aa r X i v : . [ m a t h . P R ] J a n ON HARMONIC FUNCTIONS OF SYMMETRIC L ´EVYPROCESSES
ANTE MIMICA
Abstract.
We consider some classes of L´evy processes for which the estimate ofKrylov and Safonov (as in [BL02]) fails and thus it is not possible to use the stan-dard iteration technique to obtain a-priori H¨older continuity estimates of harmonicfunctions. Despite the faliure of this method, we obtain some a-priori regularityestimates of harmonic functions for these processes. Moreover, we extend resultsfrom [ˇSSV06] and obtain asymptotic behavior of the Green function and the L´evydensity for a large class of subordinate Brownian motions, where the Laplaceexponent of the corresponding subordinator is a slowly varying function. Introduction
Recently there has been much interest in investigation of the continuity propertiesof harmonic functions with the respect to various non-local operators. An exampleof such operator L is of the form( L f )( x ) = Z R d \{ } (cid:0) f ( x + h ) − f ( x ) − h∇ f ( x ) , h i | h |≤ (cid:1) n ( x, h ) dh (1.1)for f ∈ C ( R d ) bounded. Here n : R d × ( R d \ { } ) → [0 , ∞ ) is a measurable functionsatisfying c | h | − d − α ≤ n ( x, h ) ≤ c | h | − d − α , for some constants c , c > α ∈ [0 , α ∈ (0 ,
2) H¨older regularity estimates hold for L -harmonicfunctions (see [BL02] for a probabilistic and [Sil06] for an analytic approach) . Mathematics Subject Classification.
Primary 60J45, 60J75, Secondary 60J25.
Key words and phrases. geometric stable process, Green function, harmonic function, L´evyprocess, modulus of continuity, subordinator, subordinate Brownian motion.Research supported in part by German Science Foundation DFG via IGK ”Stochastics and realworld models” and SFB 701.
For α = 0, techniques developed so far are not applicable. One of our aims is toinvestigate this case using a probabilistic approach.In many cases the operator of the form (1.1) can be understood as the infinitesimalgenerator of a Markov jump process. The kernel n ( x, h ) can be thought of as themeasure of intensity of jumps of the process.Let us describe the stochastic process we are considering. Let S = ( S t : t ≥
0) bea subordinator such that its Laplace exponent φ defined by φ ( λ ) = − log (cid:0) E e − λS (cid:1) satisfies lim λ → + ∞ φ ′ ( λx ) φ ′ ( λ ) = x α − for any x > α ∈ [0 ,
2] . Let B = ( B t , P x ) be an independent Brownian motion in R d anddefine a new process X = ( X t , P x ) in R d by X t = B ( S t ) . It is called the subordinateBrownian motion. Example 1:
Let S be a subordinator with the Laplace exponent φ satisfyinglim λ → + ∞ φ ( λ ) λ α/ ℓ ( λ ) = 1with α ∈ (0 ,
2) and ℓ : (0 , ∞ ) → (0 , ∞ ) that varies slowly at infinity (i.e. for any x > ℓ ( λx ) ℓ ( λ ) → λ → + ∞ ).Then (1.2) holds (we just use Theorem 1.7.2 in [BGT87] together with the fact that φ ′ is decreasing and φ ( λ ) = R λ φ ′ ( t ) dt ) and n ( x, h ) = n ( h ) ≍ | h | − d − α ℓ ( | h | − ) , | h | → , which means that n ( h ) | h | − d − α ℓ ( | h | − ) stays between two positive constants as | h | →
0+ .Choosing ℓ ≡ α -stable process (whose in-finitesimal generator is the fractional Laplacian L = − ( − ∆) α ) is included in thisclass. Other choices of ℓ allow us to consider processes which are not invariant undertime-space scaling.The processes described in Example 1 with ℓ ≡ ARMONIC FUNCTIONS OF L´EVY PROCESSES 3
Not much is known about harmonic functions in the case when the correspondingsubordinators belong to ’boundary’ cases, i.e. α ∈ { , } (see [ˇSSV06, Mim11,Mim12]). For the class of geometric stable processes only the non-scale invariantHarnack inequality was proved and on-diagonal heat kernel upper estimate is notfinite (see [ˇSSV06]).The following two examples belong to these ’boundary’ cases and are covered by ourapproach. Example 2:
Let φ ( λ ) = log(1 + λ ). The corresponding process X is known as thevariance gamma process. In (1.2) we have α = 0 . Moreover, it will be proved (seeTheorem 4.1) that n ( x, h ) = n ( h ) ≍ | h | − d , | h | → . (1.3)This example can be generalized in various ways. For example, we can take k ∈ N and consider φ k = φ ◦ . . . ◦ φ | {z } k times . Then (see Theorem 4.1 or Subsection 7.1) n k ( x, h ) = n k ( h ) ≍ | h | − d log · · · log | {z } k − | h | · . . . · log log | h | · log | h | − , | h | → . Another generalization is to consider φ ( λ ) = log(1 + λ β/ ) for some β ∈ (0 , X is known as the β -geometric stable process and the behavior of n is givenalso by (1.3). Example 3:
Let φ ( λ ) = λ log(1+ √ λ ) . Then in (1.2) we have α = 2 and (see Theorem4.1) n ( x, h ) = n ( h ) ≍ | h | − d − (cid:16) log | h | (cid:17) − , | h | → . This behavior shows that small jumps of this process have higher intensity thansmall jumps of any stable process.A measurable bounded function f : R d → R is said to be harmonic in an open set D ⊂ R d if for any relatively compact open set B ⊂ B ⊂ Df ( x ) = E x f ( X τ B ) for any x ∈ B, where τ B = inf { t > X t B } .The main theorem is the following regularity result, which covers cases α ∈ [0 , α = 0. By B r ( x ) we denote the openball with center x ∈ R d and radius r > ANTE MIMICA
Theorem 1.1.
Let S be a subordinator such that its L´evy and potential measureshave decreasing densities. Assume that the Laplace exponent of S satisfies (1.2) with α ∈ [0 , . Let X be the corresponding subordinate Brownian motion and let d ≥ .There is a constant c > such that for any r ∈ (0 , ) and any bounded function f : R d → R which is harmonic in B r (0) , | f ( x ) − f ( y ) | ≤ c k f k ∞ φ ( r − ) φ ( | x − y | − ) for all x, y ∈ B r (0) . Applying Theorem 1.1 to Example 1, we obtain expected H¨older regularity esti-mates. Within this example the result is new when the scaling is lost, e. g. φ ( λ ) = λ α [log(1 + λ )] − α . The situation is more interesting in Example 2, e.g. for the geometric stable process.For this process we obtain logarithmic regularity estimates: | f ( x ) − f ( y ) | ≤ c k f k ∞ log( r − ) 1log( | x − y | − ) , It is still unknown whether H¨older regularity estimates hold for harmonic functionsof this process (or, generally, of the processes belonging to the case α = 0).Let us explain why known analytic and probabilistic techniques do not work in thecase α = 0. The main idea in the proof of the a priori H¨older estimates of harmonicfunctions relies on the estimate of Krylov and Safonov.In probabilistic setting this estimate can be formulated as follows. There is a con-stant c > A ⊂ B r (0) and x ∈ B r (0) P x ( T A < τ B (0 ,r ) ) ≥ c | A || B r (0) | , (1.4)where T A = τ A c is the first hitting time of A and | A | denotes the Lebesgue measureof the set A .Performing a computation similar to the one in the proof of Proposition 3.4 in [BL02](see also Lemma 3.4 in [SV04]) we deduce P x ( T A < τ B r (0) ) ≥ c r − φ ′ ( r − ) φ ( r − ) | A || B r (0) | . ARMONIC FUNCTIONS OF L´EVY PROCESSES 5 If α ∈ (0 , r − φ ′ ( r − ) φ ( r − ) ≍ r →
0+ . This gives estimateof the form (1.4) and thus the standard Moser’s iteration procedure for obtaininga-priori H¨older regularity estimates of harmonic functions can be applied (see theproof of Theorem 4.1 in [BL02] for a probabilistic version).The situation is quite different for α = 0. To find a counterexample we will use thefollowing result. Proposition 1.2.
Let S be as subordinator such that its L´evy and potential measureshave decreasing densities and whose Laplace exponent satisfies (1.2) with α ∈ [0 , .Let X be the corresponding subordinate Brownian motion and let d ≥ .There is a constant c > such that for every r ∈ (0 , and x ∈ B r (0) P x ( X τ B r ∈ B r (0) \ B r (0)) ≤ c r − φ ′ ( r − ) φ ( r − ) . For α = 0 it will follow that lim r → r − φ ′ ( r − ) φ ( r − ) = 0 (see (2.10)). Therefore in this case(1.4) does not hold, sincelim r → P ( T B r (0) \ B r (0) < τ B r (0) ) ≤ lim r → P ( X τ B r ∈ B r (0) \ B r (0)) = 0 . Considering process X in the setting of metric measure spaces (as in [CK08] or[BGK09]) the news feature appears. Theorem 4.1 shows that the jumping kernel ofthe process X is of the form n ( x, h ) = j ( | h | ) with j ( r ) ≍ r − φ ′ ( r − ) φ ( r − ) · E τ B r (0) | B r (0) | , r → . In the case α = 0 the term r − φ ′ ( r − ) φ ( r − ) becomes significant. This has not yet beentreated within this framework.The latter discussion shows that the question of the continuity of harmonic functionsbecomes interesting even in the case when the kernel n ( x, h ) is space homogeneous,or in other words, in the case of a L´evy process. There is no known technique thatcovers this situation in the case of a more general jump process.Our technique relies on asymptotic properties of the underlying subordinator. Thepotential density can be analyzed using the de Haan theory of slow variation (see[BGT87]). ANTE MIMICA
On the other hand, there is no known Tauberian theorem that can be applied toobtain asymptotic behavior of the L´evy density µ of the subordinator. For thispurpose we perform asymptotic inversion of the Laplace transform (see Proposition3.2) to get µ ( t ) ≍ t − φ ′ ( t − ) , t → . These techniques allow us to extend results from [ˇSSV06] to much wider class ofsubordinators whose Laplace exponents are logarithmic or, more generally, slowlyvarying functions.Although we do not obtain regularity estimates of harmonic functions for cases when α ∈ [1 , α = 2 is also new. For examplethe Green function of the process corresponding to the Example 3 above has thefollowing behavior: G ( x, y ) ≍ | x − y | − d log( | x − y | − ) , | x − y | → . We may say that such process X is ’between’ any stable process and Brownianmotion.Let us briefly comment the technique we are using to prove the regularity result. InSection 2 it will be seen that any bounded function f which is harmonic in B r (0)can be represented as f ( x ) = Z B r (0) c K B r (0) ( x, z ) f ( z ) dz, x ∈ B r (0) , where K B r (0) ( x, z ) is the Poisson kernel of the ball B r (0).The following estimate of differences of Poisson kernel is the key to the proof ofTheorem 1.1: | K B r (0) ( x , z ) − K B r (0) ( x , z ) | ≤ c | z | − d φ (( | z |− r ) − ) φ ( | x − y | − ) r < | z | ≤ r j ( | z | ) φ ( | x − y | − ) | z | > r for x , x ∈ B r (0) (see Proposition 5.3).Similar type estimate has been obtained in [Szt10] for stable L´evy processes usingscaling argument and the explicit behavior of the transition density. In our settingthere are many cases where the behavior of the transition density is not known andthe scaling argument does not work. Our idea is to establish the following Green ARMONIC FUNCTIONS OF L´EVY PROCESSES 7 function difference estimates: | G ( x , y ) − G ( x , y ) | ≤ c r − φ ′ ( r − ) φ ( r − ) r d (cid:16) ∧ | x − y | r (cid:17) for all y ∈ R d and x , x B r ( y ) (see Proposition 5.1).The paper is organized as follows. In Section 2 we introduce all concepts we needthroughout the paper. Section 3 is devoted to the study of subordinators. We obtainasymptotic properties of L´evy and potential densities. In Section 4 asymptoticalproperties of the Green function and L´evy density of the subordinate Brownianmotions are obtained. Difference estimates of the Green function and the Poissonkernel are the main subject of Section 5. This type of estimates are the mainingredient in the proof of the regularity result in Section 6. In Section 7 we applyour results to some new examples. Notation.
For two functions f and g we write f ∼ g if f /g converges to 1 and f ≍ g if f /g stays between two positive constants. The n -th derivative of f (ifexists) is denoted by f ( n ) .The logarithm with base e is denoted by log and we introduce the following notationfor iterated logarithms: log = log and log k +1 = log ◦ log k for k ∈ N .We say that f : R → R is increasing if s ≤ t implies f ( s ) ≤ f ( t ) and analogously fora decreasing function.The standard Euclidian norm and the standard inner product in R d are denotedby | · | and h· , ·i , respectively. By B r ( x ) = { y ∈ R d : | y − x | < r } we denote theopen ball centered at x with radius r >
0. The Gamma function is defined byΓ( ρ ) = R ∞ t ρ − e − t dt for ρ >
0. 2.
Preliminaries
L´evy processes and their potential theory.
A stochastic process X =( X t : t ≥
0) with values in R d ( d ≥
1) defined on a probability space (Ω , F , P ) is saidto be a L´evy process if it has independent and stationary increments, its trajectoriesare P -a.s. right continuous with left limits and P ( X = 0) = 1 .The characteristic function of X t is always of the form E exp { i h ξ, X t i} = exp {− t Φ( ξ ) } , ANTE MIMICA where Φ is called the characteristic (or L´evy) exponent of X . It has the followingL´evy-Khintchine representationΦ( ξ ) = i h γ, ξ i + 12 h Aξ, ξ i + Z R d (cid:0) − e i h x,ξ i + i h x, ξ i {| x |≤ } (cid:1) Π( dx ) . Here γ ∈ R d , A is a non-negative definite symmetric d × d real matrix and Π is ameasure on R d , called the L´evy measure of X , satisfyingΠ( { } ) = 0 and Z R d (1 ∧ | x | )Π( dx ) < ∞ . The Brownian motion B = ( B t : t ≥
0) in R d with transition density p ( t, x, y ) =(4 πt ) − d/ exp n − | x − y | t o is an example of a L´evy process with the characteristic ex-ponent Φ( ξ ) = | ξ | .A subordinator is a stochastic process S = ( S t : t ≥
0) which is a L´evy process in R such that S t ∈ [0 , ∞ ) for every t ≥
0. In this case it is more convenient to considerthe Laplace transform of S t : E exp {− λS t } = exp {− tφ ( λ ) } , λ > . The function φ : (0 , ∞ ) → (0 , ∞ ) is called the Laplace exponent of S and it has thefollowing representation φ ( λ ) = γλ + Z (0 , ∞ ) (1 − e − λt ) µ ( dt ) . Here γ ≥ µ is also called the L´evy measure of S and it satisfies the followingintegrability condition: R (0 , ∞ ) (1 ∧ t ) µ ( dt ) < ∞ .The potential measure of the subordinator S is defined by U ( A ) = E (cid:20)Z ∞ { S t ∈ A } dt (cid:21) for a measurable A ⊂ [0 , ∞ ) . The Laplace transform of U is then L U ( λ ) := Z (0 , ∞ ) e − λt U ( dt ) = 1 φ ( λ ) . (2.1)Assume that the processes B and S just described are independent. We define anew stochastic process X = ( X t : t ≥
0) by X t = B ( S t ) and call it the subordinate ARMONIC FUNCTIONS OF L´EVY PROCESSES 9
Brownian motion. It is a L´evy process with the characteristic exponent Φ( ξ ) = φ ( | ξ | ) and the L´evy measure of the form Π( dx ) = j ( | x | ) dx with j ( r ) = Z (0 , ∞ ) (4 πt ) − d/ exp (cid:26) − r t (cid:27) µ ( dt ) . (2.2)The process X has the transition density and it is given by p ( t, x, y ) = Z [0 , ∞ ) (4 πs ) − d/ exp (cid:26) − | x − y | s (cid:27) P ( S t ∈ ds ) . (2.3)When X is transient, we can define the Green function of X by G ( x, y ) = Z (0 , ∞ ) p ( t, x, y ) dt, x, y ∈ R d , x = y . The Green function can be considered as the density of the Green measure definedby G ( x, A ) = E x (cid:20)Z ∞ { X t ∈ A } dt (cid:21) , A ⊂ R d measurable , since G ( x, A ) = R A G ( x, y ) dy .Using (2.3) we can rewrite it as G ( x, y ) = g ( | y − x | ) with g ( r ) = Z (0 , ∞ ) (4 πt ) − d/ exp (cid:26) − r t (cid:27) U ( dt ) . (2.4)Let D ⊂ R d be a bounded open set. We define the process killed upon exiting DX D = ( X Dt : t ≥
0) by X Dt = (cid:26) X t t < τ D ∂ t ≥ τ D , where ∂ is an extra point adjoined to D .Using the strong Markov property we can see that the Green measure of X D is G D ( x, A ) = E x (cid:20)Z ∞ { X Dt ∈ A } dt (cid:21) = E x (cid:20)Z ∞ { X t ∈ A } dt (cid:21) − E x (cid:20)Z ∞ τ D { X t ∈ A } dt (cid:21) = G ( x, A ) − E x [ G ( X τ D , A ); τ D < ∞ ] . Thus in the transient case the Green function of X D can be written as G D ( x, y ) = G ( x, y ) − E x [ G ( X τ D , y ); τ D < ∞ ] , x, y ∈ D, x = y . Since X is, in particular, an isotropic L´evy process it follows from [Szt00] that P x ( X τ Br (0) ∈ ∂B r (0)) = 0 , x ∈ B r (0) . for any r > x ∈ B r (0) . This allows us to use the Ikeda-Watanabe formula(see Theorem 1 in [IW62]): P x ( X τ Br (0) ∈ F ) = Z F Z B r (0) G B r (0) ( x, y ) j ( | z − y | ) dy dz , (2.5)for x ∈ B r (0) and F ⊂ B r (0) c .Defining a function K B r (0) : B r (0) × B r (0) c → [0 , ∞ ) by K B r (0) ( x, z ) = Z B r (0) G B r (0) ( x, y ) j ( | z − y | ) dy (2.6)the Ikeda-Watanabe formula (2.5) reads P x ( X τ Br (0) ∈ F ) = Z F K B r (0) ( x, z ) dz . (2.7)The function K B r (0) will be called the Poisson kernel for the ball B r (0) .2.2. Bernstein functions and subordinators.
A function φ : (0 , ∞ ) → (0 , ∞ ) issaid to be a Bernstein function if φ ∈ C ∞ (0 , ∞ ) and ( − n φ ( n ) ≤ n ∈ N .Every Bernstein φ function has the following representation: φ ( λ ) = γ + γ λ + Z (0 , ∞ ) (1 − e − λt ) µ ( dt ) , (2.8)where γ , γ ≥ µ is a measure on (0 , ∞ ) satisfying R (0 , ∞ ) (1 ∧ t ) µ ( dt ) < ∞ .Using the elementary inequality ye − y ≤ − e − y , y > φ satisfies: λφ ′ ( λ ) ≤ φ ( λ ) for any λ > . (2.9)There is a strong connection between subordinators and Bernstein functions. To bemore precise, φ is a Bernstein function such that φ (0+) = 0 (i.e. γ = 0 in (2.8)) ifand only if it is the Laplace exponent of some subordinator. If φ (0+) > φ can beunderstood as the Laplace exponent of a subordinator killed with rate φ (0+) (seeChapter 3 in [Ber96]). ARMONIC FUNCTIONS OF L´EVY PROCESSES 11
A Bernstein function φ is a complete Bernstein function if the L´evy measure in (2.8)has a completely monotone density, i.e. µ ( dt ) = µ ( t ) dt , where µ : (0 , ∞ ) → (0 , ∞ ), µ ∈ C ∞ (0 , ∞ ) and ( − n µ ( n ) ≥ n ∈ N .Let us mention some properties of complete Bernstein function (see [SSV10]). Thecomposition of two complete Bernstein function is a complete Bernstein functionand if φ is a complete Bernstein function, then φ ⋆ ( λ ) = λφ ( λ ) is also a completeBernstein function.Assume that S is a subordinator with the Laplace exponent φ and infinite L´evymeasure. Then φ ⋆ is a Bernstein function if and only if the potential measure U has a decreasing density u with respect to the Lebesgue measure. Moreover, if ν denotes the L´evy measure of the subordinator with the Laplace exponent φ ⋆ , then u ( t ) = ν ( t, ∞ ) for any t > Regular variation.
A function f : (0 , ∞ ) → (0 , ∞ ) varies regularly (at infin-ity) with index ρ ∈ R if lim λ → + ∞ f ( λx ) f ( λ ) = x ρ for every x > . If ρ = 0, then we say that f is slowly varying. Regular (slow) variation at 0 isdefined similarly.If f varies regularly with index ρ ∈ R , then there exists a slowly varying function ℓ so that f ( λ ) = λ ρ ℓ ( λ ) .Let ℓ be a slowly varying function such that L ( λ ) = R λ ℓ ( t ) t dt exists for all λ > L is slowly varying, lim λ → + ∞ L ( λ ) ℓ ( λ ) = + ∞ (2.10)and lim λ → + ∞ L ( λx ) − L ( λ ) ℓ ( λ ) = log x for every x > . (see Proposition 1.5.9 a and p. 127 in [BGT87]). Subordinators
Let S = ( S t : t ≥
0) be a subordinator with the Laplace exponent φ satisfying thefollowing conditions: (A-1) there is α ∈ [0 ,
2] such that φ ′ varies regularly at infinity with index α − λ → + ∞ φ ′ ( λx ) φ ′ ( λ ) = x α − for every x > , (A-2) the L´evy measure is infite and has a decreasing density µ , (A-3) the potential measure has a decreasing density u .When α = 2 we additionaly assume: (A-4) λ (cid:16) λφ ( λ ) (cid:17) ′ varies regularly at infinity with index − Remark . (a) The most important assumption is (A-1) (and (A-4) when α =2). Other assumptions hold for a large class of subordinators (e. g. when φ is a complete Bernstein function with infinite L´evy measure).(b) By Karamata’s theorem (see Theorem 1.5.11 in [BGT87]) it follows from(A-1) that φ varies regularly at infinity with index α . Proposition 3.2.
Let α ∈ [0 , and let S be a subordinator satisfying (A-1) and(A-2). Then µ ( t ) ≍ t − φ ′ ( t − ) , t → . Proof.
Let ε >
0. By a change of variable φ ( λ + ε ) − φ ( λ ) = Z ∞ ( e − λt − e − ( λ + ε ) t ) µ ( t ) dt = λ − Z ∞ e − t (1 − e − ελ − t ) µ ( λ − t ) dt . (3.1)Since µ is decreasing, the following holds φ ( λ + ε ) − φ ( λ ) ≥ λ − µ ( λ − ) Z e − t (1 − e − ελ − t ) dt . Now we can apply Fatou lemma to deduce φ ′ ( λ ) = lim ε → φ ( λ + ε ) − φ ( λ ) ε ≥ λ − µ ( λ − ) Z te − t dt = λ − µ ( λ − )(1 − e − ) . ARMONIC FUNCTIONS OF L´EVY PROCESSES 13
By setting λ = t − we get the upper bound µ ( t ) ≤ t − f ′ ( t − )1 − e − for every t > . (3.2)Now we prove the lower bound. Using (3.1), for any r ∈ (0 ,
1) and ε > φ ( λ + ε ) − φ ( λ ) = I + I (3.3)with I = λ − Z r e − t (1 − e − ελ − t ) µ ( λ − t ) dtI = λ − Z ∞ r e − t (1 − e − ελ − t ) µ ( λ − t ) dt . Since µ is decreasing, the dominated convergence theorem yieldslim sup ε → I ε ≤ λ − µ ( λ − r ) Z ∞ r te − t dt = ( r + 1) e − r λ − µ ( λ − r ) . (3.4)To handle I first we use the theorem of Potter (see Theorem 1.5.6 (iii) in [BGT87])to conclude that there are constants c > δ > φ ′ ( λt − ) φ ′ ( λ ) ≤ c t δ for all λ ≥ t ≤ . (3.5)Therefore, by (3.2), (3.5) and the dominated convergence theoremlim sup ε → I ε ≤ lim sup ε → − e − Z r e − t − e − ελ − t ελ − t t − φ ′ ( λt − ) dt ≤ φ ′ ( λ ) c − e − Z r t δ − e − t dt . (3.6)Combining (3.3), (3.4) and (3.6) we deduce φ ′ ( λ ) ≤ φ ′ ( λ ) c − e − Z r t δ − e − t dt + ( r + 1) e − r λ − µ ( λ − r ) . Furthermore, by choosing r ∈ (0 ,
1) so that c − e − R r t δ − e − t dt ≤ we get µ ( λ − r ) ≥ e r r + 1) λ φ ′ ( λ ) for every λ ≥ . By (A-1) we can find t ∈ (0 , r ) so that φ ′ ( rt − ) φ ′ ( t − ) ≥ r α − for any t ∈ (0 , t ). The lowerbound nwo follows: µ ( t ) ≥ r α e r r + 1) t − φ ′ ( t − ) for every t ∈ (0 , t ) . (cid:3) Remark . The precise asymptotical behavior of µ when α ∈ (0 ,
2) can be obtainedby Karamata’s Tauberian theorem. This is not the case when α = 0.Under an additional assumption t ta at µ ( t ) is monotone on (0 , T ) for some a ≥ T > , it is possible to prove the following precise asymptotics in the case α = 0 : µ ( t ) ∼ t − φ ′ ( t − ) , t → . Proposition 3.4.
Let α ∈ [0 , and let S be a subordinator satisfying (A-1) and(A-3). Then u ( t ) ∼ (cid:0) − α (cid:1) t − φ ′ ( t − ) φ ( t − ) , t → . Remark . It can be proved that u ( t ) ≍ (cid:0) − α (cid:1) t − φ ′ ( t − ) φ ( t − ) , t → . similarly as in Proposition 3.2. It is enough to note ψ ( λ + ε ) − ψ ( λ ) = Z ∞ ( e − λt − e − ( λ + ε ) t ) u ( t ) dt with ψ ( λ ) = − φ ( λ ) .The main reason why we need precise asymptotics of u is to be able to handle thecase α = 2 by duality. Proof of Proposition 3.4.
Let us first consider the case α = 0. In this case ℓ ( λ ) = λφ ′ ( λ ) varies slowly (at infinity) and thus it follows from Subsection 2.3 that φ ( λ ) = R λ ℓ ( t ) t dt also varies slowly andlim λ → + ∞ φ ( λx ) − φ ( λ ) λφ ′ ( λ ) = log x for every x > . ARMONIC FUNCTIONS OF L´EVY PROCESSES 15
This and (2.1) imply L U (cid:0) λx (cid:1) − L U (cid:0) λ (cid:1) λ φ ′ (cid:0) λ (cid:1) φ (cid:0) λ (cid:1) = φ (cid:0) λ (cid:1) − φ (cid:0) λx (cid:1) λ φ ′ (cid:0) λ (cid:1) φ (cid:0) λ (cid:1) φ (cid:0) λx (cid:1) → log x, λ → x >
0. Now we can apply de Haan’s Tauberian theorem (see 0 –version of[BGT87, Theorem 3.9.1]) to deduce U ( λx ) − U ( λ ) λ φ ′ (cid:0) λ (cid:1) φ (cid:0) λ (cid:1) → log x, λ → . If we apply de Haan’s monotone density theorem (see [BGT87, Theorem 3.6.8]) wefinally obtain u ( t ) ∼ t − φ ′ ( t − ) tφ ( t − ) , t → . The case α ∈ (0 ,
2) is already known. We give the proof for the sake of completenessand adapt the result to the formula obtained in the case α = 0 .Since L U ( λ ) = 1 φ ( λ )varies regularly at infinity with index − α , Karamata’s Tauberian theorem (see The-orem 1.7.1 in [BGT87]) implies U ([0 , t ]) ∼ (cid:0) − α (cid:1) φ ( t − ) , t → . Then by Karamata’s monotone density theorem (see Theorem 1.7.2 in [BGT87]) wededuce u ( t ) ∼ α (cid:0) − α (cid:1) tφ ( t − ) ∼ (cid:0) − α (cid:1) t − φ ′ ( t − ) φ ( t − ) , t → . (cid:3) Now we consider the case α = 2. Proposition 3.6.
Let α = 2 and let S be a subordinator satisfying (A-1), (A-2)and (A-4). Then µ ( t ) ∼ t − (cid:0) tφ ( t − ) − φ ′ ( t − ) (cid:1) , t → . Proof.
Since potential density exists by (A-3) we see that φ it follows that φ ⋆ ( λ ) = λφ ( λ ) defines the Laplace exponent of a (possibly killed) subordinator, which we de-note by T (see Section 2).Note that the subordinator T corresponds to the case of α = 0. If we denotepotential density of T by v , then v ( t ) = µ ( t, ∞ ) , t > . Proposition 3.4 yields Z ∞ t µ ( s ) ds ∼ t − ( φ ⋆ ) ′ ( t − ) φ ⋆ ( t − ) , t → . (3.7)By assumption (A-4) we know that t ( φ ⋆ ) ′ ( t − ) varies regularly at 0 with index1 and thus t t − ( φ ⋆ ) ′ ( t − ) φ ⋆ ( t − ) varies regularly at 0 with index − Z t − µ ( s − ) dss ∼ t − ( φ ⋆ ) ′ ( t − )( φ ⋆ ( t − )) , t → . This gives ( r = t − ): Z r µ ( s − ) dss ∼ r ( φ ⋆ ) ′ ( r ) φ ⋆ ( r ) , r → ∞ . Note that the right-hand side is now regularly varying at infinity with index 1 andthus by Karamata’s monotone density theorem (see Theorem 1.7.2 in [BGT87]) wededuce µ ( r − ) r ∼ r ( φ ⋆ ) ′ ( r ) φ ⋆ ( r ) .r → ∞ , Going back ( t = r − ) we conclude µ ( t ) ∼ t − ( φ ⋆ ) ′ ( t − ) φ ⋆ ( t − ) , t → . (cid:3) Proposition 3.7.
Let α = 2 and let S be a subordinator satisfying (A-1) and (A-3).Then the following is true u ( t ) ∼ φ ′ ( t − ) ∼ tφ ( t − ) , t → . ARMONIC FUNCTIONS OF L´EVY PROCESSES 17
Proof.
By (2.1) we get L U ( λ ) = 1 φ ( λ ) ∼ λφ ′ ( λ ) , λ → + ∞ and thus by Karamata’s Tauberian theorem (see Theorem 1.7.1 in [BGT87]) itfollows that U ([0 , t ]) ∼ tφ ′ ( t − ) , t → λ λφ ′ ( λ ) varies regualrly at infinity with index 1. By applying Karamata’smonotone density (see Theorem 1.7.2 in [BGT87]) theorem we deduce u ( t ) ∼ φ ′ ( t − ) , t → . (cid:3) L´evy density and Green function
Let S be a subordinator as in Section 3 and let X be the corresponding subordinateBrownian motion in R d with d ≥
3. Our aim is to establish asymptotical behaviorof the L´evy density and Green function of X .Recall that the L´evy density of X is of the form j ( | x | ), where j is given by (2.2). Theorem 4.1.
Assume that S satisfies (A-1) with some α ∈ [0 , and (A-2). If α ∈ [0 , , then j ( r ) ≍ r − d − φ ′ ( r − ) , r → . If α = 2 and (A-4) holds, then j ( r ) ≍ r − d − (cid:0) r φ ( r − ) − φ ′ ( r − ) (cid:1) , r → . Proof.
This result follows directly from Proposition 3.2 and Proposition 3.6 togetherwith Lemma A.1, where a = , b = 1 + α .For α ∈ [0 ,
2) the slowly varying function is given by ℓ ( t ) = t α − φ ′ ( t − ). When α = 2 we take ℓ ( t ) = tφ ( t − ) − φ ′ ( t − ) = t ( φ ⋆ ( t − )) ′ φ ⋆ ( t − ) , which varies slowly, since φ ⋆ ( λ ) = λφ ( λ ) is slowly varying by (A-1) and Karamata’sTheorem (see Theorem 1.5.11 in [BGT87]) and has a derivative that varies regularlywith index 1 by (A-4). (cid:3) The Green function of X is of the form G ( x, y ) = g ( | y − x | ), where g is given by(2.4). Theorem 4.2.
Assume that S satisfies (A-1) with some α ∈ [0 , and (A-3). Let d ≥ . If α ∈ [0 , , then g ( r ) ≍ r − d − φ ′ ( r − ) φ ( r − ) , r → . If α = 2 , then g ( r ) ≍ r − d +2 φ ′ ( r − ) ≍ r − d φ ( r − ) , r → . Proof.
We use Lemma A.1 with a = , b = 1 − α , Proposition 3.4 and Proposition3.7.When α ∈ [0 ,
2) we define ℓ ( t ) = t − φ ′ ( t − ) φ ( t − ) which varies slowly at 0 .In the case α = 2, we let ℓ ( t ) = φ ′ ( t − ) or ℓ ( t ) = tφ ( t − ) which both vary slowly at0. (cid:3) Using asymptotical results from this section we can now prove the proposition thatgives a counterexample for the estimate of the Krylov and Safonov.
Proof of Proposition 1.2.
By (2.5), P x ( X τ B r ∈ B r (0) \ B r (0)) = Z B r (0) \ B r (0) Z B r (0) G B r ( x, y ) j ( | z − y | ) dy = I + I . ARMONIC FUNCTIONS OF L´EVY PROCESSES 19
Using Theorems 4.1 and 4.2 it follows that I = Z B r (0) \ B r (0) Z B r (0) G B r ( x, y ) j ( | z − y | ) dy dz ≤ j ( r ) | B r (0) \ B r (0) | Z B r (0) g ( | y | ) dy dz ≤ c r − φ ′ ( r − ) φ ( r − ) . On the other hand, I = Z B r (0) \ B r (0) Z B r (0) \ B r (0) G B r ( x, y ) j ( | z − y | ) dy dz ≤ g ( r ) Z B r (0) \ B r (0) Z B r ( z ) j ( | y | ) dy dz . (4.1)To estimate the inner integral, note that B r ( z ) ⊂ B (0) \ B | z |− r (0) for any z ∈ B r (0) \ B r (0) and so, by Theorem 4.1, Z B r ( z ) j ( | y | ) dy ≤ c Z | z |− r s − φ ′ ( s − ) ds ≤ c φ (( | z | − r ) − ) . (4.2)Thus, by Theorem 4.2 and (4.1) I ≤ c r − d − φ ′ ( r − ) φ ( r − ) Z r r φ (( t − r ) − ) t d − dt ≤ c r − φ ′ ( r − ) φ ( r − ) Z r φ ( s − ) ds ≤ c r − φ ′ ( r − ) φ ( r − ) . In the last equality we have used Karamata’s theorem (see Theorem 1.5.11 in[BGT87]) and the fact that α ∈ [0 ,
1) . (cid:3) Difference estimates
Let X be the stochastic process in R d as in Section 4 and assume that d ≥
3. Inparticular X is transient.In this section we prove the difference estimates of the Green function and thePoisson kernel.Although we are slightly abusing notation, we set G ( x ) := G (0 , x ) = g ( | x | ). Proposition 5.1.
There is a constant c > such that for every r ∈ (0 , | G ( x ) − G ( y ) | ≤ cg ( r ) (cid:16) ∧ | x − y | r (cid:17) for all x, y B r (0) . Proof.
Assume first that | x − y | < r . By the mean value theorem it follows that forany t > ϑ = ϑ ( x, y, t ) ∈ [0 ,
1] such that (cid:12)(cid:12)(cid:12)(cid:12) e − | x | t − e − | y | t (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x + ϑ ( y − x ) | t e − | x + ϑ ( y − x ) | t | x − y |≤ | x − y |√ t e − | x + ϑ ( y − x ) | t , where in the last line the following elementary inequality was used se − s < e − s , s > . Then | x + ϑ ( y − x ) | ≥ | x | − ϑ | y − x | ≥ r implies (cid:12)(cid:12)(cid:12)(cid:12) e − | x | t − e − | y | t (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x − y |√ t e − r t . (5.1)By (5.1) | G ( x ) − G ( y ) | ≤ (4 π ) − d/ Z ∞ t − d/ (cid:12)(cid:12)(cid:12)(cid:12) e − | x | t − e − | y | t (cid:12)(cid:12)(cid:12)(cid:12) u ( t ) dt ≤ π ) − d/ | x − y | Z ∞ t − d/ − / e − r t u ( t ) dt . Since u is non-increasing and varies regularly at 0 with index α −
1, by Lemma A.1we see that there is a constant c > Z ∞ t − d/ − / e − r t u ( t ) dt ≤ c r − d +1 u ( r ) for every r ∈ (0 , . Theorem 4.2 yields | G ( x ) − G ( y ) | ≤ c g ( r ) | x − y | r . ARMONIC FUNCTIONS OF L´EVY PROCESSES 21
When | x − y | ≥ r , | G ( x ) − G ( y ) | ≤ G ( x ) + G ( y ) ≤ g ( r )since | x | , | y | ≥ r . (cid:3) Proposition 5.2.
There is a constant c > such that for all R ∈ (0 , , r ∈ (0 , R ] , y ∈ B R (0) and x , x ∈ B R (0) \ B r ( y ) | G B R (0) ( x , y ) − G B R (0) ( x , y ) | ≤ cg ( r ) (cid:16) ∧ | x − x | r (cid:17) . Proof.
By symmetry of the Green function, G B R (0) ( x i , y ) = G B R (0) ( y, x i ) = G ( x i − y ) − E y [ G ( X τ BR (0) − x i )]= G ( x i − y ) − E y [ G ( X τ BR (0) − x i )] , for i ∈ { , } . Now the result follows from Proposition 5.1. (cid:3) Proposition 5.3.
There is a constant c > such that for any r ∈ (0 , and x, y ∈ B r (0) : (i) if z ∈ B r (0) \ B r (0) , then (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) ≤ c | z | − d φ (( | z | − r ) − ) φ ( | x − y | − ) ;(ii) if z B r (0) , then (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) ≤ c j (cid:16) | z | (cid:17) φ ( | x − y | − ) . Proof.
In the estimate (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) ≤ Z B r (0) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dv we split the integral into three parts: I = Z B | x − y | ( x ) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dvI = Z B r ( x ) \ B | x − y | ( x ) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dvI = Z B r (0) \ B r ( x ) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dv . For the first part we obtain I ≤ Z B | x − y | ( x ) G B r (0) ( x, v ) j ( | z − v | ) dv + Z B | x − y | ( y ) G B r (0) ( y, v ) j ( | z − v | ) dv ≤ j (cid:16) | z | (cid:17) Z B | x − y | (0) G ( v ) dv ≤ c j ( | z | ) φ ( | x − y | − ) , (5.2)for any z B r (0). We have used Theorem 4.2 to get the last inequality in (5.2).In order to estimate I we split the integral in the following way. We let N = j log r | x − y | log 2 k and write I ≤ N X n =1 Z B n +1 | x − y | ( x ) \ B n | x − y | ( x ) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dv . Now, for each n ∈ { , . . . , N } we can apply Proposition 5.2 (with the correspondingradii (2 n − | x − y | and r ) to get Z B n +1 | x − y | ( x ) \ B n | x − y | ( x ) (cid:12)(cid:12) G B r (0) ( x, v ) − G B r (0) ( y, v ) (cid:12)(cid:12) j ( | z − v | ) dv ≤ c g ((2 n − | x − y | )2 n − Z B n +1 | x − y | ( x ) j ( | z − v | ) dv . By Theorem 4.2 g ((2 n − | x − y | ) g ( | x − y | ) ≤ c η ((2 n − | x − y | ) η ( | x − y | ) for all n ∈ { , , . . . , N } , with η ( r ) = r − d − φ ′ ( r − ) φ ( r − ) .Noting that η varies regularly at zero with index α − d <
0, the uniform convergencetheorem for regularly varying functions (see Theorem 1.5.2 in [BGT87]) gives η ((2 n − | x − y | ) η ( | x − y | ) ≤ c (2 n − α − d for all n ∈ N and | x − y | ≤ . ARMONIC FUNCTIONS OF L´EVY PROCESSES 23
By Theorem 4.2 and (2.9) g ( | x − y | ) ≤ c φ ( | x − y | − ) and so I ≤ c N X n =1 (2 n − α − d − g ( | x − y | )(2 n +1 | x − y | ) d j (cid:16) | z | (cid:17) ≤ c j (cid:16) | z | (cid:17) φ ( | x − y | − ) N X n =1 ( α − n ≤ c − α − j (cid:16) | z | (cid:17) φ ( | x − y | − ) for every z B r (0) . It remains to estimate I . Applying Theorem 5.2 we get I ≤ c g ( r ) | x − y | r Z B r ( z ) j ( | v | ) dv ≤ c | x − y | φ ( | x − y | − ) rφ ( r − ) r − d φ ( | x − y | − ) Z B r ( z ) j ( | v | ) dv ≤ c r − d φ ( | x − y | − ) Z B r ( z ) j ( | v | ) dv . (5.3)In the last inequality we have used the theorem of Potter (cf. [BGT87, Theorem1.5.6 (iii)]) to conclude that for δ < − α there is a constant A δ > | x − y | φ ( | x − y | − ) rφ ( r − ) ≤ A δ (cid:18) | x − y | r (cid:19) − α − δ ≤ A δ , since r rφ ( r − ) varies regularly at zero with index 1 − α .Since j ( | v | ) ≥ j (cid:16) | z | (cid:17) for all v ∈ B r ( z ) and z ∈ B r (0) c it follows from (5.3) that I ≤ c j (cid:16) | z | (cid:17) φ ( | x − y | − ) , On the other hand, for z ∈ B r (0) \ B r (0) we deduce from B r ( z ) ⊂ B (0) \ B | z |− r (0)(similarly as in (4.2)) that Z B r ( z ) j ( | v | ) dv ≤ c φ (cid:0) ( | z | − r ) − (cid:1) . By (5.3) I ≤ c | z | − d φ (( | z | − r ) − ) φ ( | x − y | − ) for all z ∈ B r (0) \ B r (0) . (cid:3) Regularity of harmonic functions
Recall that (2.7) gives the representation for any bounded function f : R d → R thatis harmonic in B r ( x ): f ( x ) = E x h f (cid:16) X τ Br ( x (cid:17)i = Z B r ( x ) c K B r ( x ) ( x, z ) f ( z ) dz, x ∈ B r ( x ) . (6.1) Proof of Theorem 1.1.
By (6.1) | f ( x ) − f ( y ) | ≤ k f k ∞ Z B r (0) c (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) dz . (6.2)It remains to estimate the integral in (6.2), which we split in the following way I = Z B r (0) \ B r (0) (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) dzI = Z B (0) \ B r (0) (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) dzI = Z B (0) c (cid:12)(cid:12) K B r (0) ( x, z ) − K B r (0) ( y, z ) (cid:12)(cid:12) dz In order to estimate I we use Proposition 5.3 (i). More precisely, I ≤ c φ ( | x − y | − ) Z B r (0) \ B r (0) | z | − d φ (cid:0) ( | z | − r ) − (cid:1) dz = c φ ( | x − y | − ) Z r r t − φ (cid:0) ( t − r ) − (cid:1) dt ≤ c φ ( | x − y | − ) (2 r ) − Z r φ (cid:0) t − (cid:1) dt ≤ c φ ( | x − y | − ) φ ( r − ) , where in the last inequality we have used Karamata’s theorem (see the 0-version ofTheorem 1.5.11 in [BGT87]). ARMONIC FUNCTIONS OF L´EVY PROCESSES 25
We estimate I and I with the help of Proposition 5.3 (ii). Since the L´evy measureis finite away from the origin, I ≤ c φ ( | x − y | − ) Z B (0) c j (cid:16) | z | (cid:17) dz ≤ c φ ( | x − y | − ) . Also, I ≤ c φ ( | x − y | − ) Z B (0) \ B r (0) j (cid:16) | z | (cid:17) dz ≤ c φ ( r − ) φ ( | x − y | − ) , where in the last inequality we have used Theorem 4.1. (cid:3) Examples
In this section is to illustrate our results by some examples.7.1. (Iterated) Geometric stable processes.
This class of examples belongs tothe case of α = 0.Let β ∈ (0 , { φ n : (0 , ∞ ) → (0 , ∞ ) : n ∈ N } recursively by φ ( λ ) = log(1 + λ β/ ) , λ > φ n +1 = φ ◦ φ n , n ∈ N . The function φ is a complete Bernsetin function. Since complete Bernstein functionsare closed under operation of composition, φ n belongs to this class for every n ∈ N .Let S n be a subordinator with the Laplace exponent φ n . S is known as the geomet-ric β -stable subordinator. We call S n the iterated geometric β -stable subordinator.The corresponding subordinate Brownian motions X n will be called (iterated) geo-metric β -stable processes.As already remarked in [ˇSSV06], these processes show quite different behavior com-pared to the one of stable processes. Our contribution to this class of examples isthat now we can obtain behavior of the L´evy density as a special case of Theorem4.1(even for iterated geometric stable processes). The L´evy density of X n is comparable to1 | x | d · n − Y k =1 k ( | x | − ) as | x | → , which is almost integrable. We can say that (intially) this process jumps slower thanany stable processes.This can be also seen from the behavior of the Green function: G ( x, y ) ≍ | x − y | d log n ( | x − y | − ) · n − Y k =1 k ( | x − y | − ) as | x − y | → . As a consequence, E τ B r (0) ≍ n ( r − ) as r → X n needs (on average)more time to exit ball B r (0) than any stable process or Brownian motion.Theorem 1.1 implies the following a-priori local regularity estimates of harmonicfunctions: | f ( x ) − f ( y ) | ≤ c k f k ∞ log n ( r − ) 1log n ( | x − y | − ) for all x, y ∈ B r (0)and any bounded function f which is harmonic in B r (0) .This tells us that the modulus of continuity is bounded by a logarithmic term. It isstill an open problem whether these harmonic functions satify a-priori local H¨oldercontinuity estimates.7.2. Conjugates of (iterated) geometric stable processes.
This class of ex-amples corresponds to the case α = 2.Let ψ n ( λ ) = λφ n ( λ ) , where φ n are as in Subsection 7.1.Since φ n are complete Bernstein functions, ψ n are also complete Bernstein func-tions. Therefore, there exist (killed) subordinators T n with the Laplace exponent ψ n . Killing will not affect the behavior of the L´evy and potential densities of T n near zero.In this case the L´evy density of the corresponding subordinate Brownian motion Y n behaves near the origin as1 | x | d +2 log n ( | x | − ) · n − Y k =1 k ( | x | − ) as | x | → . ARMONIC FUNCTIONS OF L´EVY PROCESSES 27
Note that the integrability conditions of the L´evy measure are barely satisfied inthis case.Comparing this behavior to the behavior of the small jumps of the α -stable process,we see that small jumps of Y n are more intensive.Another interesting feature of this process is the following behavior of the Greenfunction: G ( x, y ) ≍ | x − y | − d log n ( | x − y | − ) as | x − y | → . In this sense the process Y n is ’between’ stable processes and Brownian motion,since their Green functions are given by G ( α ) = c α | x − y | α − d and G (2) = c α | x − y | − d . Appendix A. Asymptotical properties
In the appendix we prove a technical lemma which is used throughout the paper.
Lemma A.1.
Let w : (0 , ∞ ) → (0 , ∞ ) be a decreasing function satisfying w ( t ) ≍ t − b ℓ ( t ) , t → , for a function ℓ : (0 , ∞ ) → (0 , ∞ ) that varies slowly at and b ≥ .If p > and a > , then I ( r ) = Z ∞ t − p e − art w ( t ) dt, r > , satisfies I ( r ) ≍ a − p − b +1 r − p +1 w ( r ) , r → . Proof.
Change variables yields I ( r ) = ( ar ) − p +1 Z ∞ e − t t p − w (cid:16) art (cid:17) dt (A.1)By assumptions, there are constants c , c > r > c a − b ≤ w ( ar ) w ( r ) ≤ c a − b for every r ∈ (0 , r ) . (A.2) Let us first prove the upper bound. Using the fact that w is decreasing, by (A.1)and (A.2) we get I ( r ) ≤ ( ar ) − p +1 Z e − t t p − w ( ar ) dt + ( ar ) − p +1 Z ∞ e − t t p − w (cid:16) art (cid:17) dt ≤ c ( ar ) − p +1 a − b w ( r ) (cid:20)Z e − t t p − dt + Z ∞ e − t t p + b − dt (cid:21) ≤ c ′ a − p − b +1 r − p +1 w ( r )for every r ∈ (0 , r ) .The lower bound follows similarly: I ( r ) ≥ ( ar ) − p +1 Z ∞ e − t t p − w ( ar ) dt ≥ c ( ar ) − p +1 a − b w ( r ) Z ∞ e − t t p − dt = c ′ a − p − b +1 r − p +1 w ( r )for every r ∈ (0 , r ) . (cid:3) References [Ber96] J. Bertoin,
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Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Germany
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