt ) µ ( dt ) where a, b ≥ µ is a measure on (0 , ∞ ) satisfying R (0 , ∞ ) (1 ∧ t ) µ ( dt ) < ∞ ; a is called the killing coefficient, b the drift and µ the L´evy measure of the Bernstein function. A Bernstein function φ is called a completeBernstein function if the L´evy measure µ has a completely monotone density µ ( t ), i.e., ( − n D n µ ≥ n . Here and below, by abuse of notation we will denote the L´evy density by µ ( t ).For more on Bernstein and complete Bernstein functions we refer the readers to [27].First, we show that φ being a complete Bernstein function implies that its L´evy density cannot decreasetoo fast in the following sense: Lemma 2.1
Suppose that φ is a complete Bernstein function with L´evy density µ . Then there exists C > such that µ ( t ) ≤ C µ ( t + 1) for every t > . roof. Since µ is a completely monotone function, by Bernstein’s theorem ([27, Theorem 1.4]), thereexists a measure m on [0 , ∞ ) such that µ ( t ) = R [0 , ∞ ) e − tx m ( dx ) . Choose r > R [0 ,r ] e − x m ( dx ) ≥ R ( r, ∞ ) e − x m ( dx ) . Then, for any t >
1, we have Z [0 ,r ] e − tx m ( dx ) ≥ e − ( t − r Z [0 ,r ] e − x m ( dx ) ≥ e − ( t − r Z ( r, ∞ ) e − x m ( dx ) ≥ Z ( r, ∞ ) e − tx m ( dx ) . Therefore, for any t > µ ( t + 1) ≥ Z [0 ,r ] e − ( t +1) x m ( dx ) ≥ e − r Z [0 ,r ] e − tx m ( dx ) ≥ e − r Z [0 , ∞ ) e − tx m ( dx ) = 12 e − r µ ( t ) . ✷ Suppose that S = ( S t : t ≥
0) is a subordinator with Laplace exponent φ , that is E (cid:2) e − λS t (cid:3) = e − tφ ( λ ) , ∀ t, λ > . The Laplace exponent of a subordinator is always a Bernstein function. Let U ( A ) := E R ∞ { S t ∈ A } dt denote the potential measure of S . If φ is a complete Bernstein function with infinite L´evy measure, thenthe potential measure U has a completely monotone density u ( t ) (see, e.g., [27, Remark 10.6 and Corollary10.7]).Recall that a function ℓ : (0 , ∞ ) → (0 , ∞ ) is slowly varying at infinity iflim t →∞ ℓ ( λt ) ℓ ( t ) = 1 , for every λ > . In the remainder of this paper we assume that φ is a complete Bernstein function and we will alwaysimpose the following Assumption (H):
There exist α ∈ (0 ,
2) and a function ℓ : (0 , ∞ ) → (0 , ∞ ) which is measurable, locallybounded and slowly varying at infinity such that φ ( λ ) ≍ λ α/ ℓ ( λ ) , λ → ∞ . (2.1) Remark 2.2 (a) The precise interpretation of (2.1) will be as follows: There exists a positive constant c > c − ≤ φ ( λ ) λ α/ ℓ ( λ ) ≤ c for all λ ∈ [1 , ∞ ) . The choice of the interval [1 , ∞ ) is, of course, arbitrary. Any interval [ a, ∞ ) would do, but with a differentconstant. This follows from the assumption that ℓ is locally bounded. Moreover, by choosing a > a, ∞ ) for a large enough. Although the choice of interval[1 , ∞ ) is arbitrary, it will have as a consequence the fact that all relations of the type f ( t ) ≍ g ( t ) as t → ∞ (respectively t → c − ≤ f ( t ) /g ( t ) ≤ ˜ c for t ≥ < t ≤
1) for an appropriate constant ˜ c .(b) The assumption (H) is an assumption about the behavior of φ at infinity. We make no assumption on φ near zero. As a consequence, we will be able to obtain information about the small scale behavior of thesubordinate process, but almost nothing can be inferred about its large scale behavior.7c) The main assumption in [20] was that φ is a complete Bernstein function such that φ ( λ ) = λ α/ ℓ ( λ ) , for all λ > , (2.2)where α ∈ (0 ,
2) and ℓ is a slowly varying function at infinity. This assumption allows us to obtain exactasymptotic behavior of various functions. More precisely, some of the results in [20] were of the form f ( t ) ∼ g ( t ), while with the assumption (2.1) we can obtain only the corresponding results in the weakerform f ( t ) ≍ g ( t ). Proofs of these weaker results can be found in [21]. We note that our current assumptionsare indeed strictly weaker than the ones in [20]: There exists a complete Bernstein function satisfying (2.1)which is not regularly varying at infinity, see [21, Example 2.8].(d) We briefly comment on the other assumptions from [20] which are now removed. The assumption A1 in [20] needed for transience in case d ≤ A2 and A3 in [20]used for the technical lemma [6, Lemma 5.32] are redundant - see [21, Lemma 3.1]. Assumption A4 in [20]is always valid for complete Bernstein function as proved here in Lemma 2.1. Finally, the assumption (2.5)in [20, Proposition 2.2] is no longer needed as [20, Proposition 2.2] is now replaced by Proposition 2.6 below.It follows from (2.1) that lim λ →∞ φ ( λ ) /λ = 0 and lim λ →∞ φ ( λ ) = ∞ , implying that φ has no drift andits L´evy measure is infinite. Therefore, the potential measure U of the corresponding subordinator S has acompletely monotone density u .The behavior of u ( t ) and the density µ ( t ) of the L´evy measure can be inferred from the following result. Proposition 2.3 ([34, Theorem 7])
Suppose that ψ is a completely monotone function given by ψ ( λ ) = Z ∞ e − λt f ( t ) dt, where f is a nonnegative decreasing function. Then f ( t ) ≤ (cid:0) − e − (cid:1) − t − ψ ( t − ) , t > . If, furthermore, there exist δ ∈ (0 , and a, t > such that ψ ( rλ ) ≤ ar − δ ψ ( λ ) , r ≥ , t ≥ /t , (2.3) then there exists C = C ( w, f, a, t , δ ) > such that f ( t ) ≥ C t − ψ ( t − ) , t ≤ t . We first apply the above proposition to ψ ( λ ) = φ ( λ ) − = R ∞ e − λt u ( t ) dt to obtain the behavior of u nearzero: u ( t ) ≍ t − φ ( t − ) − ≍ t α/ − ℓ ( t − ) , t → . (2.4)Condition (2.3) follows from (2.1) by use of Potter’s theorem (cf. [2, Theorem 1.5.6]). By applying (2.4) tothe complete Bernstein function λ λ/φ ( λ ) ([27, Proposition 7.1]) one obtains the following behavior of µ ( t ) near zero: µ ( t ) ≍ t − φ ( t − ) ≍ t − α/ − ℓ ( t − ) , t → . (2.5)We refer the reader to [21, Theorem 2.9, Theorem 2.10] for the detailed proofs of (2.4) and (2.5). Thecorresponding precise asymptotics are given in [20, p. 1603] under the assumption (2.2).A consequence of the asymptotic behavior (2.5) of µ ( t ) is that for any K > c = c ( K ) > µ ( t ) ≤ c µ (2 t ) , ∀ t ∈ (0 , K ) . (2.6)8he behavior of µ ( t ) at infinity has already been determined in Lemma 2.1: There exists a constant c > µ ( t ) ≤ c µ ( t + 1) , ∀ t > . (2.7)This property of µ was assumed in [20] as A4 , but we have shown in Lemma 2.1 that it always holds true.We consider now one-dimensional subordinate Brownian motions. Let B = ( B t : t ≥
0) be a Brownianmotion in R , independent of S , with E h e iθ ( B t − B ) i = e − tθ , ∀ θ ∈ R , t > . The subordinate Brownian motion Z = ( Z t : t ≥
0) in R defined by Z t = B S t is a symmetric L´evy processwith the characteristic exponent Φ( θ ) = φ ( θ ) for all θ ∈ R . Let Z t := sup { ∨ Z s : 0 ≤ s ≤ t } be the supremum process of Z and let L = ( L t : t ≥
0) be a local timeof Z − Z at 0. L is also called a local time of the process Z reflected at the supremum. The right continuousinverse L − t of L is a subordinator and is called the ladder time process of Z . The process Z L − t is also asubordinator and is called the ladder height process of Z . (For the basic properties of the ladder time andladder height processes, we refer our readers to [1, Chapter 6].)Let χ be the Laplace exponent of the ladder height process of Z . It follows from [13, Corollary 9.7] that χ ( λ ) = exp (cid:18) π Z ∞ log(Φ( λθ ))1 + θ dθ (cid:19) = exp (cid:18) π Z ∞ log( φ ( λ θ ))1 + θ dθ (cid:19) , ∀ λ > . (2.8)Using (2.8), it was proved in [20, Proposition 2.1] that, if φ is a special Bernstein function, so is χ , i.e., λ λ/χ ( λ ) is also a Bernstein function. The next result tells us that such relation is also true for completeBernstein functions. For the proof we will need the following fact, see [27, Theorem 6.10]: If φ is a completeBernstein function, then there exist a real number γ and a [0 , η on (0 , ∞ ) such thatlog φ ( λ ) = γ + Z ∞ (cid:18) t t − λ + t (cid:19) η ( t ) dt. (2.9) Proposition 2.4
Suppose φ , the Laplace exponent of the subordinator S , is a complete Bernstein function.Then the Laplace exponent χ of the ladder height process of the subordinate Brownian motion Z t = B S t isalso a complete Bernstein function. Proof.
By (2.8) and (2.9), we havelog χ ( λ ) = γ π Z ∞ Z ∞ (cid:18) t t − λ θ + t (cid:19) η ( t ) dt dθ θ . By using 0 ≤ η ( t ) ≤
1, we have η ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t t − λ θ + t (cid:12)(cid:12)(cid:12)(cid:12)
11 + θ ≤
11 + t
11 + θ (cid:18) λ θ + t + λ θ tλ θ + t (cid:19) ≤
11 + t (cid:18) λ θ + t + λ tλ θ + t (cid:19) . Since Z ∞ λ θ + t dθ = 1 t Z ∞ λ θ t + 1 dθ = 1 t √ tλ Z ∞ γ + 1 dγ = π λ √ t ,
9e can use Fubini’s theorem to getlog χ ( λ ) = γ Z ∞ (cid:18) t t ) − √ t ( λ + √ t ) (cid:19) η ( t ) dt (2.10)= γ Z ∞ (cid:18) t t ) − t ) (cid:19) η ( t ) dt + Z ∞ (cid:18) t ) − √ t ( λ + √ t ) (cid:19) η ( t ) dt = γ + Z ∞ (cid:18) s s − λ + s (cid:19) η ( s ) ds , Applying [27, Theorem 6.10] we get that χ is a complete Bernstein function. ✷ Remark 2.5
The above result has been independently proved in [23, Lemma 4].The next result relates the behavior of χ with that of φ . It will be used to obtain the asymptotic behaviorof χ at infinity. Proposition 2.6
Suppose that φ , the Laplace exponent of the subordinator S , is a complete Bernsteinfunction. Then the Laplace exponent χ of the ladder height process of Z satisfies e − π/ p φ ( λ ) ≤ χ ( λ ) ≤ e π/ p φ ( λ ) , for all λ > . (2.11) Proof.
By the representations (2.9) and (2.10), we get that for all λ > (cid:12)(cid:12)(cid:12)(cid:12) log χ ( λ ) −
12 log φ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:18)(cid:16) t t − √ t ( λ + √ t ) (cid:17) − (cid:16) t t − λ + t (cid:17)(cid:19) η ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ λ ( √ t + λ )( λ + t ) √ t ( λ + √ t ) dt = 12 Z ∞ λ ( λ + t ) √ t dt = π . This implies that − π/ ≤ log χ ( λ ) − log φ ( λ ) ≤ π/ λ >
0, which is (2.11) ✷ Remark 2.7
We note that for the last two propositions we only need to assume that φ is a completeBernstein function; the assumption (H) is not used.Let V denote the potential measure of the ladder height process of Z . We will also use V to denote thecorresponding renewal function, V ( t ) := V ((0 , t )). It follows from (2.1) and (2.11) that lim λ →∞ χ ( λ ) /λ = 0and lim λ →∞ χ ( λ ) = ∞ . Therefore, the ladder height process of Z has no drift and has infinite L´evy measure.This suffices to conclude that the potential measure V has a density denoted by v , and the renewal functioncan be written as V ( t ) = R t v ( s ) ds . Since χ is a complete Bernstein function, v is completely monotone.We record these facts as Corollary 2.8
Suppose φ , the Laplace exponent of the subordinator S , is a complete Bernstein functionsatisfying the assumption (H) . Then the potential measure of the ladder height process of the subordinateBrownian motion Z t = B S t has a completely monotone density v . In particular, v and the renewal function V are C ∞ functions. The smoothness of the renewal function V of the ladder height process Z will be used later in this paper.Similarly to the case of the density u of the potential measure U of the subordinator S in (2.4), by usingProposition 2.6, we can obtain the asymptotic behavior of the renewal function V and its density v of theladder height process of Z : V ( t ) ≍ φ ( t − ) − / ≍ t α/ ( ℓ ( t − )) / , t → , (2.12)10 ( t ) ≍ t − φ ( t − ) − / ≍ t α/ − ( ℓ ( t − )) / , t → , (2.13)see [21, Proposition 3.9]. The corresponding precise asymptotics are given in [20, Proposition 2.7] under theassumption (2.2).We next consider multidimensional subordinate Brownian motions. Let W = ( W t = ( W t , . . . , W dt ) : t ≥
0) be a Brownian motion in R d with E h e iθ · ( W t − W ) i = e − t | θ | , ∀ θ ∈ R d , t > , and let S be a subordinator independent of W with Laplace exponent φ . In the remainder of this paper wewill use X = ( X t : t ≥
0) to denote the subordinate Brownian motion defined by X t = W S t . The process X is a (rotationally) symmetric L´evy process with the characteristic exponent given by Φ( θ ) = φ ( | θ | ). It iseasy to check that when d ≥ X is transient. This follows from the criterion of Chung-Fuchstype (e.g., [26, p. 252]) which for the subordinate Brownian motion X translates to the following: X istransient if and only if Z λ d/ − φ ( λ ) dλ < + ∞ . Since transience is a global property of the process, it cannot be inferred from the behavior of φ at infinity.For example, φ ( λ ) = log(1 + λ ) + λ α/ , α ∈ (0 , d ≤
2, there exists γ ∈ [0 , d/
2) such thatlim inf λ → φ ( λ ) λ γ > . (2.14)An immediate consequence of this assumption and [21, Corollary 2.6] is that the potential density u of S satisfies u ( t ) ≤ ct γ − for all t ≥
1, where c > A1 from [20]).Transience of the process X ensures that the Green function G ( x, y ), x, y ∈ R d , is well defined. By spatialhomogeneity we may write G ( x, y ) = G ( x − y ), where the function G is radial and given by the followingformula, G ( x ) = Z ∞ (4 πt ) − d/ e −| x | / (4 t ) u ( t ) dt, x ∈ R d . (2.15)Since u is decreasing, we see that G is radially decreasing and continuous in R d \ { } .The L´evy measure of the process X has a density J , called the L´evy density, given by J ( x ) = Z ∞ (4 πt ) − d/ e −| x | / (4 t ) µ ( t ) dt, x ∈ R d . Thus J ( x ) = j ( | x | ) with j ( r ) := Z ∞ (4 πt ) − d/ e − r / (4 t ) µ ( t ) dt, r > . (2.16)Note that the function r j ( r ) is continuous and decreasing on (0 , ∞ ). We will sometimes use the notation J ( x, y ) for J ( x − y ).We discuss now the behavior of G and j near the origin. Under the assumption (2.2), the preciseasymptotic behavior was obtained in [20, Theorem 3.1, Theorem 3.2] by using precise asymptotic behaviorof the potential density u , and, respectively, L´evy density µ . These two results were proved by use of [6,Lemma 5.32], which required additional assumptions which were stated as A2 and A3 in [20]. It turned outthat by using Potter’s theorem ([2, Theorem 1.5.6]) one can circumvent those assumption and still obtain11he conclusion of the lemma. The details are provided in [21, Lemma 3.1]. The lemma combined with (2.4)(resp. (2.5)) and representation (2.15) (resp. (2.16)) gives the following asymptotic behavior of G (resp. J under the assumption (2.1): Theorem 2.9 ([21, Theorem 3.2])
Suppose that the Laplace exponent φ is a complete Bernstein functionsatisfying the assumption (H) and that α ∈ (0 , ∧ d ) . In the case d ≤ , we further assume (2.14) . Then G ( x ) ≍ | x | d φ ( | x | − ) ≍ | x | d − α ℓ ( | x | − ) , | x | → . Remark 2.10
Since α is always assumed to be in (0 , α ∈ (0 , ∧ d ) in the theorem abovemakes a difference only in the case d = 1. Theorem 2.11 ([21, Theorem 3.4])
Suppose that the Laplace exponent φ is a complete Bernstein functionsatisfying the assumption (H) . Then J ( x ) = j ( | x | ) ≍ φ ( | x | − ) | x | d ≍ ℓ ( | x | − ) | x | d + α , | x | → . Using (2.6) and (2.7), and repeating the proof of [24, Lemma 4.2] we get that(a) For any
K >
0, there exists c = c ( K ) > j ( r ) ≤ c j (2 r ) , ∀ r ∈ (0 , K ) . (2.17)(b) There exists c > j ( r ) ≤ c j ( r + 1) , ∀ r > . (2.18)Therefore by [30, Theorem 2.2 and Section 3.1] (see also [6, Theorem 5.66], [21, Theorem 4.7, Corollary 4.8]and [24]) the Harnack inequality is valid for the process X . Before we state the Harnack inequality, we recallthe definition of harmonic functions.For any open set D , we use τ D to denote the first exit time of D , i.e., τ D = inf { t > X t / ∈ D } . Definition 2.12
Let D be an open subset of R d . A function u defined on R d is said to be (1) harmonic in D with respect to X if E x [ | u ( X τ B ) | ] < ∞ and u ( x ) = E x [ u ( X τ B )] , x ∈ B, for every openset B whose closure is a compact subset of D ; (2) regular harmonic in D with respect to X if it is harmonic in D with respect to X and for each x ∈ D , u ( x ) = E x [ u ( X τ D )] . Theorem 2.13 (Harnack inequality)
There exists C > such that, for any r ∈ (0 , / , x ∈ R d , andany function u which is nonnegative on R d and harmonic with respect to X in B ( x , r ) , we have sup y ∈ B ( x ,r/ u ( y ) ≤ C inf y ∈ B ( x ,r/ u ( y ) . ¿From now we will always assume that φ is a complete Bernstein function satisfying the assumption ( H )for α ∈ (0 , ∧ d ) and the additional (2.14) in the case d ≤
2. We will no longer explicitly mention theseassumptionsFor any open set D in R d , we will use G D ( x, y ) to denote the Green function of X in D . Using thecontinuity and the radial decreasing property of G , we can easily check that G D is continuous in ( D × D ) \ ( x, x ) : x ∈ D } . We will frequently use the well-known fact that G D ( · , y ) is harmonic in D \ { y } , and regularharmonic in D \ B ( y, ε ) for every ε > X , we know that for every bounded open subset D and every f ≥ x ∈ D , E x [ f ( X τ D ); X τ D − = X τ D ] = Z D c Z D G D ( x, z ) J ( z − y ) dzf ( y ) dy. (2.19)Now we prove the following version of the Harnack inequality for X . Theorem 2.14
Let
L > . There exists a positive constant C = C ( L ) > such that the following is true:If x , x ∈ R d and r ∈ (0 , are such that | x − x | < Lr , then for every nonnegative function u which isharmonic with respect to X in B ( x , r ) ∪ B ( x , r ) , we have C − u ( x ) ≤ u ( x ) ≤ C u ( x ) . Proof.
By [21, Proposition 4.10] (see also [20, Proposition 3.8]), for every r ∈ (0 , x ∈ R d and every y ∈ B ( x, r ) it holds that K B ( x, r ) ( x, y ) := Z B ( x, r ) G B ( x, r ) ( x, z ) J ( z − y ) dz ≥ c j ( | x − y | ) r α ℓ ( r − ) , (2.20)with some positive constant c > r ∈ (0 , x , x ∈ R d be such that | x − x | < Lr and let u be a nonnegative function which isharmonic in B ( x , r ) ∪ B ( x , r ) with respect to X .If | x − x | < r , then since r <
1, the theorem is true from Theorem 2.13. Thus we only need to considerthe case when r ≤ | x − x | ≤ Lr with L > .Let w ∈ B ( x , r ). Because | x − w | ≤ | x − x | + | w − x | < ( L + ) r ≤ Lr , by the monotonicity of j and (2.20) K B ( x , r ) ( x , w ) ≥ c j (2 Lr ) r α ℓ ( r − ) . (2.21)For any y ∈ B ( x , r ), u is regular harmonic in B ( y, r ) ∪ B ( x , r ). Since | y − x | < r , by Theorem 2.13 u ( y ) ≥ c u ( x ) , y ∈ B ( x , r , (2.22)for some constant c >
0. Therefore, by (2.19) and (2.21)–(2.22), u ( x ) = E x h u ( X τ B ( x , r ) i ≥ E x h u ( X τ B ( x , r ); X τ B ( x , r ∈ B ( x , r i ≥ c u ( x ) P x (cid:16) X τ B ( x , r ∈ B ( x , r (cid:17) = c u ( x ) Z B ( x , r ) K B ( x , r ) ( x , w ) dw ≥ c u ( x ) (cid:12)(cid:12)(cid:12) B ( x , r (cid:12)(cid:12)(cid:12) j (2 Lr ) r α ℓ ( r − ) = c u ( x ) j (2 Lr ) r α + d ℓ ( r − ) . Thus, by Theorem 2.11 and the inequality above, there exists a constant c > r ∈ (0 , u ( x ) ≥ c u ( x ) j (2 Lr ) j ( r ) . The right-hand side is by (2.17) greater than c c log 2 L/ log 26 u ( x ) where c = C − (4 L ) ∈ (0 , ✷ In [20] we have established the boundary Harnack principle under the assumption (2.2) (and the additionaltransience assumption in case d ≤
2) for κ -fat open sets. Even though we only explicitly stated the results for d ≥ d = 1 also. Under the current assumptions,the same result is reproved in [21]. 13 heorem 2.15 ([20, Theorem 4.8], [21, Theorem 4.22]) Suppose that D is a bounded κ -fat open set withthe characteristics ( R , κ ). There exists a constant C = C ( d, α, ℓ, R , κ ) > r ≤ R ∧ and Q ∈ ∂D , then for any nonnegative functions u, v in R d which are regular harmonic in D ∩ B ( Q, r ) withrespect to X and vanish in D c ∩ B ( Q, r ), we have C − u ( A r ( Q )) v ( A r ( Q )) ≤ u ( x ) v ( x ) ≤ C u ( A r ( Q )) v ( A r ( Q )) , x ∈ D ∩ B ( Q, r . Before concluding this section, we give some examples satisfying our assumptions.
Example 2.16
Suppose that 0 < β < α <
2, 0 < γ < − α and define φ ( λ ) = λ α/ , φ ( λ ) = ( λ + 1) α/ − , φ ( λ ) = λ α/ + λ β/ ,φ ( λ ) = λ α/ (log(1 + λ )) γ/ and φ ( λ ) = λ α/ (log(1 + λ )) − β/ . Then φ i , i = 1 , . . . ,
5, are complete Bernstein functions which can be written as φ i ( λ ) = λ α/ ℓ i ( λ ) , i = 1 , . . . , , with ℓ ( λ ) = 1 , ℓ ( λ ) = (cid:16) ( λ + 1) α/ − (cid:17) λ − α/ , ℓ ( λ ) = 1 + λ ( β − α ) / ,ℓ ( λ ) = (log(1 + λ )) γ/ and ℓ ( λ ) = (log(1 + λ )) − β/ . As already mentioned in the introduction, the subordinate Brownian motion corresponding to φ is a sym-metric α -stable process, the subordinate Brownian motion corresponding to φ is a relativistic α -stableprocess and the subordinate Brownian motion corresponding to φ is the sum of a symmetric α -stable pro-cess and an independent symmetric β -stable process. The subordinate Brownian motions corresponding to φ and φ were discussed in [6].In the case d ≥
3, the only condition on the complete Bernstein function φ is (1.2), so we can useProposition 7.1, Corollary 7.9, Propositions 7.10–7.11, Corollary 7.12 of [27] to come up with infinitely manyexamples of such functions, e.g.: (i) λ α/ (log(1 + log(1 + λ γ/ ) δ/ )) β/ , α, γ, δ ∈ (0 , , β ∈ (0 , − α ]; (ii) λ α/ (log(1 + log(1 + λ γ/ ) δ/ )) − β/ , α, γ, δ ∈ (0 , , β ∈ (0 , α ].All of the listed example satisfy the stronger condition (2.2). As already mentioned, [21, Example 2.8]provides an example of a complete Bernstein function which satisfies (2.1), but not (2.2). κ -fat open sets In this section we will establish sharp two-sided Green function estimates for X in any bounded κ -fat opensubset D of R d , d ≥
1. Our standing assumption is that φ is a complete Bernstein function satisfying theassumption (H) for α ∈ (0 , ∧ d ) and the additional assumption (2.14) when d ≤ Lemma 3.1
There exist R = R ( ℓ ) > , L > and C > such that G ( x ) − G ( L x ) ≥ C | x | d − α ℓ ( | x | − ) for every | x | < R . Proof.
By Theorem 2.9 there exists a constant c > x ∈ R d \ { } with | x | < c − | x | d − α ℓ ( | x | − ) ≤ G ( x ) ≤ c | x | d − α ℓ ( | x | − ) . L = (4 c ) d − α ∨ c /L d − α ≤ /
4. Since ℓ is slowly varying at infinity, there exists r < ℓ ( | x | − ) ℓ ( | L x | − ) ≤ < | x | < r . Let R = r ∧ L − . Then for x ∈ R d \ { } we have G ( x ) − G ( L x ) ≥ c − | x | d − α ℓ ( | x | − ) − c | L x | d − α ℓ ( | L x | − )= c − | x | d − α ℓ ( | x | − ) (cid:18) − c L d − α ℓ ( | x | − ) ℓ ( | L x | − ) (cid:19) ≥ c | x | d − α ℓ ( | x | − ) . ✷ Proposition 3.2
For every bounded open set D , there exists a constant C > such that G D ( x, y ) ≤ C | x − y | d − α ℓ ( | x − y | − ) , for x, y ∈ D , (3.1) and G D ( x, y ) ≥ C − | x − y | d − α ℓ ( | x − y | − ) , for x, y ∈ D with L | x − y | ≤ δ D ( x ) ∧ δ D ( y ) , (3.2) where L is the constant in Lemma 3.1. Proof.
Since G D ( x, y ) ≤ G ( x, y ), D is bounded and ℓ locally bounded, (3.1) is an immediate consequenceof Theorem 2.9. Now we show (3.2). Without loss of generality, we assume δ D ( y ) ≤ δ D ( x ), and let M := diam( D ). We consider three cases separately:(a) δ D ( y ) ≤ R : Since | x − y | ≤ δ D ( y ) /L , | X τ B ( y,δD ( y )) − y | ≥ δ D ( y ) ≥ L | x − y | . Thus by the monotonicityof G and Lemma 3.1, G D ( x, y ) ≥ G B ( y,δ D ( y )) ( x, y ) = G ( x, y ) − E x h G ( X τ B ( y,δD ( y )) , y ) i ≥ G ( x − y ) − G ( L ( x − y )) ≥ c | x − y | d − α ℓ ( | x − y | − ) , ∀| x − y | ≤ δ D ( y ) L . (b) δ D ( y ) > R and | x − y | ≤ R /L : In this case, | X τ B ( y,R − y | ≥ R ≥ L | x − y | and, by the monotonicityof G and Lemma 3.1, we get G D ( x, y ) ≥ G B ( y,R ) ( x, y ) = G ( x, y ) − E x h G ( X τ B ( y,R , y ) i ≥ G ( x − y ) − G ( L ( x − y )) ≥ c | x − y | d − α ℓ ( | x − y | − ) , ∀| x − y | ≤ R L . (c) δ D ( y ) > R and | x − y | > R /L : Choose a point w ∈ ∂B ( y, R /L ). Then from the argument in (b),we get G D ( w, y ) ≥ G ( w, y ) − E w h G ( X τ B ( y,R , y ) i ≥ c R /L ) d − α ℓ (( R /L ) − ) . Since D is bounded and G D ( · , y ) is harmonic with respect to X in B ( x, R / (2 L )) ∪ B ( w, R / (2 L )), byTheorem 2.14 we have G D ( x, y ) ≥ c G D ( w, y ) ≥ c R /L ) d − α ℓ (( R /L ) − )15 c ℓ (( R /L ) − ) inf R L ≤ s ≤ M ℓ ( s − ) ! | x − y | d − α ℓ ( | x − y | − ) , ∀| x − y | > R L . ✷ Lemma 3.3
For every
L > and bounded open set D , there exists C > such that for every | x − y | ≤ L ( δ D ( x ) ∧ δ D ( y )) , G D ( x, y ) ≥ C | x − y | d − α ℓ ( | x − y | − ) . (3.3) Proof.
Without loss of generality, we assume δ D ( x ) ≤ δ D ( y ). Moreover, by Proposition 3.2 we can assumethat L > /L and we only need to show (3.3) for L δ D ( x ) ≤ | x − y | ≤ Lδ D ( x ).Choose a point w ∈ ∂B ( x, δ D ( x ) /L ). Then by Proposition 3.2, we get G D ( x, w ) ≥ c δ D ( x ) /L ) d − α ℓ (( δ D ( x ) /L ) − ) . Since | y − w | ≤ | x − y | + | x − w | ≤ ( L + 1) δ D ( x ) and G D ( x, · ) = G D ( · , x ) is harmonic with respect to X in B ( y, δ D ( x ) /L ) ∪ B ( w, δ D ( x ) /L ), by Theorem 2.14 we have G D ( x, y ) ≥ c G D ( x, w ) ≥ c δ D ( x ) /L ) d − α ℓ (( δ D ( x ) /L ) − ) ≥ c (cid:18) ℓ ( | x − y | − ) ℓ (( δ D ( x ) /L ) − ) (cid:19) | x − y | d − α ℓ ( | x − y | − ) . (3.4)By the uniform convergence theorem ([2, Theorem 1.2.1]), we can choose a small r = r ( ℓ, L ) > λ ∈ [1 , L ] ℓ (( λr ) − ) ℓ ( r − ) ≥ , ∀ r ≤ r . (3.5)If L δ D ( x ) ≤ | x − y | ≤ Lδ D ( x ) ≤ r , by (3.4)–(3.5), G D ( x, y ) ≥ c (cid:18) inf r ≤ r ,λ ∈ [1 , L ] ℓ (( λr ) − ) ℓ ( r − ) (cid:19) | x − y | d − α ℓ ( | x − y | − ) ≥ c | x − y | d − α ℓ ( | x − y | − ) . On the other hand, if L δ D ( x ) ≤ | x − y | ≤ Lδ D ( x ) and δ D ( x ) ≥ r /L , then | x − y | ≥ r LL . Thus from(3.4), we see that G D ( x, y ) ≥ c inf r ∈ [ r LL ,M ] ℓ ( r − ) ! inf r ∈ [ r LL ,M ] ℓ ( r − ) − ! | x − y | d − α ℓ ( | x − y | − )where M = diam( D ). ✷ For the remainder of this section, we assume that D is a bounded κ -fat open set with characteristics( R , κ ). Without loss of generality we may assume that R ≤ /
4. We recall that for each Q ∈ ∂D and r ∈ (0 , R ), A r ( Q ) denotes a point in D ∩ B ( Q, r ) satisfying B ( A r ( Q ) , κr ) ⊂ D ∩ B ( Q, r ). Recall also that G D ( · , z ) is regular harmonic in D \ B ( y, ε ) for every ε > D . From Theorem 2.15, weget the following boundary Harnack principle for the Green function of X in D . Theorem 3.4
There exists a constant C > such that for any Q ∈ ∂D , r ∈ (0 , R ] and z, w ∈ D \ B ( Q, r ) , we have C − G D ( z, A r ( Q )) G D ( w, A r ( Q )) ≤ G D ( z, x ) G D ( w, x ) ≤ C G D ( z, A r ( Q )) G D ( w, A r ( Q )) , x ∈ D ∩ B (cid:16) Q, r (cid:17) . R ≤ R such that12 ≤ min ≤ λ ≤ κ − ℓ (( λr ) − ) ℓ ( r − ) ≤ max ≤ λ ≤ κ − ℓ (( λr ) − ) ℓ ( r − ) ≤ , if r ≤ R . (3.6)Let us fix z ∈ D with κR < δ D ( z ) < R and let ε := κR /
24. For x, y ∈ D , we define r ( x, y ) := δ D ( x ) ∨ δ D ( y ) ∨ | x − y | and B ( x, y ) := ((cid:8) A ∈ D : δ D ( A ) > κ r ( x, y ) , | x − A | ∨ | y − A | < r ( x, y ) (cid:9) if r ( x, y ) < ε { z } if r ( x, y ) ≥ ε . (3.7)Note that for every ( x, y ) ∈ D × D with r ( x, y ) < ε δ D ( A ) ≤ δ D ( x ) ∨ δ D ( y ) ∨ | x − y | ≤ κ − δ D ( A ) , A ∈ B ( x, y ) . (3.8)So by (3.6), if r ( x, y ) < ε ≤ ℓ (( δ D ( A )) − ) ℓ (( r ( x, y )) − ) ≤ , A ∈ B ( x, y ) . (3.9)Let C := C d − α δ D ( z ) − d + α sup δ D ( z ) / ≤ r ≤ M ℓ ( r − ) − ! . (3.10)It follows from Proposition 3.2 that G D ( · , z ) is bounded above by C on D \ B ( z , δ D ( z ) / g ( x ) := G D ( x, z ) ∧ C . (3.11)Note that if δ D ( z ) ≤ ε , then | z − z | ≥ δ D ( z ) − ε ≥ δ D ( z ) / ε < δ D ( z ) /
4, and therefore g ( z ) = G D ( z, z ).The two lemmas below follow immediately from Theorem 2.14. Lemma 3.5
There exists C = C ( κ ) > such that for all x, y ∈ D , C − g ( A ) ≤ g ( A ) ≤ C g ( A ) forall A , A ∈ B ( x, y ) . Lemma 3.6
There exists C > such that for every x ∈ { y ∈ D : δ D ( y ) ≥ κ ε / } , C − ≤ g ( x ) ≤ C . We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2 . Since the proof is an adaptation of the proofs of [4, Proposition 6] and [16,Theorem 2.4], we only give the proof when δ D ( x ) ≤ δ D ( y ) ≤ κ | x − y | .In this case, we have r ( x, y ) = | x − y | . Let r := ( | x − y |∧ ε ). Choose Q x , Q y ∈ ∂D with | Q x − x | = δ D ( x )and | Q y − y | = δ D ( y ). Pick points x = A κr/ ( Q x ) and y = A κr/ ( Q y ) so that x, x ∈ B ( Q x , κr/
2) and y, y ∈ B ( Q y , κr/ | z − Q x | ≥ r and | y − Q x | ≥ r . So by Theorem 3.4, wehave c − G D ( x , y ) g ( x ) ≤ G D ( x, y ) g ( x ) ≤ c G D ( x , y ) g ( x )for some c >
1. On the other hand, since | z − Q y | ≥ r and | x − Q y | ≥ r , applying Theorem 3.4 again, c − G D ( x , y ) g ( y ) ≤ G D ( x , y ) g ( y ) ≤ c G D ( x , y ) g ( y ) . c − G D ( x , y ) g ( x ) g ( y ) ≤ G D ( x, y ) g ( x ) g ( y ) ≤ c G D ( x , y ) g ( x ) g ( y ) . Moreover, | x − y | < | x − y | < | x − y | and | x − y | ≤ κ ε ( δ D ( x ) ∧ δ D ( y )). Thus by Lemma 3.3, wehave 2 − d + α c − c − g ( x ) g ( y ) 1 | x − y | d − α ℓ ( | x − y | − ) ≤ G D ( x, y ) g ( x ) g ( y ) ≤ d − α c c g ( x ) g ( y ) 1 | x − y | d − α ℓ ( | x − y | − ) (3.12)for some c > r = ε /
2, then r ( x, y ) = | x − y | ≥ ε . Thus g ( A ) = g ( z ) = C and δ D ( x ) ∧ δ D ( y ) ≥ κr/ κε / C − c − ≤ g ( A ) g ( x ) g ( y ) ≤ C c (3.13)for some c > r < ε /
2, then r ( x, y ) = | x − y | < ε and r = r ( x, y ). Hence δ D ( x ) , δ D ( y ) ≥ κr/ κr ( x, y ) / | x − A | , | y − A | ≥ r ( x, y ). So by applying Theorem 2.13 to g twice, c − ≤ g ( A ) g ( x ) ≤ c and c − ≤ g ( A ) g ( y ) ≤ c (3.14)for some constant c = c ( D ) >
0. Combining (3.12)-(3.14), we get c − g ( x ) g ( y ) g ( A ) | x − y | d − α ℓ ( | x − y | − ) ≤ G D ( x, y ) ≤ c g ( x ) g ( y ) g ( A ) | x − y | d − α ℓ ( | x − y | − ) , A ∈ B ( x, y ) . If | x − y | < | x − y | < | x − y | ≤ R , by (3.6), ≤ ℓ ( | x − y | − ) ℓ ( | x − y | − ) ≤
2. On the other hand, if | x − y | < | x − y | < | x − y | and 2 | x − y | > R , we have that R ≤ | x − y | ≤ M and R ≤ | x − y | ≤ M . Thus fromthe local boundedness of ℓ on (0 , ∞ ), we see that c − ≤ ℓ ( | x − y | − ) ℓ ( | x − y | − ) ≤ c for some c > ✷ C , -open sets In this section we refine the estimates from Theorem 1.2 in the case of bounded C , open sets.Recall that X = ( X t : t ≥
0) is the d -dimensional subordinate Brownian motion defined by X t = W S t where W = ( W , . . . , W d ) is a d -dimensional Brownian motion and S = ( S t : t ≥
0) an independentsubordinator with the Laplace exponent φ which is a complete Bernstein function satisfying assumption (H) for α ∈ (0 , ∧ d ) and the additional assumption (2.14) when d ≤
2. Let Z = ( Z t : t ≥
0) be theone-dimensional subordinate Brownian motion defined as Z t := W dS t . Recall that the potential measure ofthe ladder height process of Z is denoted by V and its density by v . We also use V to denote the renewalfunction of the ladder height process of Z . In Corollary 2.8 we have established that both V and v are C ∞ function. Recall the notation R d + := { x = ( x , . . . , x d − , x d ) := (˜ x, x d ) ∈ R d : x d > } for the half-space.The next result, which follows from [28, Theorem 2], is very important in this paper.Define w ( x ) := V (( x d ) + ). 18 heorem 4.1 The function w is harmonic in R d + with respect to X and, for any r > , regular harmonicin R d − × (0 , r ) (in (0 , r ) when d = 1 ) with respect to X . Proof.
Since Z t := W dS t has a transition density, it satisfies the condition ACC in [28], namely the resolventkernels are absolutely continuous. The assumption in [28] that 0 is regular for (0 , ∞ ) is also satisfied since Z is symmetric and has infinite L´evy measure. Indeed, if 0 were irregular for (0 , ∞ ), it would be, by symmetry,irregular for ( −∞ ,
0) as well. But then Z would be a compound Poisson process which contradicts thefact that it has infinite L´evy measure. Further, again by symmetry of Z , the notions of coharmonic andharmonic functions coincide. Let Z (0 , ∞ ) (respectively X R d + ) denote the process Z killed upon exiting (0 , ∞ )(respectively X killed upon exiting R d + ). By [28, Theorem 2], the renewal function V of the ladder heightprocess of Z is invariant for Z (0 , ∞ ) . Thus w is invariant for X R d + . Being invariant for X R d + , w is also harmonicfor X R d + , and consequently, harmonic in R d + with respect to X . We show now that w is regular harmonic for X in R d − × (0 , r ) for any r >
0. First note that since V is continuous at zero and V (0) = 0, it follows thatlim x d → w ( x ) = lim x d → w (˜ x, x d ) = lim x d → V ( x d ) = 0 . (4.1)Thus, by harmonicity of w and (4.1) w ( x ) = w (˜ x, x d ) = lim ε → E x h w (cid:16) X τ R d − × ( ε,r ) (cid:17)i = E x h w (cid:16) X τ R d − × (0 ,r ) (cid:17)i , x d > . ✷ Proposition 4.2
For all positive constants r and L , we have sup x ∈ R d : 0 By Theorem 4.1 and (2.19), for every x ∈ R d + , w ( x ) ≥ E x (cid:20) w ( X τ B ( x,r / ∩ R d + ) : X τ B ( x,r / ∩ R d + ∈ B ( x, r ) c ∩ R d + (cid:21) = Z B ( x,r ) c ∩ R d + Z B ( x,r / ∩ R d + G B ( x,r / ∩ R d + ( x, z ) j ( | z − y | ) w ( y ) dz dy . (4.2)Since | z − y | ≤ | x − z | + | x − y | ≤ r + | x − y | ≤ | x − y | for ( z, y ) ∈ B ( x, r / × B ( x, r ) c , using (2.17) and(2.18), we have j ( | z − y | ) ≥ c j ( | x − y | ). Thus, combining this with (4.2), we obtain thatsup x ∈ R d : 0 L > x d ≥ r / (64) and e x = 0, w ( x ) E x [ τ B ( x,r / ∩ R d + ] ≤ V ( L ) E [ τ B (0 ,r / (64)) ] . Suppose x d < r / (64) and e x = 0. Let U := B (( e , r ) , r ). By the L´evy system, we have P x (cid:18) X τ B (0 ,r / ∩ R d + ∈ U (cid:19) = Z U Z B (0 ,r / ∩ R d + G B (0 ,r / ∩ R d + ( x, z ) j ( | z − y | ) dzdy ≤ c E x [ τ B (0 ,r / ∩ R d + ] . w ( x ) E x [ τ B ( x,r / ∩ R d + ] ≤ c w ( x ) P x ( X τ B (0 ,r / ∩ R d + ∈ U ) ≤ c w ( x ) P x ( X τ B (0 ,r / ∩ R d + ∈ U ) ≤ c V ( r / (16))where x = ( e , r / (16)). We have thus proved the claim. ✷ We now define the operator ( A , D ( A )) by the following formula: A f ( x ) := P . V . Z R d ( f ( y ) − f ( x )) j ( | y − x | ) dy := lim ε ↓ Z { y ∈ R d : | x − y | >ε } ( f ( y ) − f ( x )) j ( | y − x | ) dy D ( A ) := ( f : R d → R : lim ε ↓ Z { y ∈ R d : | x − y | >ε } ( f ( y ) − f ( x )) j ( | y − x | ) dy exists and is finite ) . (4.3)It is well known that C ⊂ D ( A ), where C is the collection of C functions in R d vanishing at infinity, andthat by the rotational symmetry of X , A restricted to C coincides with the infinitesimal generator of theprocess X (e.g. [26, Theorem 31.5]). Theorem 4.3 A w ( x ) is well defined and A w ( x ) = 0 for all x ∈ R d + . Proof. We first note that it follows from Proposition 4.2 and the fact that j is a L´evy density that for any L > x ∈ R d : 0 0. 20oreover, for every x ∈ R d + , z ∈ B ( x, ( ε ∧ x d ) / y ∈ B ( z, ε ) c it holds that | y − z | ≤ | y − x | ≤ | y − z | . So, using (2.17), {| y − z | >ε } (cid:12)(cid:12) ( w ( y ) − w ( z ) − {| y − z | < } ( y − z ) · ∇ w ( z )) (cid:12)(cid:12) j ( | y − z | ) ≤ c sup ε/ 0. Then, by Proposition 4.2 Z U Z U c w ( y ) j ( | x − y | ) dydx ≤ | U | sup x ∈ U Z U c w ( y ) j ( | x − y | ) dy ≤ | U | sup x ∈ U Z B ( x,r ) c w ( y ) j ( | x − y | ) dy < ∞ . (4.5)By harmonicity of w , clearly w ( X τ U ) ∈ L ( P x ) andsup x ∈ U E x (cid:2) U c ( X τ U ) w ( X τ U ) (cid:3) ≤ sup x ∈ U E x (cid:2) w ( X τ U ) (cid:3) = sup x ∈ U w ( x ) < ∞ . The last two displays show that the conditions [7, (2.4), (2.6)] are true. Thus, by [7, Lemma 2.3, Theorem2.11(ii)], we have that for any f ∈ C c ( R d + ),0 = Z R d Z R d ( w ( y ) − w ( x ))( f ( y ) − f ( x )) j ( | y − x | ) dx dy. (4.6)For f ∈ C c ( R d + ) with supp( f ) ⊂ U ⊂ U ⊂ U ⊂ R d + , Z R d Z R d | w ( y ) − w ( x ) || f ( y ) − f ( x ) | j ( | y − x | ) dxdy = Z U Z U | w ( y ) − w ( x ) || f ( y ) − f ( x ) | j ( | y − x | ) dxdy + 2 Z U Z U c | w ( y ) − w ( x ) || f ( x ) | j ( | y − x | ) dxdy ≤ c Z U × U | y − x | j ( | y − x | ) dxdy + 2 k f k ∞ | U | (cid:18) sup x ∈ U w ( x ) (cid:19) Z U c j ( | y − x | ) dy + 2 k f k ∞ Z U Z U c w ( y ) j ( | x − y | ) dydx is finite by (4.5) and the fact that j is a L´evy density. Thus by (4.6), Fubini’s theorem and the dominatedconvergence theorem, we have for any f ∈ C c ( R d + ),0 = lim ε ↓ Z { ( x,y ) ∈ R d × R d , | y − x | >ε } ( w ( y ) − w ( x ))( f ( y ) − f ( x )) j ( | y − x | ) dx dy = − ε ↓ Z R d + f ( x ) Z { y ∈ R d : | y − x | >ε } ( w ( y ) − w ( x )) j ( | y − x | ) dy ! dx = − Z R d + f ( x ) A w ( x ) dx, where we have used the fact A ε w → A w converges uniformly on the support of f . Hence, by the continuityof A w , we have A w ( x ) = 0 in R d + . ✷ x ∈ R d , let δ ∂D ( x ) denote the Euclidean distance between x and ∂D . It is well known that any C , open set D with characteristics ( R, Λ) satisfies both the uniform interior ball condition and the uniformexterior ball condition with the radius r : there exists r < R such that for every x ∈ D with δ D ( x ) < r and y ∈ R d \ D with δ D ( y ) < r , there are z x , z y ∈ ∂D so that | x − z x | = δ ∂D ( x ), | y − z y | = δ D ( y ) and that B ( x , r ) ⊂ D and B ( y , r ) ⊂ R d \ D for x = z x + r ( x − z x ) / | x − z x | and y = z y + r ( y − z y ) / | y − z y | .In the remainder of this section, we assume D is a C , open set with characteristics ( R, Λ) and D satisfiesthe uniform interior ball condition and the uniform exterior ball condition with the radius R (by choosing R smaller if necessary). Lemma 4.4 Fix Q ∈ ∂D and let h ( y ) := V ( δ D ( y )) D ∩ B ( Q,R ) ( y ) . There exist C = C ( α, Λ , R, ℓ ) > and R ≤ R/ independent of the point Q ∈ ∂D such that A h is welldefined in D ∩ B ( Q, R ) and |A h ( x ) | ≤ C for all x ∈ D ∩ B ( Q, R ) . (4.7) Proof. We first note that when d = 1, the lemma follows from Proposition 4.2 and Theorem 4.3. Infact, suppose that d = 1 and x ∈ D ∩ B ( Q, R/ Q is the origin and D ∩ B ( Q, R ) = (0 , R ) (due to uniform exterior ball condition). Since h ( y ) = w ( y ) for y ∈ D ∩ B ( Q, R ) = (0 , R ), we have A ( h − w )( x ) = lim ε ↓ Z { y ∈ R : | x − y | >ε } ( h − w )( y ) j ( | y − x | ) dy = lim ε ↓ Z { y ∈ (0 ,R ) c : | x − y | >ε } ( h − w )( y ) j ( | y − x | ) dy = − lim ε ↓ Z { y ∈ (0 ,R ) c : | x − y | >ε } w ( y ) j ( | y − x | ) dy = − lim ε ↓ Z { y ≥ R : | x − y | >ε } w ( y ) j ( | y − x | ) dy Since 0 < x < R/ y ≥ R , we have | x − y | > R/ 2, and thus |A ( h − w )( x ) | ≤ Z { y ≥ R, | x − y | >R/ } w ( y ) j ( | y − x | ) dy. Therefore, by using Theorem 4.3 (which gives A h ( x ) = A ( h − w )( x )), Proposition 4.2 and the above display,we conclude that |A h ( x ) | = |A ( h − w )( x ) | ≤ sup 4) such that for every r ≤ R ℓ ( r − )( ℓ ((2 R − ) − r − )) / ≤ ℓ ((2 R ) − )( ℓ ((2 R − ) − (2 R ) − )) / (2 R ) / r − / ≤ c r − / , (4.8)22 ( r − ) ≤ ℓ ((2 R ) − )(2 R ) q r − q ≤ c r − q , (4.9) ℓ ( r − ) − / ≤ ℓ ((2 R ) − ) / (2 R ) q r − q ≤ c r − q . (4.10)In the remainder of this proof, we fix x ∈ D ∩ B ( Q, R ) and x ∈ ∂D satisfying δ D ( x ) = | x − x | . Wealso fix the C , function ψ and the coordinate system CS = CS x in the definition of C , open set so that x = (0 , x d ) with 0 < x d < R and B ( x , R ) ∩ D = { y = ( e y, y d ) ∈ B (0 , R ) in CS : y d > ψ ( e y ) } . Let ψ ( e y ) := R − p R − | e y | and ψ ( e y ) := − R + p R − | e y | . Due to the uniform interior ball condition and the uniform exterior ball condition with the radius R , we have ψ ( e y ) ≤ ψ ( e y ) ≤ ψ ( e y ) for every y ∈ D ∩ B ( x, R ) . (4.11)Define H + := { y = ( e y, y d ) ∈ CS : y d > } and let A := { y = ( e y, y d ) ∈ ( D ∪ H + ) ∩ B ( x, R ) : ψ ( e y ) ≤ y d ≤ ψ ( e y ) } ,E := { y = ( e y, y d ) ∈ B ( x, R ) : y d > ψ ( e y ) } . Note that, since | y − Q | ≤ | y − x | + | x − Q | ≤ R/ y ∈ B ( x, R ), we have B ( x, R ) ∩ D ⊂ B ( Q, R/ ∩ D . (4.12)Let h x ( y ) := V (cid:0) δ H + ( y ) (cid:1) . Note that h x ( x ) = h ( x ). Moreover, since δ H + ( y ) = ( y d ) + in CS , by Theorem 4.3 it follows that A h x is welldefined in H + and A h x ( y ) = 0 , ∀ y ∈ H + . (4.13)We show now that A ( h − h x )( x ) is well defined. For each ε > (cid:12)(cid:12)(cid:12)(cid:12) Z { y ∈ D ∪ H + : | y − x | >ε } ( h ( y ) − h x ( y )) j ( | y − x | ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B ( x,R ) c ( h ( y ) + h x ( y )) j ( | y − x | ) dy + Z A ( h ( y ) + h x ( y )) j ( | y − x | ) dy + Z E | h ( y ) − h x ( y ) | j ( | y − x | ) dy =: I + I + I . We claim that I + I + I ≤ C (4.14)for some constant C = C ( α, Λ , R, ℓ ). This shows in particular that the limitlim ε ↓ Z { y ∈ D ∪ H + : | y − x | >ε } ( h ( y ) − h x ( y )) j ( | y − x | ) dy exists and hence A ( h − h x )( x ) is well defined, and |A ( h − h x )( x ) | ≤ C . By linearity and (4.13), we getthat A h ( x ) is well defined and |A h ( x ) | ≤ C . Therefore, it remains to prove (4.14).By the fact that h ( y ) = 0 for y ∈ B ( Q, R ) c , I ≤ sup z ∈ R d : 0 0. The last inequality is due to the fact that q < (2 − α ) / 20, which implies(1 − α/ 2) + α + 3 q − < (6 + 7 α ) / < 1, so by the dominated convergence theorem, x d Z R y d ) − α/ q | y d − x d | α +2 q − dy d (4.19)is a strictly positive continuous function in x d ∈ [0 , R ] and hence it is bounded.On the other hand, we have, using polar coordinates for e y , and by Theorem 2.11, (2.12) and (4.9)–(4.10), L ≤ c Z x d + R Z R ∧ √ Ry d − y d v ( R − p r + ( R − y d ) ) r d j (( r + | y d − x d | ) / ) dr ! dy d ≤ c Z x d + R Z R ∧ √ Ry d − y d ( R − p r + ( R − y d ) ) α/ − ℓ (( r + | y d − x d | ) − )( ℓ (( R − p r + ( R − y d ) ) − )) / ( r + | y d − x d | ) ( d + α ) / r d dr ! dy d ≤ c Z x d + R Z R ∧ √ Ry d − y d r d ( R − p r + ( R − y d ) ) − α/ q ( r + | y d − x d | ) ( d + α +2 q ) / dr ! dy d ≤ c Z x d + R Z R ∧ √ Ry d − y d R − p r + ( R − y d ) ) − α/ q ( r + | y d − x d | ) α +2 q dr ! dy d . Since, for 0 < r < R ∧ p Ry d − y d ,1 R − p r + ( R − y d ) = R + p r + ( R − y d ) ( p Ry d − y d + r )( p Ry d − y d − r ) ≤ c √ y d ( p Ry d − y d − r ) , we have L ≤ Z x d + R c ( y d ) (1 − α/ q ) / Z R ∧ √ Ry d − y d dr ( p Ry d − y d − r ) − α/ q ( r + | y d − x d | ) α +2 q dy d . Using the fact that q ≤ α , we see that with a := p Ry d − y d and b := | y d − x d | , Z R ∧ a dr ( a − r ) − α/ q ( r + b ) α +2 q = Z ( R ∧ a ) / dr ( a − r ) − α/ q ( r + b ) α +2 q + Z R ∧ a ( R ∧ a ) / dr ( a − r ) − α/ q ( r + b ) α +2 q ≤ − α/ q a − α/ q Z ( R ∧ a ) / dr ( r + b ) α +2 q + 1( b + ( R ∧ a ) / α +2 q Z R ∧ a ( R ∧ a ) / dr ( a − r ) − α/ q ≤ c a − α/ q b ( α +2 q − + + c ( R ∧ a ) α +2 q a α/ − q ≤ c ( y d ) (1 − α/ q ) / | x d − y d | ( α +2 q − + + c ( y d ) ( α +6 q ) / . L ≤ c Z R dy d ( y d ) (1 − α/ q ) | y d − x d | ( α +2 q − + + c Z R dy d ( y d ) (1+4 q ) / . (4.20)Since q < / 10, the second integral in (4.20) is bounded. And by the same argument as the one for (4.19),the first integral in (4.20) is also bounded. We have proved the claim (4.14). ✷ When d ≥ 2, define ρ Q ( x ) := x d − ψ Q ( e x ) , where ( e x, x d ) are the coordinates of x in CS Q . Note that forevery Q ∈ ∂D and x ∈ B ( Q, R ) ∩ D we have(1 + Λ ) − / ρ Q ( x ) ≤ δ D ( x ) ≤ ρ Q ( x ) . (4.21)For a, b > 0, we define D Q ( a, b ) := { y ∈ D : a > ρ Q ( y ) > , | e y | < b } when d ≥ 2. When d = 1, we simplytake D Q ( a, b ) = D Q ( a ) := B ( Q, a ) ∩ D . Lemma 4.5 There are constants R = R ( R, Λ , α, ℓ ) ∈ (0 , R / (4 p )) and C i = C i ( R, Λ , α ) > , i = 17 , , such that for every r ≤ R , Q ∈ ∂D and x ∈ D Q ( r, r ) , P x (cid:16) X τ DQ ( r,r ) ∈ D (cid:17) ≥ C V ( δ D ( x )) (4.22) and E x (cid:2) τ D Q ( r,r ) (cid:3) ≤ C V ( δ D ( x )) . (4.23) Proof. Without loss of generality, we assume Q = 0. For d ≥ 2, let ψ = ψ : R d − → R be the C , functionand CS be the coordinate system in the definition of C , open set so that B (0 , R ) ∩ D = (cid:8) ( e y, y d ) ∈ B (0 , R ) in CS : y d > ψ ( e y ) (cid:9) . Let ρ ( y ) := y d − ψ ( e y ) and D ( a, b ) := D ( a, b ) for d ≥ 2. When d = 1, D ( a, b )is simply B (0 , a ) ∩ D . The remainder of the proof is written for d ≥ 2. The interpretation in the case d = 1is obvious.Note that | y | = | e y | + | y d | < r + ( | y d − ψ ( e y ) | + | ψ ( e y ) | ) < (1 + (1 + Λ) ) r for every y ∈ D ( r, r ) . (4.24)Hence, by letting b r := R / p , D ( r, s ) ⊂ D ( b r, b r ) ⊂ B (0 , R ) ∩ D ⊂ B (0 , R ) ∩ D for every r, s ≤ b r. Define h ( y ) := V ( δ D ( y )) B (0 ,R ) ∩ D ( y ) . Let g be a non-negative smooth radial function with compact support such that g ( x ) = 0 for | x | > R R d g ( x ) dx = 1. For k ≥ 1, define g k ( x ) = 2 kd g (2 k x ) and h ( k ) ( z ) := ( g k ∗ h )( z ) := Z R d g k ( y ) h ( z − y ) dy , and let B k := (cid:8) x ∈ D ∩ B (0 , R ) : δ D ∩ B (0 ,R ) ( x ) ≥ − k (cid:9) . Since h ( k ) is C ∞ , A h ( k ) is well defined everywhere.We claim that − C ≤ A h ( k ) ≤ C on B k , (4.25)where C is the constant from Lemma 4.4. Indeed, for x ∈ B k and z ∈ B (0 , − k ) it holds that x − z ∈ D ∩ B (0 , R ). Hence, by Lemma 4.4 the following limit exists:lim ε → Z | x − y | >ε ( h ( y − z ) − h ( x − z )) j ( | x − y | ) dy 26 lim ε → Z | ( x − z ) − y ′ | >ε ( h ( y ′ ) − h ( x − z )) j ( | ( x − z ) − y ′ | ) dy ′ = A h ( x − z ) . Moreover, by the same Lemma 4.4 it holds that − C ≤ A h ( x − z ) ≤ C . Next, Z | x − y | >ε ( h ( k ) ( y ) − h ( k ) ( x )) j ( | x − y | ) dy = Z | x − y | >ε (cid:18)Z R d g k ( z )( h ( y − z ) − h ( x − z )) dz (cid:19) j ( | x − y | ) dy = Z | z | < − k g k ( z ) Z | x − y | >ε ( h ( y − z ) − h ( x − z )) j ( | x − y | ) dy ! dz. By letting ε → A h ( k ) ( x ) = Z | z | < − k g k ( z ) A h ( x − z ) dz ≤ C Z | z | < − k g k ( z ) dz = C . The left-hand side inequality in (4.25) is obtained in the same way.Using the fact that A restricted to C ∞ c coincides with the infinitesimal generator of the process X , wesee that the following Dynkin formula is true; for f ∈ C ∞ c ( R d ) and any bounded open subset U of R d , E x Z τ U A f ( X t ) dt = E x [ f ( X τ U )] − f ( x ) . (4.26)Let U ⊂ D ∩ B (0 , R ). By using (4.26) for U ∩ B k and h ( k ) , the estimates (4.25), the fact that h ( k ) arein C ∞ c ( R d ), and by letting k → ∞ we get h ( x ) ≥ E x [ h ( X τ U )] − C E x [ τ U ] and h ( x ) ≤ E x [ h ( X τ U )] + C E x [ τ U ] . (4.27)Now, we have by (4.21) and (4.27), for every λ ≥ x ∈ D ( λ − b r, λ − b r ), V ( δ D ( x )) = h ( x ) ≥ E x h h (cid:16) X τ D ( λ − b r,λ − b r ) (cid:17) ; X τ D ( λ − b r,λ − b r ) ∈ D ( b r, λ − b r ) \ D ( λ − b r, λ − b r ) i − C E x (cid:2) τ D ( λ − b r,λ − b r ) (cid:3) ≥ V (cid:0) λ − (1 + Λ ) − / b r (cid:1) P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D ( b r, λ − b r ) \ D ( λ − b r, λ − b r ) (cid:17) − C E x (cid:2) τ D ( λ − b r,λ − b r ) (cid:3) . (4.28)We also have from (4.27) V ( δ D ( x )) = h ( x ) ≤ E x h h (cid:16) X τ D ( λ − b r,λ − b r ) (cid:17)i + C E x [ τ D ( λ − b r,λ − b r ) ] ≤ V ( R ) P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D (cid:17) + C E x [ τ D ( λ − b r,λ − b r ) ] . (4.29)By (2.19) and the monotonicity of j , for every λ ≥ x ∈ D ( λ − b r, λ − b r ), P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D (cid:17) ≥ P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D ( b r, λ − b r ) \ D ( λ − b r, λ − b r ) (cid:17) ≥ P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D (3 λ − b r, λ − b r ) \ D (2 λ − b r, λ − b r ) (cid:17) = E x "Z τ D ( λ − b r,λ − b r ) Z D (3 λ − b r,λ − b r ) \ D (2 λ − b r,λ − b r ) j ( | X s − y | ) dyds Z D (3 λ − b r,λ − b r ) \ D (2 λ − b r,λ − b r ) dy ! j (10 λ − b r ) E x (cid:2) τ D ( λ − b r,λ − b r ) (cid:3) ≥ c ( λ − b r ) d j (10 λ − b r ) E x (cid:2) τ D ( λ − b r,λ − b r ) (cid:3) . Now, applying Theorem 2.11, we get for x ∈ D ( λ − b r, λ − b r ) P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D (cid:17) ≥ c ℓ (cid:0) (10 λ − b r ) − (cid:1) λ α E x (cid:2) τ D ( λ − b r,λ − b r ) (cid:3) . (4.30)Thus from (4.28)–(4.30), for every x ∈ D ( λ − b r, λ − b r ) V ( δ D ( x )) ≥ (cid:16) c V (cid:0) λ − (1 + Λ ) − / b r (cid:1) ℓ (cid:0) (10 λ − b r ) − (cid:1) λ α − C (cid:17) E x [ τ D ( λ − b r,λ − b r ) ] (4.31)and V ( δ D ( x )) ≤ c (cid:16) (cid:0) ℓ (cid:0) (10 λ − b r ) − (cid:1)(cid:1) − λ − α (cid:17) P x (cid:16) X τ D ( λ − b r,λ − b r ) ∈ D (cid:17) . (4.32)Using first (2.12) and then Potter’s Theorem ([2, Theorem 1.5.6 (1)]), we see that there exists a large λ > λ ≥ λ V (cid:0) λ − (1 + Λ ) − / b r (cid:1) ℓ (cid:0) (10 λ − b r ) − (cid:1) λ α ≥ c b r α/ (1 + Λ ) − α/ λ α/ (cid:16) ℓ (cid:16)(cid:0) λ − (1 + Λ ) − / b r (cid:1) − (cid:17)(cid:17) − / ℓ (cid:0) (10 λ − b r ) − (cid:1) ≥ C /c (4.33)and (cid:0) ℓ (cid:0) (10 λ − b r ) − (cid:1)(cid:1) − λ − α ≤ c . (4.34)Combining (4.31)–(4.34), we have proved the lemma with R := λ − b r . ✷ It is clear that every C , open set is κ -fat, i.e., for any C , open set with C , characteristics ( R, Λ),there exists a constant κ ∈ (0 , / R, Λ), such that for each Q ∈ ∂D and r ∈ (0 , R ), D ∩ B ( Q, r ) contains a ball B ( A r ( Q ) , κr ) of radius κr . In the rest of this paper, whenever we deal with C , open sets, the constants Λ, R and κ will have the meaning described above.Recall that g is defined in (3.11). Theorem 4.6 Suppose that D is a bounded C , open set in R d with C , characteristics ( R, Λ) . Then thereexists C = C ( R, Λ , α, diam ( D )) > such that C − ( V ( δ D ( x )) ∧ ≤ g ( x ) ≤ C ( V ( δ D ( x )) ∧ , x ∈ D, (4.35) or equivalently there exists C = C ( R, Λ , α, diam ( D )) > such that C − ( δ D ( x )) α/ p ℓ (( δ D ( x )) − ) ∧ ! ≤ g ( x ) ≤ C ( δ D ( x )) α/ p ℓ (( δ D ( x )) − ) ∧ ! , x ∈ D. (4.36) Proof. Since d = 1 case is simpler, we give the proof for d ≥ R is the constant in (3.6)and ε = R κ/ 24. Since g ( x ) = G D ( x, z ) ∧ C and g ( x ) = G D ( x, z ) for δ D ( x ) < ε , it suffices to showthat there exist r ∗ ∈ (0 , ε ) and c > c − ( δ D ( x )) α/ p ℓ (( δ D ( x )) − ) ≤ G D ( x, z ) ≤ c ( δ D ( x )) α/ p ℓ (( δ D ( x )) − ) , δ D ( x ) < r ∗ . (4.37)28et r ∗ := ( R / ∧ ( ε / (4 p )) and suppose that δ D ( x ) < r ∗ . Choose x ∈ ∂D satisfying δ D ( x ) = | x − x | . We fix the C , function ψ and the coordinate system CS = CS x in the definition of C , open set so that x = ( e , x d ) with 0 < x d < r ∗ , B ( x , R ) ∩ D = { y = ( e y, y d ) ∈ B (0 , R ) in CS : y d > ψ ( e y ) } . Let x := ( e , r ∗ / 2) and D ∗ := D ( r ∗ , r ∗ ) = { y ∈ D : r ∗ > y d − ψ ( e y ) > , | e y | < r ∗ } . Since B ( x , c r ∗ ) ⊂ D ∗ for small c > 0, by Theorem 2.9, Theorem 2.15 and the fact that D is bounded, G D ( x, z ) ≤ c G D ( x , z ) P x (cid:0) X τ D ∗ ∈ B ( z , ε / (cid:1) P x (cid:0) X τ D ∗ ∈ B ( z , ε / (cid:1) ≤ c G ( x , z ) P x (cid:0) X τ D ∗ ∈ B ( z , ε / (cid:1) P x (cid:16) X τ B ( x ,c r ∗ ) ∈ B ( z , ε / (cid:17) ≤ c P x (cid:0) X τ D ∗ ∈ B ( z , ε / (cid:1) ≤ c E x [ τ D ∗ ]where in the last inequality we used (2.19) and the fact that dist( D ∗ , B ( z , ε / ≥ δ D ( z ) − ε / − p r ∗ ≥ ε (see (4.24)). On the other hand, by Theorem 2.15, Lemma 3.3 and the fact that D is bounded, G D ( x, z ) ≥ c G D ( x , z ) P x (cid:0) X τ D ∗ ∈ D (cid:1) P x (cid:0) X τ D ∗ ∈ D (cid:1) ≥ c P x (cid:0) X τ D ∗ ∈ D (cid:1) . By applying (4.22)–(4.23), we have proved (4.35). The inequalities (4.36) follow from (2.12). ✷ Now we give the proof of Theorem 1.1, which is the main result of this paper. Proof of Theorem 1.1 . By [2, Theorem 1.5.3], the local boundedness and strict positivity of ℓ , there exists c > < s ≤ t ≤ D ) =: 6 Ms α ( ℓ ( s − )) ≤ c t α ( ℓ ( t − )) . Thus, if s, t, u ≤ M then c − (cid:18) s α ℓ ( s − ) ∨ t α ℓ ( t − ) ∨ u α ℓ ( u − ) (cid:19) ≤ ( s ∨ t ∨ u ) α ℓ (( s ∨ t ∨ u ) − ) ≤ c (cid:18) s α ℓ ( s − ) ∨ t α ℓ ( t − ) ∨ u α ℓ ( u − ) (cid:19) . (4.38)Also note that by (3.7)–(3.9), for every A ∈ B ( x, y )12 ∧ min κR ≤ s ≤ R ℓ ( s − )max ε ≤ s ≤ M ℓ ( s − ) ≤ ℓ (( δ D ( A )) − ) ℓ (( δ D ( x ) ∨ δ D ( y ) ∨ | x − y | ) − ) ≤ ∨ max κR ≤ s ≤ R ℓ ( s − )min ε ≤ s ≤ M ℓ ( s − ) (4.39)and (cid:18) ∧ ε R (cid:19) δ D ( A ) ≤ δ D ( x ) ∨ δ D ( y ) ∨ | x − y | ≤ κ − (cid:18) M R ∨ (cid:19) δ D ( A ) . (4.40)By Theorem 1.2 and Theorem 4.6, we have that c − ( δ D ( x ) δ D ( y )) α/ ℓ (( δ D ( A )) − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − )( δ D ( A )) α ℓ ( | x − y | − ) | x − y | d − α ≤ G D ( x, y ) ≤ c ( δ D ( x ) δ D ( y )) α/ ℓ (( δ D ( A )) − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − )( δ D ( A )) α ℓ ( | x − y | − ) | x − y | d − α . ab ∧ ba ∧ abc = ab ( a ∨ b ∨ c ) , for all a, b, c > c − ( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) ∧ ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ≤ ( δ D ( x ) δ D ( y )) α/ ℓ (( δ D ( A )) − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − )( δ D ( A )) α ≤ c ( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) ∧ ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α , to prove the theorem, we only need to show( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) ∧ ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ≤ ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! ≤ c ( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) ∧ ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! . (4.41)Since the first inequality is clear, we will proceed to the second inequality.By symmetry, we only need to consider the cases δ D ( y ) ≤ δ D ( x ) / δ D ( x ) / ≤ δ D ( y ) ≤ δ D ( x ), and,using the fact ℓ is slowly varying and [2, Theorem 1.5.3], the case δ D ( x ) / ≤ δ D ( y ) ≤ δ D ( x ) is clear.Now we assume δ D ( y ) ≤ δ D ( x ) / 3. Since | x − y | ≥ δ D ( x ) − δ D ( y ) ≥ δ D ( x ) / ℓ , δ D ( x ) δ D ( y ) ℓ ( | x − y | − ) /α ( ℓ (( δ D ( x )) − )) /α ( ℓ (( δ D ( y )) − )) /α | x − y | ≤ c δ D ( x ) δ D ( y )( ℓ (( δ D ( x )) − )) /α ( ℓ (( δ D ( y )) − )) /α (cid:18) ( ℓ (( δ D ( x )) − )) /α δ D ( x ) (cid:19) = c δ D ( y )( ℓ (( δ D ( y )) − )) /α ( ℓ (( δ D ( x )) − )) /α δ D ( x )and ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ≤ c ( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) . Thus ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! ≤ c ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! ≤ c ( δ D ( x )) α/ p ℓ (( δ D ( y )) − )( δ D ( y )) α/ p ℓ (( δ D ( x )) − ) ∧ ( δ D ( y )) α/ p ℓ (( δ D ( x )) − )( δ D ( x )) α/ p ℓ (( δ D ( y )) − ) ∧ ( δ D ( x ) δ D ( y )) α/ ℓ ( | x − y | − ) p ℓ (( δ D ( x )) − ) ℓ (( δ D ( y )) − ) | x − y | α ! . We have obtained the second inequality in (4.41). ✷ emark 4.7 Similarly as in Theorem 4.6, by use of (2.12) , the inequalities (1.3) imply the alternativeforms given in (1.4) and (1.6) . Now we give the proof of Theorem 1.3, which is a consequence of Theorems 1.1 and 2.15. Proof of Theorem 1.3 . Using the interior ball condition of D , the following holds: For every Q ∈ ∂D and r ≤ R there is a ball B = B ( z rQ , r ) of radius r such that B ⊂ D and ∂B ∩ ∂D = { Q } . In addition, itfollows from [29, Lemma 2.2] that, for each Q ∈ ∂D , we can choose a constant c = c ( d, Λ) ∈ (0 , / 8] and abounded C , open set U Q with uniform characteristics ( R ∗ , Λ ∗ ) depending only on ( R, Λ) and d such that B ( Q, c R ) ∩ D ⊂ U Q ⊂ B ( Q, R ) ∩ D and δ D ( y ) = δ U Q ( y ) for every y ∈ B ( Q, c R ) ∩ D. (4.42)Assume that r ∈ (0 , c R ], Q ∈ ∂D and u is nonnegative function in R d harmonic in D ∩ B ( Q, r ) = U Q ∩ B ( Q, r ) with respect to X and vanishes continuously on D c ∩ B ( Q, r ). Let z Q := z c RQ . By [9, Lemma4.2] and its proof, we see that u and x → G U Q ( x, z Q ) are regular harmonic in U Q ∩ B ( Q, r/ 3) with respectto X . Since the C , characteristics of U Q depend only on ( R, Λ) and d , by the boundary Harnack principle(Theorem 2.15), there exist r = r ( α, ℓ, R, Λ) ∈ (0 , / 4] and c = c ( α, ℓ, R, Λ) > r ∈ (0 , r ] we have u ( x ) u ( y ) ≤ c G U Q ( x, z Q ) G U Q ( y, z Q ) for every x, y ∈ B ( Q, r/ ∩ D. Now applying Theorem 1.1 to G U Q ( x, z Q ) and G U Q ( y, z Q ), then using (4.42), we conclude that for r ∈ (0 , ( c R ∧ r )] u ( x ) u ( y ) ≤ c δ α/ U Q ( x ) p ℓ (( δ U Q ( y )) − ) δ α/ U Q ( y ) p ℓ (( δ U Q ( x )) − ) = c δ α/ D ( x ) p ℓ (( δ D ( y )) − ) δ α/ D ( y ) p ℓ (( δ D ( x )) − ) for every x, y ∈ B ( Q, r/ ∩ D for some c = c ( α, ℓ, R, Λ) > 0. The form (1.7) given in the statement of the theorem is equivalent to theone in the display above for r ∈ (0 , ( c R ∧ r )]. Now the case r ∈ (( c R ∧ r ) , ( R ∧ / 4] follows from thecase r ∈ (0 , ( c R ∧ r )] and Theorem 2.14. ✷ References [1] J. Bertoin, L´evy Processes . Cambridge University Press, Cambridge, 1996.[2] N. H. 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Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. , (2002), 419–441.[18] P. Kim, Relative Fatou’s theorem for ( − ∆) α/ -harmonic function in κ -fat open set. J. Funct. Anal. , (2006), 70-105.[19] P. Kim and Y.-R. Lee, Generalized 3G theorem and application to relativistic stable process on non-smoothopen sets. J. Funct. Anal. (2007), 113–134.[20] P. Kim, R. Song and Z. Vondraˇcek, Boundary Harnack principle for subordinate Brownian motion. Stoch. Proc.Appl. (2009), 1601–1631.[21] P. Kim, R. Song and Z. Vondraˇcek, Potential theory of subordinate Brownian motions revisited. Preprint (2011),arXiv:1102.1369.[22] T. Kulczycki, Properties of Green function of symmetric stable processes. Probab. Math. Stat. (1997), 381–406.[23] M. Kwa´snicki, Spectral analysis of subordinate Brownian motions in half-line. Preprint (2010)[24] M. Rao, R. Song and Z. Vondraˇcek, Green function estimates and Harnack inequality for subordinate Brownianmotions. Potential Anal. (2006), 1–27[25] M. Ryznar, Estimates of Green function for relativistic α -stable process. Potential Anal. , (2002), 1–23.[26] K.-I. Sato, L´evy Processes and Infinitely Divisible Distributions . Cambridge University Press, Cambridge, 1999.[27] R. L. Schilling, R. Song and Z. Vondraˇcek, Bernstein Functions: Theory and Applications . de Gruyter Studiesin Mathematics 37. Berlin: Walter de Gruyter, 2010.[28] M. L. Silverstein, Classification of coharmonic and coinvariant functions for a L´evy process. Ann. Probab. ,539–575, 1980.[29] R. Song, Estimates on the Dirichlet heat kernel of domains above the graphs of bounded C , functions. Glas.Mat. (2004), 273–286.[30] R. Song and Z. Vondraˇcek, Harnack inequalities for some classes of Markov processes. Math. Z. , (2004),177–202.[31] R. Song and Z. Vondraˇcek, Potential theory of special subordinators and subordinate killed stable processes. J.Theoret. Probab. , (2006), 817–847.[32] R. Song and J. Wu, Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. Math. Scand. , (1967) 17–37.[34] M. Z¨ahle, Potential spaces and traces of L´evy processes on h -sets, J. Contemp. Math. Anal. (2009), 117–145.[35] Z. Zhao, Green functions for Schr¨odingers operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. , (1986) 309–334. anki Kim Department of Mathematical Sciences and Research Institute of Mathematics,Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of KoreaE-mail: [email protected] Renming Song Department of Mathematics, University of Illinois, Urbana, IL 61801, USAE-mail: [email protected] Zoran Vondraˇcek Department of Mathematics, University of Zagreb, Zagreb, CroatiaEmail: [email protected]@math.hr / } j ( | y − x | / . It follows from Proposition 4.2 and the fact that j is a L´evy density, by using the dominated convergencetheorem, that x → A ε w ( x ) is continuous for each ε . Therefore, by this and the local uniform convergence of A ε w , the function A w ( x ) is continuous in R d + .Suppose that U and U are relatively compact open subsets of R d + such that U ⊂ U ⊂ U ⊂ R d + . Let r := dist( U , U c ) >