数理解析研究所講究録
第 1669 巻 2009 年 1-13

1

Boundary Harnack principle and the quasihyperbolic
boundary condition
Expository and correction––

北海道大学大学院・理学研究院 相川弘明 (Hiroaki Aikawa)
Hokkaido University
Sapporo 060-0810, Japan

1. INTRODUCTION
This is an expository and correction of [2]. We observe that the global boundary Harnack principle holds for domains satisfying some conditions related towith $n\geq 2$ andthe quasihyperbolic metric. Let $D$ be a bounded domain in
let $\delta_{D}(x)=dist(x, \partial D)$ . We use the following notation: $B(x, r)$ (resp. $S(x,$ )stands for the open ball (resp. the sphere) center at and radius . By wedenote a positive constant which may change from one occurrence and the next.
.
if
We writeand a compact set
Consider a pair $(V, K)$ of a bounded open set
such that$\mathbb{R}^{n}$

$r)$

$x$

$J^{\cdot}\approx g$$r$$A^{-1}\leq J^{\cdot}/g\leq A$$V\subset \mathbb{R}^{n}$$K\subset R^{n}$(1)$K\subset V,$ $K\cap D\neq\emptyset$

and$K\cap\partial D\neq\emptyset$

FIGURE 1. A pair $(V, K)$ with (1).

.

$A$

2
Definition 1 (Globa] boundary Harnack princip]e). We say that $Denj(ys$ theglobal boundary Harnack principle if for all $(V, K)$ with (1), there exists
depending only on $D,$ and with thc following property: If(i)on ,and t) are positive superharmon(ii)
and are bounded, positive and harmonic in $V\cap D$ ,(iii)
and vanish on
except for a polar set,
$A_{1}$

$V$

$K$

$u$

$D$

$ic\cdot.f\dot{\iota}rnc\cdot tio\prime_{\Delta}$

$u$

$v$

$u$

$v$

$V\cap\partial D$

then$\frac{u(x)/u(\gamma)}{v(x)/v(.y)}\leq A_{1}$

for

$x,$

$y\in K\cap D$ .

Remark 1. We have the following remarks.
(i) In general, $K\cap D$ may be disconnected, so

and need to be definedon the whole $D$ and positive and superharmonic there.
(ii) Jerison-Kenig [12] and $Bass- Burdzy- Ba\tilde{n}uelos[8]$ and [9] assume thatand are positive and harmonic over the whole $D$ .
(iii) Our formulation of the boundary Harnack principle is slightly stronger.
$u$

$u$

$v$

$v$

For a Lipschitz domain the global boundary Harnack principle was provedby Ancona [4], Dahlberg [11] and Wu [14] independently. Caffarelli-FabesMortola-Salsa [10] and Jerison-Kenig [12] gave significant extensions. Fromthe probabilistic point of view, $Bass- Burdzy- Ba\tilde{n}uelos[8]$ and [9] proved theglobal boundary Harnack principle for a Holder domain, a domain whose boundary is locally given by the graph of a H\"older continuous function in.
Define the quasihyperbolic metric $k_{D}(x,y)$ by$\mathbb{R}^{n-1}$$k_{D}(x,y)= \inf_{\gamma}\int\frac{ds(z)}{\delta_{D}(z)}$,where the infimum is taken over all rectifiable curves connecting to in $D$
and $ds(z)$ stands for the line element on . Smith-Stegenga [13] said that $D$ is a
“H\"older domain“ if$\gamma$$x$$y$$\gamma$(2)$k_{D}(x, x_{0}) \leq A\log\frac{\delta_{D}(x_{0})}{\delta_{D}(x)}+A’$

for all $x\in D$with some positive constants and $A’$ . Banuelos [7] said that such a domain
is a H\"older domain of order . To avoid the confusion, we say that $D$ satisfies
the quasihyperbolic boundary condition (of’ order ) if (2) holds. One of themost significant properties of domains satisfying the quasihyperbolic boundary
$A$

$0$

$0$

3
condition is the exponential integrability of the quasihyperbolic metric: there
exists a positive constant such that$\epsilon$$\int_{l3}n-$(Smith-Stegenga [13]).
Extending (2), we consider the following condition:(3)$k_{D}(x, x_{0}) \leq A(\frac{\delta_{D}(r_{0})}{\delta_{lJ}(x)})\}+A’$

for all $x\in D$with some positive constants and $A’$ . Let us say that $D$ satisfies the quasihyperbolic boundary condition ] order if (3) holds. The above condition and(2) are interior conditions. Let us consider an exterior condition.
$A$

$0$

$\alpha$Definition 2. By Cap we denote the logarithmic capacity if $n=2$ , and the
Newtonian capacity if $n\geq 3$ . We say that the capacity density condition holds
if there exist constants $A>1$ and $r_{0}>0$ such thatCap$(B(\xi, r)\backslash D)\geq\{\begin{array}{ll}Ar if n=2,Ar^{n-2} if n\geq 3,\end{array}$wheneverand $0<r<r_{0}$ . See Armitage-Gardiner [6] for the capacity,which illustrates the inhomogeneity between $n=2$ and $n\geq 3$ .
$\xi\in\partial D$

said that $D$ is armly Holder domain of order if (3) andthe capacity density condition hold. It seems that the capacity density conditionis needed for $\alpha>0$ because of the lack of the exponential integrability of thequasihyperbolic metric.
$Ba\tilde{n}uelos[7]$$uniJ\dot{o}$$a$Remark 2. We have the following remarks:(i) Banuelos [7] showed the intrinsic ultra-contractivity for a domain sat$0\leq\alpha<2$ ,isfying the quasihyperbolic boundary condition of orderand the capacity density condition.
(ii) Bass-Burdzy-Banuelos [8] and [9] proved the boundary Harnack principle for a H\"older domain. The main tool was the so-called box argu$\alpha,$

ment.

4
(iii) They also claimed that the boundary Harnack principle may hold for adomain satisfying the quasihyperbolic boundary condition of order ,$0<\alpha<1$ , and the capacity density condition. However, no proof has
not been provided.
(iv) $Bass- B\iota iidzy- Ba\tilde{n}uelos[9]$ and [8] were very probabilistic.
(v) Aikawa [1] proved the local (or scale-invariant) boundary Harnack principle for a uniform domain by an elementary and analytic argument.
$a$Aikawa [3] gave a completely different approach to the boundary Harnack
principle: the Domar method and the equivalence between the boundary Harnack principle and the Carleson estimate.
Definition 3 (Global Carleson estimate). We say that $D$ enjoys the global Carleson estimate if for all $(V, K)$ with (1), and a point $x_{0}\in K\cap D$ , there exists
with the following property: If
depending only on $D,$ $V,$ and$A_{2}$

$K$

(i)

$u$

(ii)

$u$

(iii)

$u$

$x_{0}$is positive superharmonic on ,is bounded, positive and harmonic in $V\cap D$ ,except for a polar set,
vanishes on$D$$V\cap\partial D$

then
(4)

$u(x)\leq A_{2}u(x_{0})$for $x\in K\cap D$ .

FiGURE 2. The global Carleson estimate.

We have the following theorems.

5
Theorem 1. For arbitrary domains

Global Boundary Harnack principle$\Leftrightarrow$

Global Carleson estimate.

Theorem 2. The global boundarv $Hpl$ rnack principle holds. $fOr$ a domain satis.
fy $ing$ the quasihyperbolic boundary condition ( order ).
$0$

$CJf$

Theorem 3. Let $D$ be a bounded domain satisfy $ing$ the capacity density condition and the quasihyperbolic boundary condition $oJ^{\cdot}ordera$ ]$(?r$ some $0<\alpha<1$ .
Then the global boundary Harnack principle holds for $D$ .

Remark 3. We do not know whether the restriction $a<1$ is sharp or not.
$Ba\tilde{n}uelos[7]$ showed the intrinsic ultra-contractivity for a domain satisfying
the quasihyperbolic boundary condition of order $a,$ $0\leq\alpha<2$ , and the capacity
density condition. It is interesting to study the case $1\leq a<2$ .

2.

$\beta$-JOHN DOMAINS

Let $0<\beta\leq 1$ . We say that $D$ is a -John domain, if there is a point $x_{0}\in D$
and every point $x\in D$ can be connected to by a rectifiable curve with$\beta$$x_{0}$$\delta_{D}(y)^{\beta}\geq Af(\gamma(x,y))$for all

where $l(\gamma(x,y))$ is the arc length of the subcurvealong . See Ancona [5, 9.2].

$\gamma$$y\in\gamma$,$\gamma(x,y)$connecting

$x$

and

$y$

$\gamma$

FIGURE 3. A -John domain.
$\beta$Remark 4. We have the following remarks.
(i) If $\beta=1$ , then a -John domain is a classical John domain and thequasihyperbolic boundary condition of order is satisfied.
(ii) In general, a -John domain satisfies the quasihyperbolic boundary
condition of order $1-\beta$ .
$\beta$$0$$\beta$6
Hence, we obtain the following corollary to Theorem 3.

Corollary 1. Let $0<\beta\leq 1$ . Thencondition enjoys the global-John dontain with the capacity density
Harnack principle.

$c\iota\beta$$bo\iota rndcll’1^{\gamma}$

Remark 5. In view of Theorem 2, the capacity density condition is superfluous
in case $\beta=1$ .
A typical example of a -John domain is a -H\"older domain, whose boundary. A 1is given locally by the graph of a -H\"older continuous function in
Holder domain is a Lipschitz domain, so that the boundary Harnack principle isclassically established. A -John domain need not satisfy the capacity density
condition if $0<\beta<1$ . However, Bass-Burdzy [9] showed that the global
boundary Harnack principle holds for a -H\"older domain without the capacity
density condition for $\frac{[}{2}<\beta<1$ , and then $Bass- Bnrdzy- Ba\tilde{n}uelos[9]$ for $0<$$\beta<1$ .
$\beta$$\beta$$\mathbb{R}^{i?}-|$$\beta$$\beta$$\beta$Theorem 4. Let $0<\beta<1.$ A
Harnack principle.

$\beta- H(\dot{)}lcler$domain enjoys the global boundary

Their proof is very probabilistic. We shall give an elementary analytic proof.
for$\frac{)}{2}<\beta<1$Remark 6. There were mistakes in [2]. Here a corrected proof will be given.
cannot be covered. Its analytic simple proof
Unfortunately, theremains open.
$0< \beta\leq\frac{1}{2}$3. BOUNDARY HARNACK PRINCIPLE AND CARLESON ESTIMATE IN TERMS OF THE GREEN
FUNCTION

The Riesz decomposition theorem says that a positive superharmonic funcexcept for a polartion , which is harmonic in $D\cap V$ and vanishes on
inof a measure on
set, can be represented as the Green potential$D\cap V$ . One more geometrical observation is relevant. Let
be a closed ballis a compact set and $D\backslash V=D\cap F$ and $K\cap F=\emptyset$ .
including $D$ . Then
Consider a pair of disjoint compact sets $K$ and $F$ with$\partial D\cap V$$u$$G\mu$$D\cap\partial V$$\mu$$\overline{B}$$F=\overline{B}\backslash V$(5)$K\cap\partial D\neq\emptyset,$$K\cap D\neq\emptyset,$ $F\cap\partial D\neq\emptyset$

and$F\cap D\neq\emptyset$.

7
enjoys the global Carleson estimate in
Definition 4. We say that a domain
zerms ] the Green.firnction if for each pair of disjoint compact sets $K$ anddepending only on
with (5) and a point $x_{0}\in K\cap D$ , there exists a constant
$D$

$F$

$0$

$A_{2}$$D,$ $K,$ $F$

and$x_{0}$

such that$G(x,y)\leq A_{2}G(x_{0},y)$for $x\in K\cap D$ and $y\in F\cap D$ .

FIGURE 4. The global Carleson estimate in terms of the Green function.

The above discussion gives readily the following theorem.
Theorem 5. The global Carleson estimate and the global Carleson estimate interms $oJ^{\cdot}theGreenJi_{4}nction$ are equivalent.

Using the Riesz decomposition theorem, we similarly obtain the counterpartof the boundary Harnack principle.

Definition 5. We say that a domain $D$ enjoys the global boundary Harnack
principle in terms
GreenJimction if for each pair of disjoint compact sets
$K$
and $F$ with (5), there exists a constantdepending only on $D,$ $K$ and $F$ suchthat
for $x’\in K\cap D$ and $y,$ $y’\in F\cap D$ .
$oJ^{\cdot}the$$A_{1}$$\frac{G(x,y)/G(x’,y)}{G(x,y’)/G(x’,y’)}\leq A_{1}$$x,$Theorem 6. The global boundary Harnack principle and the global boundary
Harnack principle in terms $oJ^{\cdot}theGreen$ ]$i_{4}nction$ are equivalent.
Combine Theorems 1, 5 and 6.

Theorem 7. Let $D$ be a bounded domain inare equivalent:$\mathbb{R}^{n}$. Then thefOllowing statements

8

FIGURE 5. The global boundary Harnack principle in terms of theGreen function.
(i) $D$ enjoys the global boundary Harnack principle.
(ii) $D$ enjoys the global boundary Harnack principle in terms $of\cdot the$ Green

J’unction.

(iii) $D$ enjoys the global Carleson estimate.

(iv) $D$ enjoys the global Carleson estimate in terms$oJ^{\cdot}the$Green function.

4. LEMMAS
In order to prove the theorems we use the following notions. By $\omega(E, U)$ webe the. Letdenote the harmonic measure over an open set of
capacitary width defined by$U$$w_{\eta}(U)= \inf\{r>0$:$\frac{Cap(B(x,r)\backslash U)}{Cap(B(x,r))}\geq\eta$$E\subset\partial U$$w_{\eta}(U)$for all $x\in U\}$ .

Here $0<\eta<1$ .

Lemma 1. If $D$ satisfies the capacity density condition, then $w_{\eta}(\{x\in D$ :$\delta_{D}(x)<r\})\leq 2r$
for some .
$\eta$

The capacitary width is useful for the estimate of harmonic measure ([1,Lemma 1] .
$)$Lemma 2. There exists $A_{3}>0iJ^{\cdot}x\in U$ and $R>0$ , then$\omega^{X}(U\cap S(x,R);U\cap B(x, R))\leq\exp(2-A_{3}\frac{R}{w_{\eta}(U)})$.

Definition 6. We say that two points $x,y\in D$ are connected by a Harnack chain, andif, $k-1$ . The number is called the length
for $j=1,$
the Harnack chain.
$\{B(x_{j}, \frac{1}{2}\delta_{D}(x_{j}))\}_{j=1}^{k}$$x \in B(x_{\mathfrak{l}}, \frac{1}{2}\delta_{D}(x_{1})),$$B(x_{j+1}, \frac{1}{2}\delta_{D}(x_{j+1}))\neq\emptyset$$y \in B(y_{k}, \frac{1}{2}\delta_{D}(y_{k}))$$k$$\ldots$$B(x_{j}, \frac{1}{2}\delta_{D}(x_{j}))\cap$$oJ^{\cdot}$9
Since the shortest length of the Harnack chain connecting
comparable to the quasihyperbolic metric $k_{D}(x,y)$ , we have

$x$

and

$y$

Lemma 3. There is a cons tant $A_{4}>1$ depending only on the dimension
independent ] $D)$ such thatin $D$ is

$n$

(even

$0$

(6)
$f_{\dot{C}J}r$$\exp(-A_{4}(k_{D}(x,y)+1))\leq\frac{l\tau(x)}{h(y^{f})}\leq\exp(A_{4}(k_{D}(x,y)+1))$

every positive harmonic ]$i_{\Lambda}nctionh$

on

$D$

.

5. THE BOUNDARY HARNACK PRINCIPLE FOR DOMAINS WITH QUASIHYPERBOLIC
BOUNDARY CONDITION

We shall show the global Carleson estimate in terms of the Green function. Then Theorem 7 gives the global boundary Harnack principle. Let $D$
satisfies the capacity density condition and thebe as in Theorem 3, i.e.
andquasihyperbolic boundary condition of order $a,$ $0<\alpha<1$ . Let
be a pair of disjoint compact sets satisfying (5) and let $x_{0}\in K\cap D$ . Set$U(r)=\{x\in \mathbb{R}^{n}$ : dist $(x,$ $F)<r\}$ for $r>0$ .
Let $2R=dist(K, F)$ and put$U=U(R)$ . Observe. Let $\omega_{0}=\omega(D\cap\partial U, D\cap U)$ be the harmonic
in $D\cap U$ .
measure of.
andFirst let us compare
$D$

$K$

$F$

$\overline{U}\cap F=\emptyset$$D\cap\partial U$$\omega_{0}$$G(x_{0}, \cdot)$Lemma 4. Let $\omega_{0}=\omega(D\cap\partial U, D\cap U)$ . Then $\omega_{0}(y)\leq AG(x_{0},y)$ for $y\in F\cap D$ .

Proof. Let us employ the box argument. The main idea is to slice
to the distance from $F$

accordingand the value of the Green function. Let $R_{0}=R$ and$U$

for $j\geq 1$ .

$R_{j}=(1- \frac{3}{\pi^{2}}\sum_{k=\mathfrak{l}}^{i}\frac{1}{k^{2}})R$

Then it is easy to see that $R_{j-1}-R_{j}=(3R)/(\pi^{2}j^{2})$ , so that
(7)$\sum_{j=1}^{\infty}R_{j}=\frac{R}{2}$.

is bounded on $U\cap D$ , we may assume that $u=G(x_{0}, \cdot)/A$ is
Since
bounded by 1 on $U\cap D$ by a suitable choice of $A>1$ .
$G(x_{0}, \cdot)$

10
Let $j\geq 1$ and set$U_{j}=\{y\in U(R_{j})\cap D:0<n(y)<\exp(-2^{j})\}$ ,$D_{j}=\{y\in U(R_{j})\cap D:\exp(-2^{j\neq 1})\leq n(y)<\exp(-2^{j})\}$.

Let$q_{j}=\{\begin{array}{ll}\sup_{D;}\omega_{0}/u, if D_{j}\neq\emptyset,0 if D_{j}=\emptyset.\end{array}$Since $0<u<1$ on $U\cap D$ , it follows from (7) that
. Hence it is sufficient to show thatclosure of$\bigcup_{j=1}^{\infty}D_{j}$is included in theis bounded.

$F\cap\partial D$$q_{j}$FIGURE 6. Box argument: slice of$U$.

By (3) and (6), we havefor $y\in U_{j}$ .

$\exp(-2^{j})>u(y)\geq\exp(-A_{4}(k_{D}(y, x_{0})+1))\geq A\exp(-\frac{A}{\delta_{D}(y)^{\alpha}})$for $y\in U_{j}$ . Hence, Lemma 1 implies that. Observe that dist $(D\cap\partial U_{j-1}, U_{j})\geq R_{j-1}-R_{j}$ . Applying the, we obtainmaximum principle on

In other words,$\delta_{D}(y)\leq A2^{-1/\alpha}$$w_{\eta}(U_{j})\leq A2^{-j/\alpha}$$U_{j-1}$$\omega_{0}\leq\omega(D\cap\partial U_{j-1}\backslash \overline{D_{j-1}}, U_{j-1})+q_{j- 1}u$

Divide the both sides byyields$u$onand take the supremum over$q_{j}\leq A\exp(2^{j+}|-ARj^{-2}2^{1/\alpha})+q_{j-1}$$U_{j-1}$$D_{j}$.

. Then Lemma 2.

Since $0<a<1$ , it follows that $\sum_{j=1}^{\infty}\exp(2^{j+1}-ARj^{-2}2^{j/\alpha})<\infty$, so thatbounded. The proof is complete.

$q_{j}$

is$\square$11
Theorem 3. It is sufficient to show the global Carleson estimate in termsof the Green function by Tlieorem 7. Let and be disjoint compact sets with(5). Let $2R=dist(K, F)$ and put $U=U(R)$ . Let $\omega_{0}=\omega(D\cap\partial U, D\cap U)$ . Then
Lemma 4 gives $\omega_{0}(y)\leq AG(x_{0},y)$ for ) $’\in F\cap D$ . Since the distance between
is positive, it is obvious that $G(x,y)\leq A$ uniformly for $x\in K\cap D$ andand$y\in U\cap D$ . Fix $x\in K\cap D$ and apply the maximum principle to $G(x, \cdot)$ andto obtain $G(x,y)\leq A\omega_{0}(y)$ for $y\in U\cap D$ . Hence Lemma 4 yields$Proof\cdot oJ^{\cdot}$

$F$

$K$

$K$

$-$

$U$

$\omega_{0}$$G(x,y)\leq AG(x_{0},y)$for $y\in F\cap D$ ,as required.

口

6. THE BOUNDARY HARNACK PRINCIPLE FOR$H\ddot{O}$LDER DOMAINS

Let $0<\beta<1$ . We may assume that $D$ is above the graph of a -H\"older
and. Moreover we may assume that
continuous function in$\varphi(x)=0$ for $|x|\geq 1$ . For a point $x\in D$ we define $d(x)=x_{n}-\varphi(x’)$ , where$x=(x’, x_{n})$ . Letbe high above from the boundary. Suppose is just below
does not hold. We can assert only
and $0<d(x)<1$ . In general,$\beta$$\mathbb{R}^{\prime p-[}$$||\varphi||_{\infty}\leq 1$$\varphi$$x$$x_{0}$$\delta_{D}(x)\approx d(x)$$x_{0}$.
that
Considering the line segment connecting
estimate of the quasihyperbolic metric:$\delta_{D}(x\rangle^{\beta}\geq Ad(x)$

(8)

$x$

and

$x_{0}$, we obtain the following$k_{D}(x, x_{0}) \leq A\int_{d(.))}^{d(\iota_{0})}\frac{dt}{t^{1/\beta}}\leq Ad(x)^{1- 1/\beta}+A$.

$x_{0}$FIGURE 7. H\"older domain:$\delta_{D}(x)$The same estimate holds if $x\in D$ is close tocounterpart of Lemma 1.

$0$

and $d(x)$ .

, say $|x|<1$ . Now we have the12Lemma 5. Let $D$ be as above. $If\cdot 0<r<1$ , then $w,,(\{x\in D:d(x)<r\})\leq Ar$. foor some $\eta>0$ .
This lemma can be proved by Fubini’s theorem showing the volume density
condition. In view of Theorem 7, it is sufficient to show the globa] Carlesonandbe a pair of disjoint
estimate in terms of the Green function. Let
compact sets with (5) and let $x_{0}\in K\cap D$ . Let $U(r)=\{x\in \mathbb{R}^{n}$ : dist $(x,$ $F)<r\}$. Let
for $r>0$ . Let $2R=dist(K, F)$ and put $U=U(R)$ . Observe$\omega_{0}=\omega(D\cap\partial U, D\cap U)$ be the harmonic measure ofin $D\cap U$ . We can.
andcompare$K$$F$$\overline{U}\cap F=\emptyset$$D\cap\partial U$$\omega_{0}$$G(x_{0}, \cdot)$Lemma 6. Let $\omega_{0}=\omega(D\cap\partial U, D\cap U)$. Then $\omega_{0}(y)\leq AG(x_{0},y)$ for $y\in F\cap D$ .
The proof is similar to that of Lemma 4. We takesame way as in the proof of Lemma 4. It is sufficient to show thatBy (8) and (6), we have$Pr(oJ$.

$U_{j},$$R_{j},$$D_{j}$and in theis bounded.
$q_{j}$$q_{j}$$\exp(-2^{j})>u(y)\geq\exp(-A_{4}(k_{D}(y, x_{0})+1))\geq A\exp(-Ad(y)^{1-1/\beta})$for $y\in U_{j}$ .

In other words, $d(y)\leq A2^{-1/(I/\beta- 1)}$ for $y\in U_{j}$ . Hence, Lemma 5 implies that
, we obtain. Applying the maximum principle on$w_{\eta}(U_{j})\leq A2^{-j\beta/(1-\beta)}$$U_{j-1}$$\omega_{0}\leq\omega(D\cap\partial U_{j-1}\backslash \overline{D_{j- 1}}, U_{j-1})+q_{j- 1}u$

Divide the both sides byyields$u$onand take the supremum over$D_{j}$$q_{j}\leq Aexp(2^{j+1}-ARj^{-2}2^{\beta/(1-\beta)})+q_{j-1}$$Since\beta/(1-\beta)>1$ for- $<\beta<1$ ,$\infty$, so that$q_{j}$

it follows thatis bounded.

$U_{j- I}$.

. Then Lemma 2.

$\sum_{j=1}^{\infty}\exp(2^{j+1}-ARj^{-2}2^{j\beta/(1-\beta)})<$$\square$. Using Lemma 6, we can prove the theorem in the same
way as in the proof of Theorem 3. See [2] for details.

$Proof\cdot of\cdot Theorem4$$\square$

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