Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators

0.3 MB·English

by  Zhen-Qing Chen

Boundary Harnack principle and gradient estimates for
fractional Laplacian perturbed by non-local operators
6 Zhen-Qing Chen∗, Yan-Xia Ren† and Ting Yang‡§
1
0
2

r Abstract
a
M Suppose d ≥ 2 and 0 < β < α < 2. We consider the non-local operator Lb = ∆α/2 +Sb,
where
4 b(x,z)
2 Sbf(x):= limA(d,−β) (f(x+z)−f(x)) dy.
ε→0 Z|z|>ε |z|d+β

] Here b(x,z)is a bounded measurablefunction onRd×Rd that is symmetric in z,andA(d,−β)
R
isanormalizingconstantsothatwhenb(x,z)≡1,Sb becomesthefractionalLaplacian∆β/2 :=
P
. −(−∆)β/2. In other words,
h
t
a Lbf(x):= limA(d,−β) (f(x+z)−f(x))jb(x,z)dz,
m ε→0 Z|z|>ε
[
where jb(x,z) := A(d,−α)|z|−(d+α) + A(d,−β)b(x,z)|z|−(d+β). It is recently established in

2 Chen and Wang [12] that, when jb(x,z) ≥ 0 on Rd×Rd, there is a conservative Feller process
v
3 Xb havingLb as its infinitesimalgenerator. Inthis paper weestablish,under certainconditions
2 on b, a uniform boundary Harnack principle for harmonic functions of Xb (or equivalently, of
0
Lb) in any κ-fat open set. We further establish uniform gradient estimates for non-negative
2
0 harmonic functions of Xb in open sets.
.
1
0
AMS 2010 Mathematics Subject Classification. Primary 60J45, Secondary 31B05, 31B25.
5
1
Keywords and Phrases. Harmonic function, boundary Harnack principle, gradient estimate,
:
v
non-local operator, Green function, Poisson kernel
i
X
r
a 1 Introduction
Let d ≥ 2, 0 < β < α < 2, and b(x,z) be a bounded measurable function on Rd × Rd with
b(x,z) = b(x,−z) for x,z ∈ Rd. Consider the non-local operator Lb = ∆α/2+Sb, where
b(x,z)
Sbf(x):= limA(d,−β) (f(x+z)−f(x)) dz.
ε→0 Z|z|>ε |z|d+β
∗Department of Mathematics, University of Washington, Seattle, WA 98195, USA.E-mail: [email protected]
†LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing, 100871,
P.R. China. E-mail: [email protected]
‡Corresponding author. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081,
P.R. China. Email: [email protected]
§Beijing Key Laboratory on MCAACI, Beijing 100081, P.R. China.
1
Here A(d,−β) is a normalizing constant so that when b(x,z) ≡ 1, Sb becomes the fractional
Laplacian ∆β/2 := −(−∆)β/2; in other words, A(d,−β) = β2β−1π−d/2Γ((d + β)/2)/Γ(1 −β/2).
Thus Lb can be expressed as
Lbf(x)= lim (f(x+z)−f(x))jb(x,z)dz
ε→0Z|z|>ε
where
A(d,−α) A(d,−β)b(x,z)
jb(x,z) = + .
|z|d+α |z|d+β
Note that since b(x,z) is symmetric in z, for f ∈ C2(Rd),
b
Lbf(x)= f(x+z)−f(x)−∇f(x)·z1 jb(x,z)dz.
{|z|≤1}
Rd
Z
(cid:0) (cid:1)
Recently, Lb, the fractional Laplacian perturbed by a lower order non-local operator Sb, and its
fundamental solution have been studied in Chen and Wang [12]. It is established there that if for
every x ∈ Rd, jb(x,z) ≥ 0 (that is, b(x,z) ≥ −A(d,−α) |z|β−α) for a.e. z ∈ Rd, then Lb has a unique
A(d,−β)
jointly continuous fundamental solution pb(t,x,y), which uniquely determines a conservative Feller
process Xb on the canonical Skorokhod space D([0,+∞),Rd) such that
E f(Xb) = f(y)pb(t,x,y)dy, x ∈ Rd.
x t
Rd
h i Z
for every bounded measurable function f on Rd. The Feller process Xb is typically non-symmetric
and it has a L´evy system (Jb(x,y)dy,t) (see [12, Theorem 1.3]), where
Jb(x,y) := jb(x,y−x). (1.1)
When b takes constant value ε > 0, Xε has the same distribution as the L´evy process Y +ε1/βZ,
where Y and Z are rotationally symmetric α- and β-stable processes on Rd that are independent
of each other. Moreover, two-sided heat kernel estimates for Lb have been obtained in Chen and
Wang [12], while two-sided Dirichlet kernel estimates in C1,1 open sets have recently been obtained
in Chen and Yang [13]. We emphasize that the lower order perturbations Sb considered in [12, 13]
|b(x,z)|
and in this paper are of high intensity, in the sense that dz may be identically infinite.
Rd |z|d+β
This is the case, for example, when b(x,z) is bounded away from 0. Such high intensity non-local
R
perturbation is in stark contrast to the Feynman-Kac type non-local perturbation considered in
[5, 9, 11, 15, 16], where the kernels used in the non-local perturbation are typically integrable.
Under high intensity non-local perturbation, the potential theory for the perturbed generator Lb
can be different from that of ∆α/2. For example, the heat kernel of Lb may not be comparable to
that of ∆α/2; see [12] for details.
In this paper, we investigate boundary Harnack principle and gradient estimates for non-
negative harmonic functions of Lb in open sets. Boundary Harnack principle (BHP) asserts that
non-negative harmonicfunctions that vanish in an exterior partof a neighborhoodat the boundary
decay at the same rate. It is an important property in analysis and in probability theory on har-
monic functions. We refer the reader to the introduction of [3, 17] for a brief account on the history
2
of BHP that started with Brownian motion and then extended to subordinate Brownian motions
and to certain pure jump strong Markov processes. Since Lb is typically state-dependent and its
dual operator may not be Markovian, the BHP results in [3, 17] are not applicable to harmonic
functions of Lb. In this paper, we establish uniform boundary Harnack principle on κ-fat open sets
for non-negative harmonic functions of Lb by estimating Poisson kernels of Lb in small balls.
Gradient estimates for harmonic functions of elliptic operators and on manifolds have been
studied extensively in literature, including the celebrated Li-Yau inequality. See [14] and the ref-
erences therein and for a coupling argument. See also [21] for gradient estimates of the transition
semigroups of some L´evy processes using coupling method. Gradient estimates for harmonic func-
tions fornon-local operators arequite recently. In[4], agradient estimate for harmonicfunctions of
symmetric stable processes is obtained. Gradient estimates for harmonic functions of mixed stable
processes were derived in [22]. It has recently been extended to a class of isotropic unimodal L´evy
process in [20]. For gradient estimate for harmonic functions of the Schro¨dinger operator ∆α/2+q,
see [4] for α ∈ (1,2) and [19] for α ∈ (0,1]. The second main result of this paper is to establish
gradientestimates forpositiveharmonicfunctionsofLb. Asfarasweknow,thisisthefirstgradient
estimate result for non-L´evy non-local operators.
We now describe our main results in details. In this paper, we use “:=” as a way of definition.
For a,b ∈ R, a∧b := min{a,b} and a∨b := max{a,b}. Let |x−y| denote the Euclidean distance
between x and y, and B(x,r) the open ball centered at x with radius r > 0. For any two positive
c
functions f and g, f . g means that there is a positive constant c such that f ≤ cg on their
c
common domain of definition, and f ≍ g means that c−1g ≤ f ≤ cg. We also write “.” and “≍”
if c is unimportant or understood. If D ⊂ Rd is an open set, for every x,y ∈ D, define
δ (x) := dist(x,∂D) and r (x,y) := δ (x)+δ (y)+|x−y|. (1.2)
D D D D
It is easy to see that
r (x,y) ≍ δ (x)+|x−y| ≍ δ (y)+|x−y|. (1.3)
D D D
Denote by τb := inf{t > 0 : Xb 6∈ D}, the exit time from D by Xb. When there is no danger of
D t
confusion, we will drop the superscript b and simply write τ for τb.
D D
Definition 1.1. A function f defined on Rd is said to be harmonic in an open set D with respect
to Xb if it has the mean-value property: for every bounded open set U ⊂ D with U ⊂ D,
f(x)= E f(Xb ) for x ∈ U. (1.4)
x τU
h i
It is said to be regular harmonic in D if (1.4) holds for U = D.
Denote by ∂ a cemetery point that is added to D as an isolated point. We use the convention
that Xb := ∂ and any function f is extended to the cemetery point ∂ by setting f(∂) = 0. So
∞
E f(Xb ) should beunderstoodas E f(Xb ) :τ < +∞ . In Definition 1.1, we always assume
x τD x τD D
implicitly that the expectation in (1.4) is absolutely convergent.
(cid:2) (cid:3) (cid:2) (cid:3)
3
Assumption 1 Suppose M ,M ≥ 1. b(x,z) is a bounded function on Rd×Rd satisfying
1 2
kbk ≤ M and b(x,z) = b(x,−z) for x,z ∈ Rd, (1.5)
∞ 1
and there exists a positive constant ε ∈ [0,1] such that for every x,y ∈ Rd,
0
M−1Jε0(x,y) ≤ Jb(x,y) ≤ M Jε0(x,y). (1.6)
2 2
Here Jε0(x,y) = A(d,−α)|x −y|−d−α +ε A(d,−β)|x −y|−d−β. Since Jε0(x,y) depends only on
0
|x−y|, we also write Jε0(|x−y|) for Jε0(x,y).
Definition 1.2. Let κ ∈ (0,1). An open set D ⊂ Rd is said to be κ-fat if for every z ∈ ∂D and
r ∈ (0,1], there is some point x ∈ D so that B(x,κr) ⊂ D∩B(z,r).
The following is the first main result of this paper.
Theorem 1.3 (Uniform boundary Harnack inequality). Suppose Assumption 1 holds and D is
a κ-fat open set in Rd with κ ∈ (0,1). There exist constants r = r (d,α,β,M ) ∈ (0,1] and
1 1 1
C = C (d,α,β,κ,M ,M ) ≥ 1 such that for every z ∈ ∂D and r ∈ (0,r /2], and all non-
1 1 1 2 0 1
negative functions u,v that are regular harmonic in D∩B(z ,2r) with respect to Xb and vanish in
0
Dc∩B(z ,2r), we have
0
u(x) u(y)
≤ C for x,y ∈ D∩B(z ,r).
1 0
v(x) v(y)
We call the above property uniform boundary Harnack principle because the constants r and
1
C in the above theorem are independent of ε ∈ [0,1] appeared in condition (1.6). We next study
1 0
the gradient estimates for non-negative harmonic functions in open sets. We write ∂ or ∂ for ∂
xi i ∂xi
and ∇ for (∂ ,··· ,∂ ).
x1 xd
Theorem 1.4. Let D be an arbitrary open set in Rd. Under Assumption 1, there is a constant
C = C (d,α,β,M ,M ) > 0 such that for any non-negative function f in Rd which is harmonic
2 2 1 2
in D with respect to Xb, ∇f(x) exists for every x ∈ D, and we have
f(x)
|∇f(x)|≤ C for x ∈ D.
2
1∧δ (x)
D
For x = (x ,··· ,x )∈ Rd and 1 ≤ i ≤ d, we write x˜i for (x ,··· ,x ,x ,··· ,x ) ∈Rd−1.
1 d 1 i−1 i+1 d
Assumption 2. Suppose there is i ∈{1,··· ,d} so that for every x ∈Rd,
b(x,z) = ϕ(x˜i)ψ(|z|) a.e. z ∈ Rd,
where ϕ : Rd−1 → R is a non-negative measurable function, and ψ : R → R is a measurable
+
function such that
ψ(r)
is non-increasing in r > 0. (1.7)
rd+β
4
Theorem 1.5. Suppose Assumption 1 and Assumption 2 hold. Let D = x ∈ Rd : x > Γ(x˜i) be
i
a special Lipschitz domain, where Γ : Rd−1 → R is a Lipschitz function with Lipschitz constant λ
(cid:8) (cid:9) 0
(that is, |Γ(x˜i)−Γ(y˜i)| ≤ λ |x˜i −y˜i| for every x˜i,y˜i ∈ Rd−1). Then there are positive constants
0
R = R (d,α,β,λ ,M ,M ) and C = C (d,α,β,λ ,M ,M ) ≥ 1 such that for every r ∈ (0,R ],
1 1 0 1 2 3 3 0 1 2 1
there isaconstant η = η (d,α,β,λ ,M ,M ,r)∈ (0,r/2) sothat foreveryz ∈ ∂D and everynon-
1 1 0 1 2 0
negative functionf that is harmonic inD∩B(z ,r) with respect to Xb and vanishes inDc∩B(z ,r),
0 0
f(x) f(x)
C−1 ≤ |∇f(x)|≤ C for x ∈ D∩B(z ,η ). (1.8)
3 δ (x) 3δ (x) 0 1
D D
Obviously Assumption 2 is implied by
Assumption 3. There exists a measurable function ψ : R → R satisfying (1.7) such that for
+
every x ∈ Rd,
b(x,z) = ψ(|z|) a.e. z ∈ Rd.
Definition 1.6. AnopensetD ⊂ Rd issaidtobeLipschitzifforeveryz ∈ ∂D,thereisaLipschitz
0
function Γ : Rd−1 → R, an orthonormal coordinate system CS and a constant R > 0 such
z0 z0 z0
that if y = (y ,··· ,y ,y ) in CS coordinates, then
1 d−1 d z0
D∩B(z ,R ) = {y : y > Γ (y ,··· ,y )}∩B(z ,R ).
0 z0 d z0 1 d−1 0 z0
If there exist positive constants R and λ so that R can be taken to be R for all z ∈ ∂D
0 0 z0 0 0
and the Lipschitz constants of Γ are not greater than λ , we call D a Lipschitz open set with
z0 0
characteristics (λ ,R ).
0 0
Clearly, if D is a Lipschitz open set with characteristics (λ ,R ), then it is κ-fat for some
0 0
κ = κ(λ ,R )∈ (0,1). The following theorem follows directly from Theorem 1.5.
0 0
Theorem 1.7. Let D be a Lipschitz open set in Rd with characteristics (λ ,R ). Under Assump-
0 0
tions 1 and 3, there are positive constants R = R (d,α,β,λ ,R ,M ,M ) and
2 2 0 0 1 2
C = C (d,α,β,λ ,R ,M ,M ) ≥ 1 such that for every r ∈ (0,R ], there is a constant η =
4 4 0 0 1 2 2 2
η (d,α,β,λ ,R ,M ,M ,r) ∈ (0,r/2) so that for every z ∈ ∂D and every non-negative function
2 0 0 1 2 0
f that is harmonic in D∩B(z ,r) with respect to Xb and vanishes in Dc∩B(z ,r),
0 0
f(x) f(x)
C−1 ≤ |∇f(x)|≤ C for x ∈ D∩B(z ,η ).
4 δ (x) 4δ (x) 0 2
D D
Results inTheorem1.4, Theorem 1.5andTheorem1.7 can becalled uniformgradientestimates
because the constants C , 2≤ k ≤ 4, and η , 1 ≤ i ≤ 2, are independent of ε of (1.6).
k i 0
Therestofthepaperisorganizedasfollows. PreliminaryresultsonGreenfunctionsandPoisson
kernels are presented in Section 2. The proof of the uniform boundary Harnack principle is given
in Section 3. Section 4 is devoted to the proof of Theorem 1.4, while the proof of Theorem 1.5
is given in Section 5. In this paper, we use capital letters C ,C ,··· to denote constants in the
1 2
statements of results. The lower case constants c ,c ,··· , will denote the generic constants used
1 2
in proofs, whose exact values are not important, and can change from one appearance to another.
We use e to denote the unit vector along the positive direct of x -axis.
k k
5
2 Preliminaries
Recall the L´evy system (Jb(x,y)dy,t) from (1.1), which describes the jumps of Xb: for any non-
negative measurable function f on R ×Rd ×Rd with f(s,y,y) = 0 for all y ∈ Rd, x ∈ Rd and
+
stopping time T (with respect to the filtration of Xb),
T
E f(s,Xb ,Xb) = E f(s,Xb,y)Jb(Xb,y)dyds . (2.1)
xs≤T s− s  x(cid:20)Z0 ZRd s s (cid:21)
X
 
Suppose D is a Greenian open set of Xb. Let Gb (x,y) denote the Green function of D, that is,
D
τD
f(y)Gb (x,y)dy = E f(Xb)ds
D x s
ZD (cid:20)Z0 (cid:21)
for every bounded measurable function f on D and x ∈ D. It follows from (2.1) that for every
bounded open set D in Rd, every f ≥ 0, and x ∈ D,
E f(Xb ) :Xb 6= Xb = f(z) Gb (x,y)Jb(y,z)dy dz. (2.2)
xh τD τD− τDi ZD¯c (cid:18)ZD D (cid:19)
Define
Kb (x,z) := Gb (x,y)Jb(y,z)dy for (x,z) ∈D×D¯c. (2.3)
D D
ZD
We call Kb (x,z) the Poisson kernel of Xb on D. Then (2.2) can be written as
D
E f(Xb ) :Xb 6= Xb = f(z)Kb (x,z)dz.
xh τD τD− τDi ZD¯c D
For any λ > 0, define
b (x,z) := λβ−αb(λ−1x,λ−1z) for x,z ∈ Rd.
λ
It is not hard to show that
Jbλ(x,y) = λ−(d+α)Jb(λ−1x,λ−1y) for x,y ∈Rd. (2.4)
and
{λXb ;t ≥ 0} has the same distribution as {Xbλ;t ≥ 0}.
λ−αt t
So for any λ > 0, we have the following scaling properties:
Gb (x,y) = λd−αGbλ (λx,λy) for x,y ∈ D, (2.5)
D λD
Kb (x,z) = λdKbλ (λx,λz) for x ∈ D, z ∈ D¯c. (2.6)
D λD
If u is harmonic in D with respect to Xb, then for any λ > 0, v(x) := u(x/λ) is harmonic in λD
with respect to Xbλ.
When b(x,z) ≡ 0, X0 is simply an isotropic symmetric α-stable process on Rd, which we will
denote as X. We will also write J for J0. It is known that if d> α, the process X is transient and
its Green function is given by
Γ(d/2)
G(x,y) = |x−y|α−d for x,y ∈ Rd. (2.7)
2απd/2Γ(α/2)2
6
It is shown in Blumenthal et al. [1] that the Green function of X in a ball B(0,r) is given by
Γ(d/2) z
G (x,y) = (u+1)−d/2uα/2−1du|x−y|α−d for x,y ∈ B(0,r), (2.8)
B(0,r) 2απd/2Γ(α/2)2
Z0
wherez = (r2−|x|2)(r2−|y|2)|x−y|−2 andr > 0. Theabove formulayieldsthefollowing two-sided
estimates (see, for example, [6]): Suppose B is an arbitrary ball in Rd with radius r > 0. Then
there is a universal constant c = c (d,α) > 1 so that for every x,y ∈ B,
1 1
α/2 α/2
G (x,y) ≍c1 |x−y|α−d 1∧ δB(x) 1∧ δB(y) . (2.9)
B
|x−y| |x−y|
(cid:18) (cid:19) (cid:18) (cid:19)
Since for a,b > 0, a∧b ≍ ab and 1∧ a ≍ a , in view of (1.3) we can rewrite (2.9) as
a+b b a+b
δ (x)α/2δ (y)α/2
G (x,y) ≍ |x−y|α−d B B . (2.10)
B r (x,y)α
B
It follows immediately from (2.9) that there is a positive constant c = c (d,α) > 1 so that for
2 2
B = B(x ,r),
0
c−1rα ≤ E τ ≤ c rα for x ∈ B(x ,r/2).
2 x B 2 0
Riesz (see [1]) derived the following explicit formula for the Poisson kernel K (x,z) of X on
B(0,r)
B(0,r).
Γ(d/2)sin(πα/2) (r2−|x|2)α/2
K (x,z) = for |x| < r and |z| > r, (2.11)
B(0,r) πd/2+1 (|z|2 −r2)α/2|x−z|d
We also know from [2, Lemma 6] that, for rotationally symmetric α-stable process X,
P (X ∈ ∂D) = 0 and so P (X 6= X ) = 1
x τD x τD τD−
foreveryx ∈D ifD isadomainthatsatisfiesuniformexterior conecondition. (Althoughtheresult
of [2, Lemma 6] is stated only for bounded domain satisfying uniform exterior cone condition, its
proof works for any domain satisfying uniform exterior cone condition.)
3 Boundary Harnack principle
Recall that we write X and J for X0 and J0, respectively. First we record the following gradient
estimate on the Green function G of symmetric α-stable process X from [4].
D
Lemma 3.1. ([4, Corollary 3.3]) Let D be a Greenian domain in Rd of X. Then
G (x,y)
D
|∇G (x,y)| ≤ d for x,y ∈ D, x 6= y. (3.1)
D
|x−y|∧δ (x)
D
7
For x 6= y in D, define
α/2
|x−y|α−β−d 1∧ δD(y) if α > 2β,
|x−y|
 (cid:16) β(cid:17)
hD(x,y) := |x−y|β−d 1∧ δD(y) 1∨log |x−y| if α = 2β, (3.2)
 |x−y| δD(x)

 (cid:16) (cid:17) (cid:16)α/2 (cid:17) β−α/2
|x−y|α−β−d 1∧ δD(y) 1∨ |x−y| if α < 2β,
 |x−y| δD(x)

 (cid:16) (cid:17) (cid:16) (cid:17)
The following two results are established in [13].
Lemma 3.2. ([13, Theorems 4.11 and Lemma 4.13]) Suppose b is a bounded function satisfying
(1.5), and that for every x ∈ Rd, Jb(x,y) ≥ 0 a.e. y ∈ Rd. There exist positive constants r =
1
r (d,α,β,M ) ∈ (0,1] and C = C (d,α,β,M ) such that for any x ∈ Rd and any ball B =
1 1 5 5 1 0
B(x ,r) with radius r ∈ (0,r ], we have for x,y ∈ B,
0 1
1 3
G (x,y) ≤ Gb (x,y) ≤ G (x,y) and |SbGb (x,y)| ≤ C h (x,y) (3.3)
2 B B 2 B x B 5 B
Moreover, P Xb ∈ ∂B = 0 for every x ∈ B. In this case, for every non-negative measurable
x τB
function f,
(cid:0) (cid:1)
E f(Xb ) = f(z)Kb(x,z)dz for x ∈B.
x τB ZB¯c B
Lemma 3.3. ([13, Lemma 4.2]) Let D be a bounded open set in Rd. There exists a constant
C = C (d,α,β,diam(D),M ) > 0 such that for any bounded function b satisfying (1.5), and that
6 6 1
for every x ∈ Rd, Jb(x,y) ≥ 0 for a.e. y ∈ Rd, we have
Gb (x,y) ≤ C |x−y|α−d for x,y ∈ D.
D 6
Note that the constant C below is independent of ε ∈[0,1] appeared in (1.6).
7 0
Theorem 3.4 (Uniform Harnack inequality). Let r ∈ (0,1] be the constant in Lemma 3.2. Under
1
Assumption 1, there exists a constant C = C (d,α,β,M ,M ) ≥ 1 such that for every x ∈ Rd,
7 7 1 2 0
r ∈ (0,r ], and every non-negative function u which is regular harmonic in B(x ,r), we have
1 0
sup u(y) ≤ C inf u(y).
7
y∈B(x0,r/2) y∈B(x0,r/2)
Proof. Let u∗(x) := E u(Xε0 ) . Then u∗ is regular harmonic in B(x ,r) with respect to
x τ 0
B(x0,r)
the mixed stable processhes Xε0. In viiew of Lemma 3.2 and Assumption 1, for every x ∈ Rd and
0
r ∈ (0,r ], the Poisson kernel Kb (x,z) on B(x ,r) of Xb is comparable to that of Xε0. Thus
1 B(x0,r) 0
for every x ∈ B(x ,r/2), u(x) is comparable to u∗(x). Theorem 3.4 then follows from the uniform
0
Harnack inequality for mixed stable processes; see [7, (3.40)].
Lemma 3.5 (Harnack inequality). Under Assumption 1, there exists a constant C = C (d,α,β,
8 8
M ,M ) > 0 such that the following statement is true: If x ,x ∈ Rd, r ∈ (0,r ] and k ∈ N are
1 2 1 2 1
such that |x −x | < 2kr, then for every non-negative function u which is harmonic with respect to
1 2
Xb in B(x ,r)∪B(x ,r), we have
1 2
C−12−k(d+α)u(x )≤ u(x ) ≤ C 2k(d+α)u(x ). (3.4)
8 2 1 8 2
8
Proof. Without loss of generality, we may assume |x − x | ≥ r/4. Note that for every x ∈
1 2
B(x ,r/8) ⊂ B(x ,r/8)c, we have |x−x | < 2k+1r. Thus by Lemma 3.2 and Assumption 1, we
2 1 1
have
1
Kb (x ,x) ≥ G (x ,y)J(y,x)dy
B(x1,r/8) 1 2M B(x1,r/8) 1
2 ZB(x1,r/8)
1
= K (x ,x)
2M B(x1,r/8) 1
2
c
≥ 1 2−αrα(2−k−1r)−d−α =c r−d2−k(d+α). (3.5)
2
2M
2
Recall that by Theorem 3.4, we have u(x) ≥ c u(x ) for every x ∈ B(x ,r/8). Thus by (3.5),
3 2 2
u(x ) ≥ u(x)Kb (x ,x)dx
1 B(x1,r/8) 1
ZB(x2,r/8)
≥ c c u(x )r−d2−k(d+α) dx
2 3 2
ZB(x2,r/8)
≥ c 2−k(d+α)u(x ),
4 2
and (3.4) follows by symmetry.
Proof of Theorem 1.3. Note that there are constants R = R (d,α,β,M ) ∈ (0,r ) and c =
0 0 1 1
c(d,α,β,M )> 1 so that
1
c−1 c
≤ Jb(x,y) ≤
|x−y|d+α |x−y|d+α
for all |x−y|≤ R and b(x,z) satisfying (1.5). Thus using (3.3), we can get uniform estimates on
0
the Poisson kernel
Kb (x,z) = Gb (x,y)Jb(y,z)dy
B(x0,r) B(x0,r)
ZB(x0,r)
of any ball B(x ,r) with respect to Xb with r ∈ (0,R /3), x ∈ B(x ,r) and r < |z −x | < 2r.
0 0 0 0
Specifically, for r < |z −x | < 2r, Kb (x,z) is uniformly comparable to K (x,z). Using
0 B(x0,r) B(x0,r)
the explicit formula (2.11) for the Poisson kernel K , (3.3), Theorem 3.4 and (1.6), we can
B(x0,r)
adapt the arguments in [8, Theorem 2.6] to get our uniform boundary Harnack principle 1.3 (cf.
the proof of [7, Theorem 3.9]). Since the proof is almost identical to those in [8, Section 3], we omit
the details here.
Lemma 3.6. Suppose Assumption 1 holds and D is a Lipschitz open set with characteristics
(λ ,R ). Let r ∈ (0,1] be the constant in Lemma 3.2. There is a positive constant C =
0 0 1 9
C (d,α,β,λ ,R ,M ,M ) ≥ 1 such that for every z ∈ ∂D, r ∈ (0,r /2), and every non-negative
9 0 0 1 2 0 1
harmonic function u that is regular harmonic in D∩B(z ,2r) with respect to Xb and vanishes in
0
Dc∩B(z ,2r),
0
E u(Xb ): Xb ∈ Bc ≤ C 2−kαu(x), x ∈D∩B , (3.6)
x τD∩Bk τD∩Bk 0 9 k
h i
for B := B(z ,2−kr) and k ≥ 1.
k 0
9
Proof. Without loss of generality, we may assume z = 0. By the uniform inner cone property of
0
a Lipschitz open set, one can find a point z˜ ∈ D ∩B(0,r) and κ = κ(λ ,R ) ∈ (0,1) such that
0 0 0
B := B(z˜ ,κ2−kr)⊂ B ∩D for every z˜ := 2−kz˜ and k ≥ 0. Define
k k k k 0
e u (x) := E u(Xb ) :Xb ∈ Bc .
k x τD∩Bk τD∩Bk 0
h i
Since u = u, (3.6) is clearly true for k = 0. Henceforth we assume k ≥ 1. Note that u ≥ 0 is
0 k
regular harmonic with respect to Xb in D∩B , and u (x) ≤ u (x) for all x ∈ Rd. Define
k k k−1
I (x) := E u(Xb ): Xb ∈ Bc .
k x τBk τBk 0
h i
Clearly by definition u (z˜ ) ≤ I (z˜ ). For any k ≥ 1, by Lemma 3.2 and (2.5), we have
k k k k
Kb (z˜ ,y) = Gb (z˜ ,z)Jb(z,y)dz
Bk k Bk k
ZBk
3
≤ G (z˜ ,z)Jb(z,y)dz
2 Bk k
ZBk
3
= G (2−(k−1)z˜ ,z)Jb(z,y)dz
2 2−(k−1)B1 1
Z2−(k−1)B1
3
= 2−(k−1)d G (2−(k−1)z˜ ,2−(k−1)w)Jb(2−(k−1)w,y)dw
2 2−(k−1)B1 1
ZB1
3
= 2−(k−1)α G (z˜ ,w)Jb(2−(k−1)w,y)dw. (3.7)
2 B1 1
ZB1
Note that for any y ∈Bc and w ∈ B ,
0 1
|y−w| |y|+|w|
≤ ≤ 3.
|y−2−(k−1)w| |y|−2−k+1|w|
Thus by (1.6) we have
Jb(2−(k−1)w,y) ≤ M Jε0(|y−2−(k−1)w|) ≤ 3d+αM Jε0(|y−w|) ≤ 3d+αM2Jb(w,y). (3.8)
2 2 2
It follows from (3.7), (3.8) and Lemma 3.2 that for any y ∈ Bc,
0
3d+α+1
Kb (z˜ ,y) ≤ M22−(k−1)α G (z˜ ,w)Jb(w,y)dw
Bk k 2 2 B1 1
ZB1
≤ 3d+α+1M22−(k−1)α Gb (z˜ ,w)Jb(w,y)dw
2 B1 1
ZB1
= c 2−kαKb (z˜ ,y).
1 B1 1
Now we have for k ≥ 1
I (z˜ ) = u(y)Kb (z˜ ,y)dy
k k Bk k
ZB0c
≤ c 2−kα u(y)Kb (z˜ ,y)dy
1 B1 1
ZB0c
10