A R K I V F O R MATEMATIK Band 6 nr 1
1.65 04 2 Communicated 24 February 1965 by TRYGVE •AGELL and LENNART CARLESON
E s t i m a t e s o f h a r m o n i c m e a s u r e s
By
KERSTI I-IALISTEIntroduction
E s t i m a t e s of h a r m o n i c measures in terms of E u c l i d e a n quantities are useful in m a n y situations. I n the two-dimensional case one can a p p l y m e t h o d s of conformal m a p p i n g a n d extremal lengths, a n d m a n y sharp results are well known. Different means of h a r m o n i c measures can be studied in the n-dimensional case. This paper is i n t e n d e d to provide a s u r v e y of m e t h o d s available to estimate h a r m o n i c measures.
T h e two-dimensional case is t r e a t e d in Chapter I. The second p a r a g r a p h contains well-known distortion inequalities f r o m the t h e o r y of conformal mapping, a n d w 3 contains well-known results f r o m t h e t h e o r y of extremal lengths. I n w 4 we prove two s y m m e t r i z a t i o n theorems with the aid of a result f r o m w 3. I n w 5 we a p p l y results f r o m w 3 to comb domains.
Chapter I I gives n-dimensional methods. I n w 6 a m e t h o d of Carleman [6.1] is applied to h a r m o n i c measures. T h e derivation of Carleman's m e t h o d in T h e o r e m 6.1 follows t h a t of Dinghas [6.3]. The estimates of h a r m o n i c measures in Theorems 6.2 a n d 6.3 are new in t h e case n > 2 . I n w 7 we t r e a t N e v a n l i n n a ' s m e a n value in a special case. I n w 8 we prove some s y m m e t r i z a t i o n results with probabilistic methods.
The m a i n p r o b l e m is to provide u p p e r b o u n d s for h a r m o n i c measures. Lower b o u n d s are discussed in w 2 a n d w 7.
Bearing in m i n d the possibility of exhausting a given d o m a i n with more regular domains we have n o t aimed at generality in assumptions a b o u t the domains con- sidered.
The subject of this paper was suggested b y Professor L. Carleson, to w h o m I a m deeply grateful for all his advice.
1. Definitions
R n is the n-dimensional Euclidean space, n >~ 2, with points z = @1, Yl ... Yn-1) = (xl, y). I n Chapter I we t r e a t the case n = 2 a n d prefer to write
z=x+iy.
Thefollowing definitions are t h e n to be u n d e r s t o o d with Re z instead of x 1.
D denotes a d o m a i n (open connected set) a n d ~D the b o u n d a r y of D.
Ox-{z[xl=x, zeD}.
D x is the s u b d o m a i n of
{zlxl<x , zED}
t h a t contains a given p o i n t %.Ox={z]zeOx, ze~D~}.
W i t h o u t D being specified D~ denotes a d o m a i n in {z]x~<~} with p a r t of its b o u n d a r y on {z Ix x =~} a n d O~ t h e n denotes the interior of {z]x~ = ~ , z E ~D~}.
1: i 1
KERSTI HALISTE:
Estimates of harmonic measures
Given D~, Ox={ZlXl=X ,zED~}, x<~.
0~, i= 1, 2 ... n(x),
are the c o m p o n e n t s or unions of c o m p o n e n t s of 0z(v~x) t h a t separate t w o given points or surfaces. (Cf. w 2.)O(x), O(x), O(x), O~(x)
are the measures of the respective sets.0~ will be used in preference to v~ a n d 0~, if D is such t h a t all the Ox are connected.
0~ can also be used to denote a set in the ( n - 1)-dimensional y-space.
e)(z; ~; D) denotes the h a r m o n i c measure at the point z of a ~ ~D with respect to D.
c m a y denote various constants.
S y m m e t r i z a t i o n of a n n-dimensional open set A with respect to an ( n - 1 ) - d i m e n - sional h y p e r p l a n e p (Steiner s y m m e t r i z a t i o n ) [1.1, p. 5, pp. 151-152] means the fol- lowing: A is t r a n s f o r m e d into A* so t h a t a n y straight line perpendicular to iv t h a t intersects A also intersects A*. B o t h intersections have the same measure (length) a n d the intersection with A* is a single line-segment s y m m e t r i c with respect to p.
W h e n n = 2 this reduces to the definition of s y m m e t r i z a t i o n with respect to a straight line.
A continuous function / is s y m m e t r i z e d with respect to a h y p e r p l a n e iv b y sym- metrizing the sets
{zi/(z )
> a } , inf/(z) ~<a < sup/(z), in the m a n n e r described above.S y m m e t r i z a t i o n of an n-dimensional open set, n > 2 , with respect to a straight line 1 (Schwarz s y m m e t r i z a t i o n ) [1.1, pp. 151-152] means the following: A is trans- f o r m e d into A* so t h a t a n y (n 1)-dimensional h y p e r p l a n e perpendicular to l t h a t intersects A also intersects A*. B o t h intersections h a v e the same measure a n d t h e intersection with 2t* is a sphere with its centre on 1.
Chapter I. The two-dimensional case
2. Distortion theorems in the theory of conformal mapping
P r o b l e m s of distortion in the t h e o r y of conformal m a p p i n g h a v e been widely studied. W e shall refer to the s u r v e y given b y L e l o n g - F e r r a n d [2.2, Ch. VI, in partic- ular pp. 185-202, pp. 216-217].
L e t D be a s i m p l y connected d o m a i n in the z-plane n o t containing the point at infinity. L e t A a n d B be two accessible b o u n d a r y points. W e limit the discussion to the following situation: A a n d B are the only b o u n d a r y points of D at infinity a n d Re z -+ - 0% when z - + A , z E D, a n d Re z - + + 0% when z--->B, z E D. L e t L be a J o r d a n arc in D joining A a n d B. 0~, i = 1, 2 ... n(x), are the segments of Ox t h a t separate A a n d B. 01 is the first of the 0~ t h a t is m e t when m o v i n g along L f r o m A to B. 0~
can also be defined as t h a t segment a m o n g the 0~. t h a t separates the largest sub- d o m a i n of D f r o m A. Cf. Fig. 2.1.01(x), the length of 01, is lower semicontinuous. A detailed discussion of the definition of 01 is given b y L e l o n g - F e r r a n d [2.2, pp. 185- 186].
D is m a p p e d eonformally o n t o G in the w-plane,
w =u +iv.
G = {wI I
v ] < 89 a n d A corresponds to u = - ~ a n d B to u = + oo. yx is the image of 0~. ua a n d u2 are defined b yul(x )
= inf u, u2(x ) = sup u.W E ~ x W E ~ x
Let be the subdomain of D separated from B by0 . G a = { w i u < a , IvI and I a = {w ]u = a, Iv ] < 89 W e use conformal invariance of harmonic measures, the
6 - A
ARKIV FOR MATEMATIK. B d 6 n r 1
F i g . 2.1
B - - >
extension principle, and explicit harmonic measures in G~ to establish the following relations:
max a)(z;
0~; D~)< eo(u2(x);
/~,(~); G~,~)) ze(~. . . . arctg exp (- ul(_~ ) +
4 u2(x)), us(x )
< ul(~);(2.1)
max
w(z; 0~; D~) >1
eo(ui(x); /~.~);G~,~(~))=
4 - arctg exp ( - u2(~ ) +Ul(X))-
(2*2)Ahlfors' first distortion inequality [2.1, pp. 7-I2], [2.2, pp. 187-190], [2.3, pp.
93-100], states that
fl
ul(~ ) - us(x ) > ~ 01(t )
- 4~, when ~l~t i > 2. (2.3) By (2.1) this yields an upper bound for~o(z; 0~; D~).
(Also cf. [2.3, pp. 76=78]~) However, a more general result is proved in Theorem 3.2 with the method of extremal lengths. In connection with estimates of harmonic measures, distortion inequalities in the other direction arc more useful, since there are few other methods for finding lower bounds of harmonic mcasures.Distortion inequalities in the other direction requil'e various restrictive assumptions about D. Ahlfors' original second inequality [2.1, pp. 12-17] is contained (with a different constant term) in an inequality by Lelong-Ferrand [2.2, pp. 194=198].
Another variant was proved by Warschawski [2.4, pp. 291-296], [2.2, p. 202]. W i t h (2.2) this yields the following theorem.
Theorem 2.1.
Let D be bounded by the curves
y:-~02(x )and
y=q~l(X), ~2(x)>~01(x), - - ~ < x < ~ .Let ql and q~2 have' bounded derivatives;
[Vi(x)[ < m , [ ~ ( x ) [ < m ,- - C<:) < X ~: cx:z, ~)(X) - - ol.((~l(X) "002(X)), (~)(X):::(p2(,~) -- ~91(X ).
Then
KERSTI HALISTE,
Estimates of harmonic measures
m a x w ( z ; O ~ ; D ~ ) > / c e x p - ~ O ~ - ~ [ 0 ~ + 1 2 0 ( 0 ]
at ,
ZeOx
where
c = exp ( - 8z(1 + I m2)) 9A n o t h e r m e t h o d for determining lower b o u n d s of h a r m o n i c measures will be discussed in w 7. Lower b o u n d s of the f o r m c exp ( - ~ j'x ~
dt/O(t))
can n o t be estab- lished if O oscillates too much. A n example illustrating this is given in w 5.3. Relations between extremal lengths and harmonic measures
This subject dates b a c k to Beurling's thesis [3.2], a n d appears to h a v e been well k n o w n to m a n y m a t h e m a t i c i a n s before an a c c o u n t of it was published b y H e r s c h [3.3].
L e t F be a family of locally rectifiable curves (denoted ?) in a d o m a i n D (i.e.
each c o m p a c t subcurve of a y is rectifiable).
Consider n o n - n e g a t i v e functions 0 in D for which
= L~(F) = inf ( 9 [ d z
[
L0
? Jv
A q =
AQ(D) = f f D o~dxdy
are defined a n d are n o t b o t h 0 or b o t h c~. (For a locally rectifiable ~,
So[dz
[ is defined as the s u p r e m u m of the integrals over subcurves of y.) The extremal length X(F) is defined b yL 2 2(F) = sup -.~.
q Aq
T h e functions Q define a conformal metric b y da = e ( z ) ] d z [ .
T h e definition of extremal length is due to Ahlfors a n d Beurling [3.1, p. 114]. T h e extremal length is a conformal invariant. Thus, if relations between some extremal lengths a n d harmonic measures are k n o w n in, for instance, the circle, these relations can be used to find estimates of h a r m o n i c measures.
I n the following, when discussing a family of curves we shall assume t h a t t h e y are locally rectifiable. L e t D be s i m p l y connected a n d let all points of OD be accessible.
L e t four points be picked on ~D, so as to divide ~D into four parts, ~x,
ill' 0~2' fi2,
in this order. D is t h e n called a quadrangle. L e t F be the family of curves joining
~1 to ~ within D. T h e n
is called the extremal distance between a 1 a n d a s in D.
W e n o w list a few well-known properties of extremal lengths a n d distances [3.1, p. 115], [3.3, pp. 305-308]. W e o n l y discuss extremal lengths different f r o m zero a n d infinity. Considering those 0 for which S~ ff
[dz[
~> 1, for all 7 E F, ;t(P) is defined b y~(F) = sup A~ 1.
Q
Such a ~ will be called admissible with respect to F.
4
AIIKIV F 6 R MATEMATIK. B d 6 n r 1
L e m m a 3.1. ~t(F) is a con/ormal invariant.
L e m m a 3.2. / / I l l C F 2 , then ~t(F1)~>~t(F2).
The first two l e m m a t a follow i m m e d i a t e l y f r o m the definition.
L e m m a 3.3. Let the F k be i n disjoint domains Dk, Ic = 1, 2 . . . n. Let I ~ = {~} be such that each ~ contains at least one ~k r Fk /or each lc, lc = 1, 2 . . . n. T h e n
;~(r)/> ~ 2(r~).
k = l
Proo/. L e t ~)k be admissible with respect to Fk, ]c = 1, 2 ... n. L e t tk, lc = 1, 2 . . . n,
~- n = Y . ~ I tk~k is admissible with respect to be positive n u m b e r s ~ lth ~k=l tk = 1. T h e n
be positive numbers ~ lth ~k=l tk = 1. Then
I~ n *-.7
F. The l e m m a is p r o v e d b y choosing tk =~( k)" (~k=l Jt(Fk)) -1, /c = 1, 2, n.
L e m m a 3.4. Let D be a rectangle with sides o~ 1 and ~2 o] length a, and sides fll and fl~ o/ length b. T h e n
~ ( ~ 1 ' ~2) -- ba-i.
Proo/. L e t D be { z I 0 < R e z < a , 0 < I m z < b } . Then, b y t h e Schwarz inequality,
I n t e g r a t i n g over x we o b t a i n
aL~ <~bAe (3.1)
a n d hence ~(el, ~2) <~ba 1.
E q u a l i t y in (3.1) holds for ~ = b -1. This proves the lemma.
L e m m a 3.5. Let D be a quadrangle with ~D divided into 6r
ill' ~2, f12
i n this order.fll and fi2 are assumed to be analytic arcs and ~1 and ~2 simple arcs. Let u be harmonic i n D with boundary values 1 on ~2, 0 on ~1, and #u/an = 0 on fix and f12. T h e n
2-i(~. ~2)= f f ]grad uI~dxdy.
Proo/. B y L e m m a 3.1 it is sufficient to prove this in the case of a rectangle. L e t
~1 in L e m m a 3.4 be on the real axis. T h e n u - y b =1 a n d the l e m m a is correct.
L e m m a 3.6. Let - denote reflection i n the real axis. Let D be symmetric with respect to the real axis and F such that ~ E F ~ ~ E F. T h e n it is su//icient to consider ~'s sym- metric with respect to the real axis to determine 2(F).
Proo/. L e t ~ be admissible with respect to F. T h e n ~ a n d 1(~ +~) are also admis- sible. F u r t h e r m o r e
5
KERSTI HALISTE~ Estimates of harmonic measures
A - 1 2 1
This proves the lemma.
Next we collect the information that we shall need about elliptic integrals. We assume that 0 < k < 1. Define
and Then
K ( k ) = f ~ ( 1 - x 2 ) - i ( 1 - k 2 x 2 ) - 8 9 K ' ( k ) = K ( ( 1 - k 2 ) ~ ) , (3.2) K'(k)
t(k)= K(k~" (3.3)
t(k)~<~ log ~, (3.4)
7~
t(k) --2 log 4=A(k)]c2;
IA(k)l~c0,
when 0 ~ ] c ~ ] c 0 < l . (3.5) (3.4) and (3.5) follow, for instance, from [3.4, p. 54] and are used b y Hersch [3.3, pp. 316-319].The following theorem giving an explicit relation between a harmonic measure and an extremal length was proved by Hersch [3.3, pp. 319-320].
Theorem 3.1. Let D be simply connected and let all points o~ ~D be accessible. ~D is divided into two connected parts ~ and ft. z o is a / i x e d point in D and co =o~(zo; ~; D).
Let F be the/amily o/ curves in D ~oining points on ~ and separating z o/rom ft. t is de/ined by (3.2) and (3.3). Then
2(F) = 2t (sin 2~~ ) . (3.6)
Proo/. D is mapped conformally onto G, the interior of the unit circle in the w-plane, so that z=z0 corresponds to w = O and to Iwl = 1 } . f i l : ~ G - ~ 1 and ~ l - { w l - 1 < R e w ~ 0 , Im w - 0 } . F 1 is the family of curves 71 in G joining points on ~1 and separating w = 0 from ill. C is the family of curves c joining
~1 and zq in G I = G - ~ I .
B y Lemma 3.6 it is sufficient to consider ~ symmetric with respect to the real axis to determine ~t(l~l) and ~t(C). A curve 71 E I~I contains two curves c' and c" in C.
Let - denote reflection in the real axis. Then, for a symmetric Q,
Hence and
L o(F1) = 2Lq(C)
~(F1) = 4~(C). (3.7)
ARKIV FOR MATEMATIK. B d 6 n r 1
Pxe
Z =~
Fig. 3.1 Fig. 3.2
2(C) =2G,(~1, ~1) is now determined by a conformal mapping of the quadrangle G 1 onto a rectangle. By virtue of conformal invarianee the theorem follows.
The following corollary will be useful in the next paragraph.
Corollary 3.1.
Let D~ (containing z0) satisfy the assumptions of Theorem 3.1 with 0~ = ~ and let ~D~be piecewise smooth. D~ is mapped conformally onto G = {w I I w I < 1}
so that
z = z ocorresponds to w = 0 and 0~ to { w [ - 7 ~ < a r g w<zee), Iw] =1}, where co=co(z0; 0~; D~). ~x denotes {w[0~<Re w < l , I m w = 0 } and g is the image of ~ l i n D~. Let u be harmonic in D~-~r with boundary values 1 on ~, 0 on OD~-0~, and
~u/On
= 0 on 0~ (except at the endpoint of u). Then
2t(sin2)=ff~lgradul~dxdy. (3.8)
Proo/.
We consider G. Then by (3.6) and (3.7)
1 -- 1 1 .
~c-~,(zl, fl~)=2t(sin ~ ( 1 m ) ) = 2 t - ( s i n ~ - ) The corollary now follows by Lemma 3.1 and Lemma 3.5.
Corollary 3.2.
With the notation of Theorem 3.1 o)(z0; a; D)~<4 exp ( - 4 J t ( l ~ ) ) .
Proo/.This follows from {3.6) and (3.4).
Theorem 3.2.
Let D be s i m p l y connected and let all p o i n t s o~ ~ D be accessible. A s s u m e that D has no boundary p o i n t at i n / i n i t y w i t h / i n i t e R e z. Let D~ (containing z o = x o + iyo) be such that O~ consists o/ one segment. Let O~x, i = 1, 2 . . . n(x), separate z o and 0~.A s s u m e that n(x) = O, x < x o. T h e n
KERSTI HALISTE~ Estimates of harmonic measures
/'~ /~(~) 1 \ \
- ~ - - d x
(A)(ZO;O';D')~4exp :7~Jxoti=lO~(x) ) )"
Proo/. The 0~ cover the shaded area in Fig. 3.2. L e t F be the f a m i l y of curves in D~ joining points on 0~ a n d separating z 0 f r o m ~D~-O~. W e use Corollary 3.2 with :r = 0~ a n d D = D~. T h u s
A simple estimate of 2(P) is o b t a i n e d b y the following choice of ~:
O~(x)' z ~ Cx, i = 1, 2 , . . . , n(x), xo < x <
~(z)
0 otherwise.
W e note t h a t each 0f (x) is lower semicontinuous. N o w
~t(F) >~ r ~ a 1 ) 4 dx,
a i
a n d the t h e o r e m is t h u s proved.
Remark 1. T h e possibility of such a choice of metric (in the case of n ( x ) - - l ) was n o t e d b y Hersch [3.3, pp. 325-326].
Remark 2. L e t the D in T h e o r e m 3.2 have a b o u n d a r y point B such t h a t Re z - ~ § ~ , z - ~ B , z E D. L e t 0~, i = 1, 2 ... n(x), separate z 0 a n d B. D~ is the s u b d o m a i n of D separated from B b y 0~. W e drop the a s s u m p t i o n t h a t n(x)=0, x < x o. L e t I be the interval generated b y the 0~ separating z 0 a n d a fixed 0~. T h e n
, 1
where ' means t h a t the s u m is to be t a k e n over the 0~ separating z 0 a n d 0~.
Remark 3. The estimate of o)(%; 0~; D~) in T h e o r e m 3.2 in terms of the lengths of the 0~ separating z 0 a n d 0~ can be generalized to higher dimensions, cf. T h e o r e m 6.2.
Remark 4. I t does n o t a p p e a r possible to e x t e n d the m e t h o d of w 3 to higher dimen- sions. I n T h e o r e m 3.1 )~(F) differs little from 42D(~ 1, ~), w h e n D is a quadrangle with two opposite b o u n d a r y arcs :r a n d :r such t h a t the distance between al a n d :r is large in comparison with the length of ~ a n d z 0 is near :% N o w let G be a vertical right cylinder of height b a n d let t h e area of the two horizontal sides ~1 a n d :r be A.
Then, in a n a l o g y with L e m m a 3.4, 2C(al, ~ ) = b A - 1 . However, the h a r m o n i c measure of ~ with respect to G depends n o t only on the size b u t also on the shape of a cross- section of the cylinder.
ARKIV FSR MATEMATIK. Bd 6 nr 1
4. Symmetrization results for harmonic measures
S y m m e t r i z a t i o n with respect to a straight line is defined in w 1. T h e following two theorems are p r o v e d with the aid of Corollary 3.1. A n o t h e r m e t h o d to prove sym- m e t r i z a t i o n results is dicussed in w 8.
Theorem 4.1.
Let D~ (containing %) be bounded by a piecewise smooth simple closed curve. Let O~ consist o/one segment. * denotes symmetrization with respect to the real axis. Then
m a x ~o(z0;
0~; D~) <~ W(Xo; 0~; D~).
R e Zo=Xo
Equality holds i/and only i/ D~ is a translate o] D~.
Proo/.
W e use the same n o t a t i o n as in Corollary 3.1. L e t u be the image of ul in D~. The slit u'={zlx0-~<Re z < ~ , I m z - 0 } is the image of ~1 in D~. L e t u be har- monic in D ~ - u with b o u n d a r y values 1 on u, 0 on~D~-O~,
a n d~u/~n=O
on 0~(except at the e n d p o i n t of u). L e t v be h a r m o n i c in D~ - u with b o u n d a r y values 1 * ' on u', 0 on
~D~-O~,
a n d~v/~n =0
on 0~ (except at the endpoint of u'). T h e n b y (3.8),2t(sin 2co)=ffD~ ~'gradu]2dxdy,
2 t ( s i n
7~2- **)=frO. ~ ,,Igradvl~dxdy.
where co = to(z0; 0~; D~) a n d ~o*=
to(xo; 0~; D~).
L e t u be s y m m e t r i z e d with respect to the real axis. The s y m m e t r i z e d function is u*. u corresponds to ~* on t h e real axis so t h a t u * = 1 on z*. I t is possible t h a t u*
extends to the left of x0. W e n o w reflect D~ a n d D~ in l = {z IRe z = ~} a n d d e n o t e the reflected domains b y /3~ a n d D~. Set G = D~ U 0~ U D~ a n d G* = D~ U 0~' U/)~. ~, ~', a n d u* (including endpoints) are also reflected in l, a n d s, s', s* denote the unions of the given slits a n d their reflections. B y reflection in l u is defined to be h a r m o n i c in G - s a n d v is defined to be harmonic in G * - s ' . T h e d o m a i n of u* is also e x t e n d e d in this way. According to a result of P61ya a n d Szeg5 [4.4, p. 186-187]
f f a_slgrad ul2dxdy>~ f f o, z. lgrad u*l~dxdy.
(4.1) W h e ns* - s' 4 r ~v/~n
= 0 on s* - s'. B y Dirichlet's principle (with free b o u n d a r y values)ffo. lgrad *l dxdy> (( Igradvl2dxdy
J J G * - s '
and hence
/fa ~'gradu]2dxdy>~ff.,-s, igradvl2dxdy"
(4.2)F r o m this we o b t a i n
KERSTI HALISTE, Estimates of harmonic measures
Fig. 4.1 Fig. 4.2
t sin :re) >it sin~re~ .
Since t(k) is strictly increasing, 0 < k < 1, we obtain the result eo ~<~o*.
The Dirichlet integrals in (4.2) are the inverted values of the modules of the doubly connected domains G - s and G*-s'. These modules are (without restrictions in the b o u n d a r y assumptions) equal if and only if G - s is a translate of G* - s ' . This follows from a result b y Jenkins [4.2, p. 106, p. 115]. Theorem 4.1 is now proved.
Remark 1. We mention another possibility of discussing equality in Theorem 4.1.
This is to make a detailed examination of a proof of (4.1) along the lines of a proof given in [4.1, pp. 416419]. Such an investigation was made b y Ohtsuk~ [4.3, p p . 202-205] in the case of circular symmetrization (cf. the following remark). B y the result of Jenkins above the case of equality can be settled for more general domains.
Remark 2. Theorem 4.1 can also be formulated for circular symmetrization with respect to the positive real axis. To define circular symmetrization, in the definition of symmetrization in w 1 straight lines are replaced b y circles with their centres at the origin [4.4, pp. 193-195].
Theorem 4.2. Let D~ (containing z o-xo) be bounded by a piecewise smooth closed curve. Let O~ consist o/ one segment. D~ is assumed to be symmetric with respect to the real axis. D~ is reflected in l: Re z =~. The reflected domain is f)~ and G = D~ [J O~ U b~.
G is symmetrized with respect to l, the symmetrized domain being G*; O~ ={zlzEG*, Re z = ~ } and n ~ = { z l z e G * , Re z<~}. Then
co(x0; 0~; D~)~< oJ(x0; 0~; D~) with equality i/ and only i/Dr = D~.
Proo/. W e use Corollary 3.1 and its notation again. L e t ~ be the image of ~1 in De.
L e t u be harmonic in D r with b o u n d a r y values 1 on x, 0 on ~ D e - 0 r a n d
~u/~n=O on 0~ (except at z = ~ ) . Set ~ * = { z l I R e z - ~ e I ~ < ~ - x o, I m z = 0 } . B y reflection in l, u is defined to be harmonic in G - u*. L e t v be harmonic in G * - ~ * with b o u n d a r y values 1 on u* a n d 0 on ~G*. Then b y (3.8)
10
ARKIV FOR MATEMATIK. B d 6 n r 1
[ zt~o *k
t ( s i n ~ ) =
fro
. . . . ]grad vledxdy, where o ) = co(z0; 0~; D~) a n d co*= co(x0; 0~; D~).L e t u be s y m m e t r i z e d with respect to 1. T h e s y m m e t r i z e d f u n c t i o n is u*. We t h e n o b t a i n in the same w a y as in the proof of T h e o r e m 4.1
f f G_ I grad u dxdy f f o.
~, I g r a d v l2 dxdy.T h e proof can n o w be completed in the same w a y as in the proof of T h e o r e m 4.1.
5. A n application to comb domains
L e t a simply connected d o m a i n D satisfy t h e following conditions. Ox =~b outside A < x < B ( - ~ < A < B ~ < ~ ) . O x - { z l R e z = x , I I m z l < ~ } for all x in A < x < B except x - Xm. T h e n u m b e r of points x,, in a finite interval is finite. E a c h Oxm consists of one b o u n d e d line-segment. W e t h e n call D a comb domain.
First we m e n t i o n an explicit example illustrating w 2. L e t D be b o u n d e d b y t h e straight lines {z ]Re z = xm = - 2ma, I I m z I >~ b}, m = 1, 2 ... (a > 0) a n d the i m a g i n a r y axis, w h e r e a = {z I R e z = 0, I I m z [ < b}. ~(z; ~; D) can be d e t e r m i n e d explicitly when z=xm. W e write exp ( - ~ b a -1) = k a n d use the n o t a t i o n in (3.2) a n d (3.3).
B y a conformal m a p p i n g of {z I - 2a < Re z < 0} o n t o a rectangle {w ] ] Re w ] < kK, 0 < I m w < kK'} in the w-plane a n d b y analytic continuation, we o b t a i n a conformal m a p p i n g of D onto a strip { w i R e w < k K , 0 < I m w < k K ' } . Hence
~o(x,; a; D)=_4 a r c t g exp zx~
at(k)"
W h e n a---~0, the t e r m on the right tends to 4~ -1 arctg exp (7tx~/2b), by (3.5).
N o w let G be a d o m a i n such t h a t Ox = {z I Re z = x, I I m z I < O ( x ) / 2 } , - ~ < x < 0 , a n d O ~ = r x~>0 L e t inf O ( x ) = 2 b be a t t a i n e d at t h e points xm, m = 1 , 2 ... T h e n eo(xn; ~; G) <<. ~o(x~; ~; D). F o r small values of a, the a b o v e estimate for eo(x~; a; G) can, for suitably chosen O(x), be considerably smaller t h a n c exp ( - ~ S~
Theorem 5.1. Let the comb domain D be bounded by the lines {z I R e z=xm,
I
I m z ~>b~}, m = l , 2 ... 0 > x l > x 2 > .... and the imaginary axis. Set ~ = { z l R e z = 0 , I m z ~<b0} , O,n={ZlI:tez=xm, I I m z l < b m } , and b i n = m a x (bm, brn_l), m = l , 2 . . . Given x, let n be such that xn+ 1 < x <~x~. Assume thatm--~ \
b'~ !
Then there is a constant c such that
( XmlX.)
~o(x; ~; D)~<c exp - ~ 2b~ "
1
11
KERSTI HALISTE~ E s t i m a t e s o f h a r m o n i c measures
Proo/. ~o=~o(x; ~; D). Let F be the family of curves in D joining points on a n d separating the point z = x from ~ D - g. B y Corollary 3.2
~o~<4exp - i 2 ( F ) .
Let C be the family of curves in { z ] z E D , Re z > x n } joining On and ~. B y the same reasoning as in the proof of (3.7),
2(r) >~4~(C).
We now estimate ~t(C). Set x 0 = 0 and O 0 = ~ . Let Dm denote {zlxm<Re Z<Xm_l} ~ r e = l , 2, .... By L e m m a 3.3
),(C) ~ ~ ~Dm(Om,
Om-1)- /'n=lSet O~ = (z I R e z = xm, I I m z I < bZ} and O r e - 1 = {Z ] Re z = xm-1, I I m z I < bZ}, where b~ = m a x (bin, b~-l), m = 1, 2 .... , n. B y L e m m a 3.2
2(O~, O~ 1) is determined explicitly b y a conformal m a p p i n g of D~ onto a rec- tangle. We write
i 2 b: 1
km exp\ xm- l - Xm/
and define tm b y (3.2) and (3.3). Then
2Din(Ore,
O~n-i) = 2tin 1]~y (3.5) 2tm 1 -
Xm-1 -- ~ xm-1 ~ Xm~2
where IBm I is less t h a n a constant depending only on M.
Remark. If, in Theorem 5.1, 2bm=O(x,~) for a suitable continuous O(x), we can, b y a limiting process, obtain a special ease of Theorem 3.2.
Chapter II. The general case 6. C a r l e m a n ' s m e t h o d
This method was first used b y Carleman in 1933 in a proof of Denjoy's conjecture concerning the n u m b e r of finite a s y m p t o t i c values of an integral function of finite order [6.1]. Denjoy's conjecture had been p r o v e d earlier b y Ahlfors who used his distortion inequality (2.3). An account of Carleman's method in two dimensions 12
ARKIV FOR MATEMATIK. B d 6 n r 1
is given in two text-books [6.4, pp. 219-224] and [6.6, pp. 121-126]. The growth of harmonic and subharmonic functions of n variables has been investigated b y several authors with Carleman's method. Besides the references mentioned here in the text see [6.9]-[6.32] in the bibliography. I n these investigations the main problem has been to study the growth of a given function; constants appearing in the estimates are allowed to depend on the function under consideration. However, a direct appli- cation to harmonic measures was given b y Tsuji [6.8, pp. 112-117] in polar coordi- nates in the plane, cf. L e m m a 6.7.
Carleman's method consists in establishing a differential inequality for the Carle- m a n mean (6.1) of a harmonic function. This can also be interpreted as a differential inequality for a certain Dirichlet integral. I n Theorem 6.1 Carleman's method is applied to harmonic measures. The proof of Theorem 6.1 follows--with some altera- t i o n s - - a proof given b y Dinghas [6.3, pp. 3-9]. We use Theorem 6.1 and some lem- m a t a to establish the estimates of harmonic measures in Theorem 6.2 and Theorem 6.3. I n the case n > 2 these estimates are new.
We assume t h a t D satisfies the condition A below.
A. D is such t h a t 0 ~ = r for x ~ 0 a n d O ~ q ~ for x > 0 . D is bounded b y a finite n u m b e r of piecewise smooth surfaces. D has no b o u n d a r y point at infinity for which x I is finite. O(x) is bounded and ~< M for all x > O. Set 0 0 - {z] x 1 = 0, z E ~D}. The mea- sure of 00 is positive.
To begin with, we collect information a b o u t the principal eigenvalues in B.
B. L e t the domain G in R n be bounded b y a finite n u m b e r of piecewise smooth surfaces. L e t V be the class of functions / such t h a t
( 1 ) / is continuous in G U ~G and piecewise continuously differentiable in G, (2) /(z)=O, z ~ a ,
(3)
/(z) =~ o.Consider the variational problem of minimizing in V the Rayleigh quotient f a l g rad
[ [2dz
R(/)= " f f dz
Let v be the first (normed) eigenfunction of
Av+2v=O
in G, v = 0 on ~G, and 2 the principal eigenvalue. I t is well known t h a t the Rayleigh quotient is minimized in V b y v and t h a t the m i n i m u m is )~ [6.2, p. 399]. ~t decreases when G increases [6.2, p. 409]. ~t varies continuously with G [6.2, p. 423].Furthermore 2 decreases when G is symmetrized with respect to an ( n - 1)-dimen- sional hyperplane (cf. w 1) [6.5, p. 419]. We also mention the F a b e r - K r a h n inequality.
L e t A be the volume of G. Let
Vn
be the volume a n d An the principal eigenvalue of the n-dimensional unit sphere. Then [6.5, p. 413]A 2 1 n ~ ~ A TZ21n
For a domain G with a less regular boundary, we define A as infn 2, where ~ is of the t y p e considered above and ~ c G.
13
KERSTI tIALISTE,
Estimates of harmonic measures
L e t z 0 = (x0, Y0) be a f i x e d p o i n t in D. W e w r i t eu(z)
= oJ(z; 0~; D~).W e shall e s t a b l i s h u p p e r b o u n d s for
u(z)
a t z = z 0 b y s t u d y i n g t h e C a r l e m a n m e a n ~, d e f i n e d b y/ a
~ ( x ) = J~l~
y)dy,
0 < x < ~ . (6.1)T h e d e f i n i t i o n of ~ is c o m p l e t e d b y s e t t i n g ~v(O)=0 a n d ~v(~)=0(~).
C. W e define 2(x) in t h e following w a y . Consider a d e c r e a s i n g sequence, {e}, of p o s i t i v e n u m b e r s w i t h l i m i t zero. W e choose t h e e so t h a t g r a d u ~ 0 on t h e surfaces u - e in D~. (The n u m b e r of v a l u e s a for w h i c h g r a d u v a n i s h e s a t p o i n t s of t h e e q u i p o t e n t i a l set u - a in D~ is e n u m e r a b l e [6.7, p. 276].) W e w r i t e u~ = m a x ( u - e , 0 ) a n d s e t
~)qz,~--.{ZlXl--X,
u ~ ( z ) > O } . ~x.~ consists of a f i n i t e n u m b e r of c o m p o n e n t s#i . . . . To e a c h 0 ~ x. ~ we define ~ . ~(x) a c c o r d i n g t o B. F i n a l l y we define
~ ( x ) = m i n 2t. ~(x), ~ ( x ) = lim 2~(x).
i e--~O
T h e e x i s t e n c e of ~t(x) will be e s t a b l i s h e d in t h e p r o o f of T h e o r e m 6.1.
L e t ~t be d e f i n e d b y (~. T h e n we define ~v b y
W e can n o w s t a t e
(6.2)
T h e o r e m 6.1.
Let D satis/y the assumptions A. Let q~ and ~ be de/ined by
(6.1)and
(6.2).Then q) is a convex/unction o/y)/or O ~ x < ~ and
~(x) <~(x)0(~)w-l(~).
(6.3)To p r o v e T h e o r e m 6.1 we c o n s i d e r
q~(x) = ,J~~ ~ u~(x, y) dy.
I n t h e following l e m m a t a 6.1-6.4 we d r o p t h e i n d e x e. T h i s m e a n s t h a t we w o r k u n d e r t h e a s s u m p t i o n s t h a t u = 0 on
~Dr
a n d t h a t u is h a r m o n i c in a n e i g h b o u r h o o d of e a c h p o i n t of ~D~ for w h i c h x 1 < ~. A c t u a l l y v~x. ~ = r for x ~< x~, v~z, ~ # r for x > x~ > 0, b u t w h e n d r o p p i n g t h e i n d e x e we also w r i t e 0 i n s t e a d of x~.L e m m a 6.1.
O(x) is continuous and ,~(x) is upper semicontinuous, 0 < x <~.
Proo/.
F o r s i m p l i c i t y we t r e a t o n l y t h e case n - 3 . S u p p o s e t h a t#(x)
is discon- t i n u o u s a t x - 0 . T h e nS - {zlu(z )
= 0) c o n t a i n s a surface e l e m e n t of p o s i t i v e m e a s u r e in t h e p l a n e x 1 = a . T h i s is i m p o s s i b l e , since a n a n a l y t i c surface h a s a t m o s t a f i n i t e n u m b e r of p o i n t s in c o m m o n w i t h a n y s t r a i g h t line (not c o n t a i n e d in t h e surface).T h u s #(x) is c o n t i n u o u s . 14
A R K I V FOR MATEMATIK. Bd 6 n r 1
When the plane x 1 = a is not tangent to S, each component v~ of # , is bounded b y a finite n u m b e r of analytic curves, and ~(a) is the principal eigenvalue of #~.
I f the plane x~ = a is tangent to S, ~#~ m a y for instance contain isolated points. I n such a case ~t~(a) is t a k e n to be infant, where ~ is contained in v% and ~ consists of a finite n u m b e r of analytic curves. T h e n ~i(a)= inf
R(f)
for / s V in #~.Given a n u m b e r A >~(a) =~((~), we can choose g2 so t h a t A is the principal eigen- value of ~ and ~2 is strictly contained in #~. Then ~ is contained in some component of #x for all x sufficiently near (~. Thus A >~(x) for all x sufficiently near a and ~ is upper semicontinuous.
L e m m a 6.2. ~ ' ( x ) = 2
~uu~dy,
0 < x < ~ .Proo/.
Let #~\v% be the set of points y belonging to v~ b u t not to v%. Then~(x + h)-q~(x)= fo~+ u2(x+ h, y)dy- fo u~(x, y)dy= fo (u2(x+ h, Y)-U2(x, y))dY
+ ( u (x+h,y)dy - ( 2(x+h,y)dy=I +Z2+I .
J,~+h\'~ J ~\'~+~,
D i v i d i n g I 1 b y h and letting h tend to zero we obtain 2
So uu~ dy.
Now consider 12 for small h. To each point (x + h, y), y E#x+h\#x, belongs some point (s, y), x ~ s ~ x § h, such t h a tu(s, y)=0.
As the measure of#x+h\# x
is O(1), 12 isO(h2).
I n the same w a y I a isO(h2).
This proves the lemma.L e m m a 6.3. ~"(x) = 2]o~ [grad u ]2
dy, 0 < x < $.
Pro@
B y L e m m a 6.2 and Green's formula'(x)=2f (f lgradnl2dy)dx
The integrand S ~ ] g r a d
ul2dy
being continuous b y L e m m a 6.1, we obtain our lemma.L e m m a 6.4.
( Carleman' s di//erential inequality.)
, i >~ I 89
cf (x) ~ 2q~ (x) 2 (x), 0 < x < ~.
Proo/.
B y L e m m a 6.3q/'(x)= 2 fo u:dy + 2 f~ ]gradyu]2 dy, O <x < ~.
The first integral is estimated b y applying the Sehwarz inequality to ~'(x) in L e m m a 6.2. Thus
f U2x dy q9
(6.4)
~ 4~v
15
KERSTI HALISTE, Estimates of harmonic measures T h e second integral is e s t i m a t e d b y B.
f~, lgrad~ul2dy>~ 2~(x) f , u2dy>~ 2(x) fo: u~ dy.
S u m m i n g o v e r i we o b t a i n
f ~ I grad~ u ]2 dy >12@) of(x). (6.5)
t t t 2
H e n c e 2~ q >~ ~ + 44, 0 < x < ~. (6.6)
B y the proof of L e m m a 6.3 ~v' is a Dirichlet integral a n d t h u s positive for 0 < x < ~ . B y the inequality (~v"/~'-~'/T)2>~0 a n d t a k i n g the square root the l e m m a now follows f r o m (6.6).
P r o o / o / Theorem 6.1. W e n o w use the index e again. B y B a n d L e m m a 6.1 {2~}
is a decreasing sequence of integrable functions. 2 89 = lim 2~ is integrable o v e r 0 < x < ~.
( I n t e g r a b i l i t y a t x = 0 is g u a r a n t e e d b y the a s s u m p t i o n t h a t the m e a s u r e of 00 is positive.) B y L e m m a 6.4
~7(x) >/2~(x) 2~(x) >i 2~(x)2~-(x), x~ < x < ~.
F o r ~0 defined b y (6.2)
~o"(x) = 2289 ~'(x) a.e.
H e n c e dx (log ~ - log ~p') ~> 0 a.e. d
Since log ~v: a n d log ~v' are absolutely continuous on a n i n t e r v a l a ~< x ~< fl, x~ < a < fl < ~,
~v~ is t h u s a c o n v e x function of v 2 on a n y such interval. B y c o n t i n u i t y a t x = 0 a n d x - ~ , ~ - l i r a ~v~ is a c o n v e x f u n c t i o n of ~ for 0 ~ x ~<~. T h u s (6.3) is t r u e a n d T h e o r e m 6.1 is proved.
Remark. ( A s y m p t o t i c e q u a l i t y in C a r l e m a n ' s differential inequality.)
E q u a l i t y in (6.4) holds if a n d only if u =/Ux, where / depends only on x. W h e n t9 x is connected a n d ~z9 x smooth, e q u a l i t y holds in (6.5) if a n d only if u = g v , where g d e p e n d s only on x a n d v is the first eigenfunetion of A v + 2 v = O in v~x, v=O on ~v~z.
N o w let D~ be a right cylinder, D~ = {z Ix 1 <~, y E O}, where 0 is s i m p l y connected a n d has a s m o o t h b o u n d a r y . L e t {~n}~ be the eigenvalues (21=4) a n d {v~}[ r t h e corresponding eigenfunctions of Av + 2 v = 0 in O, v = 0 on ~ ) . Then, b y the m e t h o d of s e p a r a t i o n of variables (the c~ denoting constants)
u(x, y) = ~ c n e x p (2~(x - ~))vn(y).
1
Thus, for large n e g a t i v e values of x - ~,
u(x, y) ~ c 1 exp (289 - ~)) vl(y ), 16
ARKIV F6R MATEMATIK. B d 6 n r 1
ux(x, y) "~ c x h ~ exp (21(x - ~)) vl(y).
% as well as q~, is n o w twice differentiable, a n d a s y m p t o t i c e q u a l i t y in (6.4) a n d (6.5) holds for q.
I n s t e a d of defining ~v as an integral over v~x as in (6.1) we n o w define ~ as an integral over 0x. I n order to estimate ~ in this case we introduce the following additional assumptions a b o u t D.
D. Ox is connected, x ~< x 0. ~D is smooth. I n a n y finite interval ~D has a finite n u m b e r of t a n g e n t h y p e r p l a n e s x 1 = c .
E. X(x) is defined as in C, b u t with respect to 0z instead of Vqx . (For x <~x o Oz is under- stood to be equal to O~).
L e m m a 6.5. Let D satis/y A and D. T h e n (6.3) in Theorem 6.1 is true ]or x - x o with % ~, and y~ defined with respect to Ox instead o/ ~9~.
Proo/. The choice of 0~ is due to Tsuji [6.8, p. 112]. I n the case of simply c o n n e c t e d 0~ a n d D~ he takes into a c c o u n t only the first of the 0~z separating 00 f r o m 0~, cf. Ahlfors' distortion inequality (2.3). However, some irrelevant c o m p o n e n t s of z9 x n o t separat- ing 00 f r o m 0~ also enter the discussion. I n Fig. 6.1 t h e 0~ cover the shaded area.
L e t x 1 = a be those hyperplanes t a n g e n t to ~D for which D , is a proper s u b d o m a i n of D~+0.
L e t t h e a belonging to the interval 0 < x < ~ be a l < a S <... < a ~ . T h e n
~ ( ~ ) < ~ ( ~ . + 0), ~ ( ~ - 0) < r + 0), ~ = 1, 2 ... m.
T h u s ~ a n d d%/dyJ h a v e positive j u m p s at x - a p , t t = l , 2 ... m. According to the proof of T h e o r e m 6.1 q~ is a convex f u n c t i o n of yJ on intervals n o t containing a n y of
the a. Thus, for x~<X<Xo<~a ~,
~v~(x) < 0 ( ~ ~(x) - ~(x~)
"~" ~(~) - ~ ( x ~ ) "
The l e m m a n o w follows b y letting s t e n d to zero.
L e m m a 6.6. u(zo)~<c~t(x0), where z o - ( x o, Yo) and c depends only on the geometry o/
D near %.
Proo/. This l e m m a is an i m m e d i a t e consequence of H a r n a c k ' s inequality for positive h a r m o n i c functions. I f a closed sphere S(r; %) with radius r a n d centre a t z o is c o n t a i n e d in D, t h e n for z ES(r/2; %)
u(z) >1 (~)~-~ U(Zo),
a n d hence q~(x) >1 Cn r n- lu2(zo),
where c~ o n l y depends on n. This proves t h e lemma.
W e n o w discuss estimating ~ f r o m
~(X) ~< ~/)(X) 0(~) ~0--1(~).
2:1
(6.3) 17
KERSTI HALISTE,
Estimates of harmonic measures
f
f
Fig. 6.1 W r i t i n g tt instead of 22 89 we h a v e instead of (6.2)
;0 (; )
v2(x ) = e x p
~(u) du dt.
Since
O(x) <<- M,
it follows b y t h e F a b e r - K r a h n inequality t h a t #(x) >~ m > 0. H e n c eW(x)=exp (f~(u)du) f~exp (- f(~(u)du)dt<m l exp (f(~(u)du). (6.7)
I n t h e s a m e way, a s s u m i n g t h a t / ~ ( x ) ~<
ml,
we o b t a i n for ~ > m l 1 log 2( ; )
~0(~ e) >~ (2ml) -1 e x p
#(u) du
a n d hence
q~(x)<~cexp(f]#(u)du).
(6.8)I t is, of course, u n s a t i s f a c t o r y to require t h a t ~ be bounded. I n general the e s t i m a t e (6.8) does n o t follow f r o m (6.3). F o r instance t a k e
O(x)~O(x)~l
and/~(x)--=2x, x > 0 . Then, for large ~o(~) ~ ( 2 ~ ) - l e x p ~2. A general result is given in the following l e m m a . L e m m a 6.7.Let D satis/y A and ,~ be de/ined by C. Then/or 0 ~ 1 < ~ - xo, z o ~ (xo, Yo),
U(Zo)<~cl
8 9 ( - f x ' ~ ' ( x ) d x ) ,where the constant c depends only on A and the geometry o / D near %. I / D also satis/ies D this is true with ~ de/ined by E.
Proo/.
W e use a trivial e s t i m a t e of y~(~). W r i t i n g # = 22 89 we h a v e for l > 0y~(~)>~exp ( ; - I t t ( u ) d u ) ; lexp ( ; _ z #(u)du)dt>~l exp ( ; - Z # ( u ) d u ) .
18ARKIV FOR MATEMATIK. B d 6 n r 1
Hence, b y (6.3) and (6.7)
Our l e m m a now follows from L e m m a 6.5 and L e m m a 6.6.
Remark. Tsuji proved this result in polar coordinates in the plane [6.8, pp. 112- 117], b u t with a more complicated derivation of the estimate of ~.
I n Theorem 6.3 we shall estimate q~ from (6.3) in some special cases. First, however, we shall m a k e the same choice among the components of ~ as in Theorem 3.2 with Carleman's method. B y L e m m a 6.2 and Green's formula ~ / 2 is a Dirichlet integral.
B y working in terms of convexity in the proof of Theorem 6.1 difficulties due to infinite Dirichlet integrals are avoided. Now we have to introduce auxiliary functions possessing finite Dirichlet integrals. We shall assume t h a t D satisfies D and D~ F.
F. 0~ is connected, and the diameter of D ~ - D~_ i is less t h a n a fixed constant.
G. Given ~, let 0~, i = 1, 2 ... n(x), denote the components or unions of com- ponents of v~, separating 00 from 0~. ~t~(x) is now defined with respect to 0~z in the same w a y as was 2(x) with respect to v~, in C. We write
n(x)
~89 = V ~ ( x ) .
i = 1
L e m m a 6.8. Let D satis/y A and D and let D~ satis/y F. Let ,~ be de/ined by G. Then /or z o = (xo, Yo), ~ > xo + 1,
u(zo) <~c ex p
(-
(6.9)where the constant c depends only on A, F, x0, and the geometry o / D near %.
Proo/. We first modify D~. Since, b y F, the diameter of D ~ - D ~ _ i is bounded, each v~x with ~ - 1 ~<x~<~ is contained in an ( n - 1 ) - d i m e n s i o n a l sphere of fixed radius R a n d centre Y. Set G ~ = { z l ~ - l < x l < ~ , l Y - Y I < R } . We consider D~tJ G~ instead of D~, b u t do not change our notation. Thus O~={z Xl=~, y - Y < R } . Set O'~={zlxl=~, l y - Y l < R / 3 } and O'~'={zlx~=~, y - Y <2R/3}. We choose a function F, twice continuously differentiable and monotonic for 8 9 with
F ( 1 ) = I and F(~) =0.
Now let / be harmonic in D~ with b o u n d a r y values 0 on ~D~-O~', F(R-11yl) on 0~'-0~, and 1 on 0~. We shall need an inequality of the following type: :
u(z)<~c/(z). (6.10)
Applying the method of separation of variables in the cylinder G~, we obtain t h a t u(z) a n d / ( z ) tend to zero in the same w a y when z Ev~_89 tends to ~v~_89 Therefore we can, with the aid of H a r n a c k ' s inequality, establish the inequality (6.10) on v~ i and hence in D~_89
Let D~. denote the subdomain of D separated from 0~ b y 0ix. We assume t h a t the O~z, i = 1, 2 ... n(x), are taken in such order t h a t D~xc D~ +i, i = 1, 2 ... n ( x ) - 1.
19
KERSTI HALISTE,
Estimates of
h a r m o n i c m e a s u r e sW e n o w define
a n d
Dr ~ = (z I z e D~, /(z) > e}, O~x., = {z ] z e O~x, /(z) > e},
D~x., = {z ]z e D~x, l(z)
> e}, 1~ = max (l - e, 0), W,.~(x) =f,,,., ]/~]2dy,
d,. ~(x) = fD'~., I grad /" l" dz"{e} is t a k e n to be a sequence of the t y p e considered in C (with respect t o / ) .
L e t x 1 = a be those h y p e r p l a n e s t a n g e n t to a D for which one of the following situa- D~+0, a n d di. ~(a) ~<di. ~(a § tions occurs for some i. I. Di, is a p r o p e r s u b d o m a i n of i
I I . D~ is a p r o p e r s u b d o m a i n of D , - o , a n d
di.,(a)<~d~,~(a-O).
I I I . T h e interior of D~'+I- D~, contains no 0~ (j = 1, 2 ...n(x)),
a n d d,. ~(a)~<d,+l,~(a). Cf. Fig. 3.2.On a n interval I (not containing a n y points a) where Dtx increases w i t h x 2di. ~(x) = ?~(x)
b y L e m m a 6.2. L e t 25 be defined b y G. B y v i r t u e of L e m m a 6.4 we o b t a i n
di'.,(x)/> 22~(x)d,. ~(x). (6.11)
On a n i n t e r v a l J (not containing a n y points a) where D~ increases with - x we o b t a i n in the s a m e w a y
- d~'. ~(x) ~> 22i (x) d~, ~(x). ~ ( 6 . 1 2 ) N o w we r u n t h r o u g h D~_ 1 f r o m z 0 to 0~_1 so t h a t each 0~ is passed once. T h e n t h e
i
Dx. ~
on intervals I a n d J according Dirichlet integral of/~ overDx, ~
increases with tto (6.11) a n d (6.12). A t the points a the Dirichlet integral of/~ has n o n - n e g a t i v e in- c r e m e n t s as described in I - I I I . W h e n running t h r o u g h D~_ 1 in this m a n n e r we i n t e g r a t e in (6.11) a n d (6.12). W r i t i n g d~ i n s t e a d of dl, ~ we o b t a i n
d~(xo) ~ d~(~ -
1) e x p ( - 2 [ ~ - 1in(x) \
\
:By G r e e n ' s f o r m u l a (with inner n o r m a l derivatives) we o b t a i n
w h e n ~--> 0. :By the m a x i m u m principle
-- fo / ~ d Y < - - foJeSudY,
where g is h a r m o n i c in G~ w i t h b o u n d a r y values / on 0~ a n d b o u n d a r y values 0 on
~ G ~ - 0~. H e n c e for sufficiently small e, d~(~) ~< %.
F i n a l l y we estimate/(Zo). Since 2d~ = ~ is increasing,
r -~. x o cp~(xo) <~ c d~(xo).
20
ARKIV FOR MATEMATIK. B d 6 n r :1 W e let e t e n d to zero a n d t h e n estimate/(Zo) b y L e m m a 6.6. T h u s we o b t a i n (6.9) for f(z0). B y (6.10), (6.9) is also valid for u(zo) in the modified d o m a i n a n d hence for t h e original U(Zo)=cO(Zo; 0~; De).
Remark. T h e original d o m a i n D e was modified so as to 1 ~ facilitate defining auxi- liary functions /, 2 ~ assure the boundedness of d~(~), a n d 3 ~ allow estimating u in terms o f / . I n the two-dimensional case we can instead begin b y considering d o m a i n s D b o u n d e d b y a finite n u m b e r of a n a l y t i c curves. / can n o w be defined in the original d o m a i n D e as above. Discussion o f / ~ is unnecessary. W e can use t h e technique of L e m m a 6.9 below. W e define N in the following way: ~ E N if a n d only if an isosceles triangle A t with base along 0~ a n d a fixed opposite angle 2~c is contained in D e.
W h e n ~ E N, the Dirichlet integral of f is b o u n d e d b y a fixed c o n s t a n t a n d further- more the inequality (6.10) is correct. I f ~ ~ N a simple expedient is to consider D e U A t instead of D e.
F r o m L e m m a 6.7 a n d L e m m a 6.8 we n o w o b t a i n t h e following
Theorem 6.2. Let D satisfy A p. 13 and let ~ be defined by C p. 14. z o = (xo, Yo) is a fixed point in D. Then for ~ > x o + 1
o~(zo; O~; D~) <c exp ( - f x ~ l ~ ( x ) d x ) . (6.13) I / D satisfies A p. 13 and D p. 17 and ~ is defined by E p. 17, (6.13) is also correct.
Finally, (6.13) is correct i/ D satisfies A p. 13, D p. 17, and D e satisfies F p. 19, and ~ is defined by G p. 19.
The constant c depends only on the geometry of D near z o and constants appearing in the conditions satisfied by D and D~; when F is used c also depends on x o.
Remark. The conditions involving smoothness in A a n d D were i n t r o d u c e d for simplicity a n d are n o t essential. If A, D, F (or some of them) - a p a r t f r o m smoothness c o n d i t i o n s - - a r e satisfied, we can e x h a u s t D e with a m o n o t o n e sequence of subdomains D~ ), v = l , 2 .... , in which the t h e o r e m a b o v e can be applied. N o w e0(z; 0~; De) a n d 2(x) in D e are defined f r o m the corresponding quantities ~o (~) a n d )~(~)(x) in D~ ~) b y a limiting process. (~(~)(x)} is a non-increasing sequence. I t follows t h a t (6.13) is valid even t h o u g h t h e s m o o t h n e s s conditions are n o t satisfied.
/
If ~t is b o u n d e d we can integrate up to ~ in (6.13) a n d let c also d e p e n d on t h e least u p p e r b o u n d of 4. We shall n o w show t h a t integration up to ~ in (6.13) is possible in some special cases when ~t is n o t bounded.
Theorem 6.3. Let D satisfy A p. 13 and @x be connected, x > 0 . Let ,~(x) be defined by C p. 14. Let (~(x) and )~(x) be continuous, x > 0 . z0-(x0, Y0) is a fixed point in D. Then
~o(zo; O~; D~) <~c exp ( - ; , , ~ ( x ) d x ) (6.14) is implied by any one o/the ]ollowing three conditions.
(a) n = 2.
(b) n > 2 , ,~89174 x > 0 , / o r some r < l .
21
KERSTI HALISTE,
Estimates of harmonic measures
(c) n > 2, ~ (x)@(x)-~Mo, ),(x)
non-decreasing, x > O.
The condition (d) implies
(6.15).(d) n > 2, 289 = ~ ( x 0 )
k(x)~ 89 (@(x)/O(xo) ) 1/)~
1),k(x) non-decreasing, x > O.
eO(Zo;O~;D~)<~ck ~ n(~)exp(-f]o2 89
(6.15)The constants c depend only on the geometry o / D near %, xo, and constants occurring in A, (b), and (c).
W e first prove a lemma.
L e m m a 6.9.
Let / be continuous and bounded, O</(x)<~K,
x ~ 0 ,and /(x)=K, 0 <~ x <~ a. Then/or p > 1 and ~ >~ a
; ( ; )
m a x ] P(x) exp -- / - l ( u ) d u
dt>~c>O,
(6.16)where e only depends on p, a,
a n d K.Proo/.
L e t P denote the curve y =/(x), x > 0 , in thexy-plane. A z
denotes a triangle with vertices(x, 0), (x,/(x)), (x /(x)
cot ~, 0), where tg ~=q
is a large constant. Set N - {xlthe interior of A x lies below P}. If we choose q >K/a, N
will certainly be non- e m p t y .N o w let ~ belong to h r. Given $, we t a k e t o - 4 - (2q)-l/(~) 9 T h e n
; l - l ( x ) d x < s ( / ( ~ ) - q ( ~ - x ) ) - l d x < . l o g 2.
o
Hence
/ l(~) f ~ e x p ( - f ) /q(u)du)dt>~(4q) -1.
(6.17)B y t a k i n g x = ~ in (6.16), the t r u t h of (6.16) follows for ~ EN.
N o w consider a ~o ~ N , ~o > a. B y c o n t i n u i t y there exists a largest b < ~o such t h a t b EN. T h e n (6.17) is valid for ~ = b a n d f u r t h e r m o r e
e x p ( _ f ~ ~ 2 4 7 l/q,
(6.18)a n d ~o b ~ < K c o t a. N o w t a k e
x = b
a n d $=~0 in (6.16) a n d use (6.17) with ~ = b . T h e n it remains to determine a fixed lower b o u n d forB y virtue of (6.18) this is done b y choosing q ~> ( p - 1) -1. T h u s the l e m m a is proved.
22
ARKIV F6R MATEMATIK. B d 6 n r 1
Proo/ o/ Theorem 6.3.
We use the technique of Lemma 6.9. B y (6.3), (6.7), and Lemma 6.6 we can reduce the proof of our theorem to determining a fixed positive lower bound forIf this cannot be done for all large ~, we can under the condition (a) obtain the desired estimate for
U(Zo)
b y a simple estimate of an auxiliary harmonic measure.Under the condition (b) we instead reduce the proof to determining a fixed positive lower bound for
;
max O--l(x) exp --2 ~.'(u)du dt. (6.19)
x ~
(a)~ (6.14). We use the technique of Lemma 6.9 w i t h / = ( 2 ~ ) - 1 @ =89 ~. (The as- sumption that / be constant for small x is not essential. I t can be satisfied by a modi- fication of D.) If ~EN, (6.17) is true, and (6.14) follows. Now consider a ~ N , ~ suf- ficiently large. B y continuity there exists a largest b <~, such that b EN. Then (6.17) is true for b instead of ~. B y the maximum principle
o~(z0; | D~)< w(z0; Oh; Db) max co(z; @~; D~).
Z~Ob
Hence it suffices to prove t h a t
exp (~/~O l(u)du)maxeo(z;O~;D~) ~c.
z~O b
(6.20) B y the definition of b
2g f~ O-I(u) du ~- ~-1
log (1 § 2~ q(~ - b) O-l(b)).If
(~-b)O-l(b)<~cl
(a fixed number), then (6.20) is correct. If(~-b)O-l(b)>cl,
we use a simple estimate of ~o(z; 0~; D~). Set
l={Z]Xl=~ }
and let G be the domain bounded by l and{zlx 1 =b, y >~
O(b)/2}. B y the extension principle and an explicit eonformal mapping of G onto a half-plane we obtainmax co(z; @~; D~) ~< ~o(b;
l; G) < c(~- b)-89189
zeO b
Hence (6.20) is correct, and this part of the lemma is proved.
(b) ~ (6.14). We use Lemma 6.9 with / = 89 Since
@(x)
~< M, x > 0, 289 is bounded from below according to the F a b e r - K r a h n inequality. The condition (b) now implies (6.16). We thus obtain a fixed lower bound for (6.19) and (6.14) follows.(c)~ (6.14), ( d ) ~ (6.15). We use the technique of Lemma 6.9 w i t h / = 8 9 -89 For a non-increasing / all sufficiently large ~ are in N and (6.17) gives the desired results.
(Instead of / being non-increasing we can require t h a t / ' exists and is bounded by a fixed constant.)
23
KERSTI HALISTE, Estimates of harmonic measures
7. The Nevanlinna mean
I n a special case lower b o u n d s of h a r m o n i c measures can be established b y s t u d y - ing t h e :Nevanlinna m e a n (7.2). Heins used this n a m e for (7.2) in t h e case of a rec- tangle or a half-plane (in polar coordinates) [7.1, p. 4].
W e consider domains D such t h a t Ox~:r - c~ < x < ~ . O 0 is a s s u m e d to be Steiner s y m m e t r i c with respect to the coordinate hyperplanes y i = 0 , i = l , 2 ... n - 1 . B y this we m e a n t h a t t h e intersection of O0 with a straight line perpendicular to y~ = 0 is either a single line-segment s y m m e t r i c a l with respect to y~ = 0 or e m p t y , i = 1, 2, ..., n - 1 . W h e n n > 2 B(~)0 is a s s u m e d to possess piecewise continuous curvature. O~ is is o b t a i n e d f r o m O0 b y t h e m a p p i n g y--~ k-l(x) y, k(0) = 1, #(x) > 0, #(x) non-decreasing a n d twice c o n t i n u o u s l y differentiable, - c o < x < co. U n d e r these assumptions we shall establish the following
Theorem 7.1.
Let ,~ be the principal eigevalue o/ 0o. Then
max c~ O~; D~) ~ (]c(~)/k(x))i-n exp (-)~89 ; k(t)dt)
W e start b y p r o v i n g some lemmata. W e use the following n o t a t i o n . L e t {v~}~
be t h e n o r m e d eigenfunctions of
Av+~v=O
in Oo, v = 0 on B~)o, a n d {2~}~ the cor- responding eigenvalues. L e t {Vx. , } ~ a n d {A~. ~}~r be defined in the same w a y with respect to O~. T h e nAx. ~ = k2(x) ~ ,
Vx, ~(y) = #(~ 1)/2(x) v~(lc(x) y).
(7.1) I n t h e following we write ~ a n d v instead of 21 a n d v 1.L e m m a 7.1..Let
m(x) be the Nevanlinna mean o/ u(x,
y ) = co(z;0~; D~) over 0~, x<~,
re(x) = j~)~ u(x, y) v(k(x) y) dy.
(7.2)Let y.
g r a dv denote the scalar product in (~o. Then
( ux(x, y)v(k(x)y)dy§ ]c'(x) je(U(X' y)
( y . g r a dv) (k(x)y)dy, x<~.
(7.3) m l ( x ) ~JO~
Proo/.
U n d e r our assumptions a b o u t k a n d BOo, u possesses continuous first deriva- tives u p to the b o u n d a r y at points where BO o is of continuous c u r v a t u r e (in the case n > 2 ) for x 1 < ~ [7.3, p. 635]. T h e same is true for v, since exp(~Xl)V(y)
is harmonic in a right cylinder with base O0. The l e m m a t h e n follows.L e m m a 7.2.
SexUx(X, y) v(k(x) y) dy <~ ~ ]c(x) m(x), x < ~.
Proo/.
L e t x be fixed a n d setO={z=(s, t)ls<x ,
tEOx}. W e write 1 ~ instead of BG a n d define ~ = F - Ox. L e t U be h a r m o n i c in G with b o u n d a r y valuesu(x, y)
on Ox a n d 0 on ~. G c D , since O0 is Steiner s y m m e t r i c with respect to t h e coordinate hyperplanes a n d k is non-decreasing. B y the m a x i m u m principle t h e inner n o r m a l derivatives of U a n d u satisfy the inequalityBU/Bn <--.Bu/Bn
on O,. W e writeV(y)
instead of
v(k(x)y).
T h e n 24ARKIV FSR MATEMATIK. B d 6 nr 1
fo u~Vdy<~- f o V ~U ~ n d y = - - f r V ~ n d a '
(7.4) where da is the area element on F. We now apply Green's formula to the last integral a n d note t h a t U = 0 on ~ a n d~V/~n =0
on Ox. (We first consider finite subdomains of G and then use some m a j o r a n t of~U/~n
in the limiting process.) Taking (7.1) into account we obtainfrv eU da= faU• ~n = -Zk~(x) f UVdz.
Hence b y (7.4),
/~ u~ (x, y) v(k(x) y) dy ~k2(x) ~ u(s, t) v(k(x)t) ds dr.
J J a
(7.5)
Now U can be represented in the following w a y in the cylinder G:
U(s, t) = (_ y)P(s, t; x, y)dy, s<x,
(7.6)where
P(s,
t; x, y) = ~ exp ((s - x) A~, ~) Vx. ~(y) Vx. n(t).We shall need the relation
fo P (s, t; x, y) v(k(x) t) dt
= exp(289 - x) k(x)) v(k(x) y), s < x. (7.7)
B y (7.6) a n d (7.7), after first considering subdomains { z = ( s , t ) [ z E G ,
- c ~ < a < s < b < x }
ofG,
we obtainf l u ( s , t) V(]C(X) t) dsdt = ~89
k-l(x).B y (7.5), this proves our lemma.
L e m m a 7.3.
Under the assumption that 0 o is Steiner symmetric with respect to the coordinate hyperplanes, the scalar product y.
gradv is non-negative.
Proo/.
We w a n t to prove t h a t v is symmetrically decreasing with respect to y~ ~ 0 , i = 1 , 2, ..., n - 1 . L e t us assume t h a t this is false for some i. We then symmetrize v with respect to y~=0 (cf. w 1) and denote the symmetrized function b yv*.
B y [7.2, pp. 184-186]f~~ f~.lgradvl~dY>~folgradv*l~dy"
25
