~= Go(r)) 0) = f,f(ry)dH~ A Maximum Principle With Applications To Subharmonic Functions in n-space

A Maximum Principle With Applications To Subharmonic Functions in n-space

RO~TALD GARIEI~u a n d J o H ~ L. L ~ w I s

1. Introduction

D e n o t e p o i n t s in n d i m e n s i o n a l E u c l i d e a n s p a c e R ~, n > 3, b y x =

@l, x 2 , . . . , x , ) . L e t r~-- Ixl a n d x l = r c o s 0 , 0 < 0 < x . A real v a l u e d f u n c t i o n f d e f i n e d on a s u b s e t E o f R" is said to b e s y m m e t r i c (with r e s p e c t to t h e x 1

axis) if f ( x ) = f ( y ) w h e n e v e r x , y C E a n d x a n d y h a v e t h e s a m e r,O c o o r d i n a t e s .

t~or r > 0 let B ( r ) = { x : Ixl < r } , S ( r ) = { x : Ix] = r } a n d S = S ( 1 ) . F o r 0 < ~ < ~ let C ( c ~ ) - - ~ S [ 7 { x : 0 < c ~ } . G i v e n a set E C R n, let E , 0E, d e n o t e t h e closure a n d b o u n d a r y o f E in R". I f E c S(r) let ~E d e n o t e t h e b o u n d a r y o f E r e l a t i v e to S(r). L e t H m d e n o t e m d i m e n s i o n a l H a u s d o r f f m e a s u r e in R n.

I f f is d e f i n e d on a set E c R n let O(r) be d e f i n e d b y H"-](C(O(r))) -- H~-l(p(S(r) 17 E))

w h e r e p d e n o t e s t h e r a d i a l p r o j e c t i o n o f R" - - {0} o n t o S. F o r 0 < 0 < O(r) let

0) =

s u p

f,f(ry)dH~

/<r,

w h e r e t h e s u p r e m u m is t a k e n o v e r all m e a s u r a b l e sets F C p ( S ( r ) n E) w i t h Hn-I(F) = Hn-I(C(O)).

L e t t9 be a b o u n d e d region in R ~ o f t h e f o r m

~= U Go(r))

r l < r < r 2

w h e r e O < r x < r 2 < ~ a n d O < O ( r ) < ~ for r x < r < r z. L e t h be a s y m - m e t r i c , b o u n d e d , h a r m o n i c f u n c t i o n in D s u c h t h a t , for r x < r < r z, h(r, O) is a n o n increasing f u n c t i o n o f 0 for 0 < 0 < O(r). T h e n

254 I ~ 0 N A L D G A R I E P Y AIqD J O H N L . L E W I S

h(r, O ) :

f

h(ry)dH"-~y in D.

J c(o)

L e t u be a s u b h a r m o n i e f u n c t i o n ( ~ - - ~ ) in B ( R ) ~ D , _ R > r 2.

we will p r o v e

I n w 3

TREOREM 1. I f h has a continuous extension to f2 - - {O} and 4 <_ h + c on O D - - { O } where c ~ O , then 4 < h + c everywhere in D.

W e n o t e t h a t B a e r n s t e i n [2, T h e o r e m A'] has o b t a i n e d a similar t h e o r e m in R 2.

W e will give t w o applications o f T h e o r e m 1. T h e f i r s t is t o a n e x t r e m a l p r o b l e m for p o t e n t i a l s . G i v e n a real n u m b e r },, 1 ~ y < ~ , let H(~) d e n o t e t h e class o f p o t e n t i a l s

p(x)

⁼

[ Ix

- - Y l 2 - " d # ( Y ) , x E R "

J S

w h e r e # is a p r o b a b i l i t y m e a s u r e on S a n d

p(x) ~ ~ w h e n e v e r x E R".

Choose ~ so t h a t t h e N e w t o n i a n c a p a c i t y o f S - - C(~) is r - 1 a n d let P C H ( y ) d e n o t e t h e corresponding e q u i l i b r i u m p o t e n t i a l . I n w 4 we p r o v e

THEOREM 2. I f q5 is a nondecreasing convex .function on

whenever r ~ O and p E H(~).

( - - ~ , ~ ) , then

T h u s , i f 2 ~_ 1, qi(u) = u x for u > _ 0, a n d q)(u) = 0 for u < 0, we h a v e

w h e n e v e r r ~ 0 a n d p E H(~). I t follows t h a t

m a x {p(x): x E S(r)} _< m a x {P(x): x E S(r)}

w h e n e v e r r ~ 0 a n d p E H ( ~ ) .

W e n o t e t h a t t h e a b o v e i n e q u a l i t y has b e e n o b t a i n e d b y Davis a n d Lewis [6].

I f u is a s u b h a r m o n i c f u n c t i o n in R", let M(r, u) ~- m a x { u ( x ) : x E S(r)}

w h e n e v e r r ~ 0 a n d M(0, u) ~-- u(0). As a second a p p l i c a t i o n of T h e o r e m 1 we p r o v e in w 5.

THEOREm 3. Given 0 < # ~ 1 and O < fl < 1, there exists ~ ~-- ~(#, fi, n) ~ 0 such that i f u is a n y subharmonic function in R" with

A 1VIAXIMUI~I :PRII~OIPLE W I T H A : P P L I C A T I O l q S TO SLTBHAI:tlYIONIC FTJI~TCTIO~TS I ~ ~,-SP-~e]] 2 5 5

H~-~(}x: u(x) > ~M(Ixl, u)} n S(r)) _ t~H~-~(S(r))

whenever r > O, then either u < 0 everywhere in R ~ or lim~+~ r-eM(r, u) exists and is positive (possibly -ff c~).

F o r 0 < fl < 1 a n d /~ = 0, D a h l b e r g [4], H i i b e r [14], a n d T a l p u r [15] h a v e all s h o w n t h e existence o f ~* ---- ~*(fi, n) > 0 for w h i c h t h e conclusion a b o v e holds.

I n w 6 we will show t h e ~ we o b t a i n is b e s t possible for 0 < # < 1 a n d 0 < fi < 1.

B a e r n s t e i n [1] has o b t a i n e d a similar r e s u l t in R ~.

T o p r o v e T h e o r e m 3 for 0 < # < 1 we use T h e o r e m 1 t o r e d u c e t h e p r o b l e m to one considered b y D a h l b e r g [5] a n d E s s e n a n d Lewis [TJ. F o r # = 0 we use T h e o r e m 1 a n d a r g u m e n t s similar to t h o s e o f H e i n s [12, p. 114, ex. 11].

2. Spherical symmetrization

G i v e n a closed set F c R% d e f i n e t h e spherical s y m m e t r i z a t i o n F * o f • as follows: I f F gl S(r) = r t h e n F * N' S(r) = r Otherwise H~-I(F * N S(r)) = H~-X(F n S(r)) a n d F * [~ S(r) is e i t h e r t h e p o i n t (r, 0 . . . . , 0 ) or t h e closed cap on S(r) c e n t e r e d a t (r, 0, . . . , 0). L e t u be s u b h a r m o n i c in B ( R ) , / ~ > 0. G i v e n t , - - ~ ~ t < ~ , let F(t)~--{x:u(x) ~_t} a n d n o t e t h a t F(t) is closed. D e f i n e an associated f u n c t i o n u* b y l e t t i n g

u*(x) = sup {t: x E F*(t)} w h e n e v e r x C B(R).

I t is easily seen t h a t u* is s y m m e t r i c a n d {x: u*(x) ~ t} = F*(t). I t follows t h a t u* is u p p e r s e m i c o n t i n u o u s , u a n d u* are e q u i m e a s u r a b l e , a n d

4(r, 0) = f u*(ry)dH~-Xy

(2.1)

J c(o)

w h e n e v e r 0 < r < R, 0 < 0 < ~. W e n o t e for l a t e r r e f e r e n c e t h a t G e h r i n g [10, l e m m a 4] has s h o w n t h a t u* is L i p s c h i t z in B(/~) w h e n e v e r u is.

Consider n o w t h e r e s t r i c t i o n o f u a n d u* (also d e n o t e d b y u a n d u*) t o S(r) for f i x e d r, 0 < r < / ~ . Assume t h a t u a n d u* are L i p s c h i t z f u n c t i o n s o n S(r).

D e f i n e a B o r e l m e a s u r e u # H ~-1 on R b y l e t t i n g u#H~-I(E) = H~-I(u-I(E))

w h e n e v e r E is a B o r e l subset o f R. D e f i n e u ~ H n-1 analogously.

L e t ~ d e n o t e t h e g r a d i e n t r e l a t i v e to t h e s p h e r e S(r), a n d let G be t h e subset o f S(r) w h e r e V u * exists. D e f i n e a f u n c t i o n g on R b y l e t t i n g

g(t) • 0 if (u*)-~(t) f'l G : r a n d

256

R O N A L D G A R I E P Y A N D J O H N L . L E W I S

g(t) = ]Vu*(x)[ for a n y x E (u*)-~(t) CI G, otherwise.

is s y m m e t r i c , g is well defined. N o t e t h a t g o u * ( x ) = IVu*(x)I

Since u* for

H ~-1

a l m o s t e v e r y x C S(r), T h u s b y [8, 2.4.18 (1)].

y ~ ,

*(tx, t2) tl

w h e r e A*(t 1, t~) = {x: t 1 < u*(x) < t2}.

Since u//H ~-~ = ' a ~ u " * ~ - ~ we see b y [8, 2.4.18 (2)] t h a t g o u is H "-~ m e a s u r a b l e a n d

g # (g o u)2dH ~-1

w h e r e A(tl, t~) = {x: tl < u(x) < t2}. H e n c e

f A (g ~ u)2dH'~-l = L 1(Tu*12dH=-l"

(tl, t2) *(tl, tz)

Using t h e coarea f o r m u l a [8, 3.2.22 (3)] a n d t h e sphericM i s o p e r i m e t r i c i n e q u a l i t y for sets o f f i n i t e p e r i m e t e r (see [8, 3.243 a n d 4.5.9 (31)] for a similar i n e q u a l i t y in t h e E u c l i d e a n case), we o b t a i n

L*(t.t~)

' V u * I 2 c l t t " - l =

s ( f (o,)-.(,) e~176

^{2) dt}

f'(f

g o udH ~-2 dt =

) s

(g o u ) l V u [ d H "-~.

1 u--i(t) (tl, t2)

F r o m H o l d e r ' s i n e q u a l i t y , it follows t h a t

T h u s

f (go u ) l ~ u [ d H ''-1 <

Aft. t,)

if

(g o u)2dHn 1

jl [f

uL2dH,-~

]1j2

A(t. t~) A(h. t~)

A p p l y i n g t h e coarea f o r m u l a again we o b t a i n

I ~ ujdH ~-2) dt

A ) i A X I ] i I U . Y l I ) R I N C I I ) L E W I T I ~ A I ~ I ~ L I C A T I O I ~ S T O S U B I I A R M O N I C F U I ~ C T I O I ~ I S I N ~ I - S P A C E 2 5 7

w h e n e v e r t~ ~ t 2. H e n c e for a l m o s t e v e r y t (with r e s p e c t to one dimensional L e b e s q u e measure)

f(,.)_~(,) ] V u * ] d H "-~ ~

f o [ JuldH

n-: (2.2) T h e coarea f o r m u l a also implies t h a t

H~-:[u-~(t) -- ~{x: u(x) > t}] --~ 0

for a l m o s t e v e r y t. Thus, for a l m o s t e v e r y t, we can replace u-~(t) b y

~{x: u(x) > t} in (2.2).

T h e a r g u m e n t a b o v e was suggested b y [10, (27)].

3. Proof of Theorem 1

T h e p r o o f is b y c o n t r a d i c t i o n . Suppose t h e r e is a n x 0 E ~ such t h a t

~(Xo) > h(xo) ~- c. L e t w(x) = h(x) -~ ~]xl 2-~ -~ ~x 1, w h e r e ~ > 0 is so small t h a t 4(x0) - - ~(Xo) = c 1 ~ c. Clearly w is s y m m e t r i c , h a r m o n i c in Y2, a n d Ow/aO ~ 0 at each p o i n t of t9 off t h e x 1 axis. Also, 4 < ~ - c , on ~Y2--{0}.

T h e r e exists a decreasing sequence {u/} o f s u b h a r m o n i c f u n c t i o n s in B(1/2(r 2 ~- R)) w i t h c o n t i n u o u s second p a r t i a l d e r i v a t i v e s t h a t converges pointwise t o u in B(1/2(r 2 Jr R)). Since u* is L i p s c h i t z in B(r2), it follows f r o m (2.1) t h a t 4j is c o n t i n u o u s in / ~ ( r 2 ) - {0}. Since

0 ~ ~j(r, O) -- ~(r, O) < ~j(r, Jr) - - ~(r, Jr),

a n d ~j(r, ~), ~(r, ~) are c o n t i n u o u s f u n c t i o n s of r on [a, 1/2(r 2 ~- R)] for 0 < a < 1/2(r 2 -+- R), it follows f r o m Dini's T h e o r e m t h a t {~j} converges u n i f o r m l y to ~ in t h e closure o f B(r2) - - B(a) w h e n e v e r 0 < a < r 2. T h u s ~ is c o n t i n u o u s on / ~ ( r 2 ) - {0}. Choose o > 0 so small t h a t ~ - 6 < c 1 on t h e closure of B(a) [3 Y2. T h e n t h e r e exist m a n d s > 0 such t h a t

~ ( x ) + sH~-I(S)Ix]2 -- ~(x) < c, w h e n e v e r x E 0[Y2 - - B(a)].

L e t v(x) ~-- Um(X ) ~- S]X[ 2 for x E D - - B(a) a n d n o t e t h a t

J c(o) de(o)

has a r e l a t i v e m a x i m u m a t a p o i n t in f2 - - B(a) w i t h coordinates (ro, 0o) , 0 ~ 0 0 ~ z.

N o t e also t h a t

/~v ~ 2ns. (3.1)

258 I % O N A L D G A R I E P Y A N D J O H N L. L E W I S

Since v* a n d w are continuous in zg, it follows t h a t

v*(ro, 0o)= w(r o,

0o) a n d for 0 -- 0 o > 0 a n d sufficiently small,

f c(o)_c(o~ <- f c(o)_c(o~

Since v* is Lipsehitz, a n d

v*(ro, O)

a n d W(ro, 0) are nonincreasing a n d decreasing functions of 0 respectively, it follows t h a t

l~v*(ro, 0)1 > I~W(ro,

0)l > 0 (3.2) for all 0 in a set F w i t h t h e property: Given a n y v > 0, t h e one dimensional Lebesgue measure of F [3 [00, 0o, -4- 3] is positive.

F o r 0 - 0o small a n d positive, let

E(O)C S

be such t h a t (i) S fl {y:

V(roy )

> v*(ro, 0)} c

E(O) c S 13

{y:

V(roy ) > v*(ro,

0)}, (ii) H"-I(E(O)) = ~n-~(C(O)),

(iii)

~(ro, O) = f V(roy)dH"-ly = f v*(roy)dH"-ly.

J E (o) dc(o)

Note t h a t all three sets in (i) h a v e t h e same H "-~ measure whenever 0 E F , a n d t h a t

E(Oo) C E(O)

whenever 0 o < 0 6 F .

L e t

a n d observe t h a t

v(r) < ~(r, 0o) -- ~(r, 0o) < V(ro),

for r sufficiently close to r o. Thus V has a relative m a x i m u m a t r o a n d d~o

Consequently given a n y 7 > 0, we h a v e

f E(o) ar -- 7

whenever 0 -- 0 o > 0 a n d sufficiently small.

F o r 2 > 0 let

L(O, 2) = {sy: r

o _ < s _ < r o + 2 , y E E ( 0 ) }

a n d

L(O) = L(O,

0). Since {v*(ro, 0): 0 6 F Cl [00, 00 -~- 3]} has positive one di- mensional measure, whenever ~ > 0 there is a n F ' C F containing 0 arbitrarily n e a r 00 a n d such t h a t (2.2) holds w i t h

v = u, t = v*(r o, 0),

a n d

aL(O)

replacing

u-l(t)

whenever 0 E F t B y [8, 3.2.22 (2)] we can assume t h a t

~L(O)

is

A M A X I M U M I ' I ~ I N C I P L E W I T t I A I ~ t ' L I C & T I O ~ S T O SUBItAtClVIO2r F U N C T I O N S I ~ n - S P A C E 2 5 9

(H ~-~, n -- 2) rectifiable whenever 0 E F ' a n d hence t h a t

OL(O, ~)

is

(H ~-~, n --

1) rectifiable whenever 0 E F ' .

Now, from (3.1),

/ *

2ne)-~Hn(L(O, ~)) ~ ~-1 ] /~vdH ~.

J L

(+. 4)

Using t h e Gauss-Green t h e o r e m [8, 4.5.6 (5)] a n d letting ~ - + 0 we o b t a i n for 0 E F ' ,

f

^av

2neHn-l(L(O)) <_ r~-" J L(+) ~

Since w is h a r m o n i c a similar a r g u m e n t gives

f

^Ow

r~-" dc~(o) ~-r~(r'~-l~r)~=~odH~-i

~ Lc~(+) ]~JwldH'-2

where

C1(0 ) ---- {roy: y E C(O)}.

Using (3.2), (2.2), (3.3), a n d the above inequalities we o b t a i n for 0 E

F',

<- ~o ) ~ ~=~ ~ ....

0( 2ns//'-i(L(0))-~-y

~-- f _ I~wldH"-: _ 2nsH"-~(L(O)) + dO c~(o)

Thus

2nsH~-I(L(O)

< y whenever

O E F '

a n d hence

2ner]-lH~-l(C(Oo)) <_ y.

Since y is a r b i t r a r y a n d 0 o > O, we h a v e reached a contradiction. H e n c e T h e o r e m 1 is true.

4. Proof of Theorem 2

L e t

7, H(y),

a n d P E H ( y ) be as in w 1. I f y--~ l, t h e n t h e conclusion of Theorem 2 is obvious since P is t h e only m e m b e r of H(1). Thus we assume t h a t 1 < y ~ ~ . T h e n 0 ~ z a n d h - - ~ - - P is subharmonic in R ~, h a r m o n i c in R n - [ S - C ( a ) ] , a n d h ~ - - - y on S - - C ( ~ ) . I t is readily seen t h a t h is s y m m e t r i c a n d t h a t

h(r, O)

is a nonincreasing f u n c t i o n of 0 for 0 ~ 0 ~ ~ a n d f i x e d r ~ 0. F r o m t h e p r o o f of Theorem 1 we see t h a t ]t is continuous in R n -- {0}.

260 R O N A L D G A R I ] N P Y A N D JO/-I/'7 iL. L E W I S

Now suppose p E H(7 ) a n d u = - p. Clearly u is subharmonic in 1t ". Given s > 0 choose R large enough t h a t 4 < s on S(R).

L e t 12 ~ B ( R ) denote t h e b o u n d e d s y m m e t r i c region in R " such t h a t B ( R ) - - ~(2 consists of the union o f S -- C(~) a n d t h e line s e g m e n t from the origin to ( - - R , 0 , . . . , 0 ) , One verifies t h a t ~ ( r , ~ ) = ] ~ ( r l u ) for 0 < r < oo. Since u* >_ h ---- -- 7 on S -- C(~), it follows t h a t 4 < s on S -- C(c~). Thus 4 _< s + on ~12 -- {0}. B y Theorem 1, ~ _< ft q- s in 12. I t follows t h a t ~ < s in II '~ -- {0}.

Note t h a t

~(r, 0) = (-- ~;)(r, 0) = ~(r, ~ -- 0) -- ~;(r, ~)

w i t h a similar relation holding b e t w e e n h a n d /5. Thus since /~(r, ~) = / 3 ( r , ~) for 0 < r < ~ , we h a v e 2~ < t 5 in t l ~ - { 0 } . I t i s k n o w n . [ l l , p. 170, 249--250]

t h a t this i n e q u a l i t y implies t h e conclusion of Theorem 2.

5. Proof of Theorem 3

I t sufficies to assume t h a t u > 0 (otherwise consider m a x {u, 0}) a n d t h a t u ~ = 0 . L e t ~ , 0 < ~ < ~ , be such t h a t H'~-I(C(~x))=DH n 1(S) a n d let

p(x)

^{= m a x}

{~M(JxJ, u), u*(x)}

whenever x C R ~ - {0}. W e observe from t h e h y p o t h e s e s of Theorem 3 t h a t p(r, O) ---- ttM(r, u) if 0 > c ~ . F o r 0 < a < z let

K((~) = (ty: 0 < t ~ co y C C(a)}

a n d let K(a, R) = B(_R) Cl K(a). Assume henceforth t h a t M ( E , u) > 0. N o t e t h a t for ~ > c% p is upper semicontinuous on aK(a, R ) - {0}, a n d continuous except on a polar set. Thus there is a unique b o u n d e d h a r m o n i c function h, in K(~,/~) such t h a t

lim sup ha(x ) < p(y) whenever y C aK(a, R) -- {0},

x--~y

a n d limx_~y h ( x ) = p(y) except on a polar set in 0K(a, R) [13, L e m m a 8.20].

Since #M(Ixl, u ) is s u b h a r m o n i c in R " ( M ( O , u ) = u ( O ) ) , it follows t h a t r u) < It(x) in K(r R). F r o m the b o u n d a r y values of h we see t h a t h is s y m m e t r i c in K(a, R).

L e t

qo(r,O) = s u p { h ( r , 01):0 < 0 1 < a} in K ( ~ , R ) .

T h e n q~ is s y m m e t r i c a n d has t h e same b o u n d a r y values as h~. Using t h e fact t h a t qo(r, 0) = h,(r, 01) for some 0~, 0 < 0~ < a, i~ is easily checked t h a t q~ is upper semicontinuous a n d satisfies a local sub mean-value p r o p e r t y in K(a, R).

A M A X I M U ~ P R I N C I P L E } V I T I t A P P L I C A T I O N S T O S U B I - I A R M O N I C F U N C T I O N S I N Y~-SPACE 261

T h u s q~ is s u b h a r m o n i c in K ( a , / ? ) a n d since it is obvious t h a t h~ < q~, it follows t h a t h~ = q~ in K ( a , R ) . H e n c e h(r, 0) is n o n i n c r e a s i n g for 0 < 0 < a a n d f i x e d r, 0 < r < R. T h e p r o o f o f this f a c t is due t o M a t t s Ess6n (oral c o m m u n i - cation).

F i x a > a a n d l e t v ( x ) = h ( x ) + s l x ] 2-~ for x E K ( a , R ) a n d e > 0. Observe t h a t v has a c o n t i n u o u s e x t e n s i o n t o K ( a , /?) - - {0} a n d t h a t ~ ~ v on S(R) f l / ~ ( a ) . Thus, if

sup {~,(y) - - ~(y): y e 0K(a, R) - - {0}} = c > 0,

t h e n 4(r, a) -- v(r, a) = c for some r w i t h 0 < r < R . H o w e v e r since u*(r, 0) _< tiM(r, u) < v(r, O) w h e n e v e r a < 0 < a, it follows t h a t

~(r, a) - - ~(r, cr > ~(r, a) - - v(r, a) = c > 0,

which c o n t r a d i c t s T h e o r e m 1. H e n c e c < 0. A p p l y i n g T h e o r e m 1 a n d l e t t i n g e - + 0 we h a v e ~ < s in K ( a , R ) w h e n e v e r a > c ~ .

L e t h~(x) = #M(Ixl, u) for x E B(R) -- K(a, R). T h e n ho is s u b h a r m o n i c in B(R) a n d if ~ < a a < % t h e n h <ho~ in B(2~). T h u s h = l i m ~ + h ~ is s u b h a r m o n i e in B(R) a n d h a r m o n i c in K ( a , R). Clearly

h(x)= #Mlxl, u)

in B(_R) - - K(~, R). Since B(R) -- K ( ~ , / ~ ) is n o t t h i n a t a n y x ~_ OK(a) 91B(R) [13, Corollary 10.5], it follows t h a t h(x) = #M(lx[, u) on OK(a) Cl B(R). Since t ~ < h in K ( a , R ) w h e n e v e r a > a , we h a v e ~ < h in K(~,/~).

L e t

/~ = PR + QR

where PR a n d QG are b o u n d e d h a r m o n i c f u n c t i o n s in K ( ~ , / ~ ) w i t h lira PR(x) = # M ( [ y l , u) w h e n e v e r y E OK(a) Cl B(JR),

x ' . y

lira PR(x) = 0 w h e n e v e r y ~. K(a) Cl S(R),

x~+y

a n d Q R = h - - P R . N o t e t h a t

lira QR(x) = 0 for y C 0K(a) Cl B(_R),

x-+y

lim QR(x) = I)(Y) for y s, K(a) ~ S(R),

x-+y

off of a polar set.

L e t 0 < 71 < 72 < 9 9 9 be t h e eigenvalues o f t h e b o u n d a r y v a l u e p r o b l e m

~r + 7 r = 0 on C(a), r on ~c(a)

where ~ is t h e B e l t r a m i o p e r a t o r d e f i n e d in t e r m s o f t h e L a p l a c i a n A b y (5.1)

262 l ~ O l q A L I ) G A R I E P Y A N D J O H N ]5. L E V V I S

~ = r ~ - ~ g r~-Xg +r-~.

L e t {~k} denote corresponding s y m m e t r i c eigenfunetions w i t h continuous second .partial derivatives in C(~) a n d

f r

1 for k 1, 2 , . . .

c(~)

L e t ~k be t h e positive root of t h e e q u a t i o n ~ k ( ~ k + n - - 2) = y k for k = 1 , 2 , . . . T h e n as in [9] we h a v e

QR(r, O) = ~ ak(r/R)~ O)

in K(~, R), (5.2)

k = l

where

ak = f

P(Ry)r J c(~)

Using t h e estimates in [7, w 8] or [4, L e m m a 2.5], t h e series

(r/R)~ 0) I

k ~ l

can be seen to converge u n i f o r m l y in

K(x, sR)

whenever 0 < s < 1. Note also t h a t

Iak[ ~_ M(R, u)Hn-~(C(~)) ~/2.

The case

# = 0. I n case # = 0 we h a v e PR = 0, QR = h , p = u*, a n d hence

---- f u*(Ry)r

(5.3)

ak

J c(~)

I t is k n o w n [3, VI w 6] t h a t r is either positive or negative in K(~, R). Assume r ~-- 0. Since r is s y m m e t r i c a n d ~r = -- Y~r it is readily seen t h a t

dr ~_ 0

^

in C(~). Using this a n d the f a c t t h a t z~ _~ h in K(~, it), we have

re(r) = c(~) u*(rY)r ~- o

0 ~ M (1, 0)d0 =

F r o m (5.2) a n d (5.3)

fc(~)

f c(~) h(ry)r

h(ry)r = ax(r/R)o1= (r/R) ~ f u*(RY)r

c(~)

A M A X I M U M 1 ) I = t I N C I P L E W I T ] ~ A P I ~ L I C A T I O I ~ S T O S I Y B H A R ! ~ O N I C F U N C T I O N S I ~ T b - S : P A C E 263

a n d hence

r-Q'm(r) <_ R-olin(R)

for 0 < r < R.

C o n s e q u e n t l y b = l i m r _ ~

r-e'm(r)

exists. W e assume t h a t lim i n f

(r-elM(r, u)) < oo.

r --> oo

Otherwise t h e p r o o f is c o m p l e t e in case # = 0 w i t h @ = @1. Since

m(r) < M(r, u)H"-I(C(~)) 1/2

we h a v e b < oo.

N o w f r o m (5.2) we d e d u c e t h a t for 0 < r <

R/2, r-e~4(r, O) < r-e~h(r, O)

= R-~ f r § R-~ ~ ak(r/R)-~176 f CkdH "-1

C(O) k = 2 C(O)

R-~Im(R) f r dH"-~ -5 R=~ u)g(r/R)

<

J c(o)

where g is c o n t i n u o u s on [0, 1] a n d g(0) = 0. Since lim infR_,~

R-~'~'M(R, u) < oo,

it follows t h a t

r-~l~(r, O) ~_ b f r "-1

in

K(a).

J c(~)

This i n e q u a l i t y a n d t h e s u b h a r m o n i c i t y o f u i m p l y t h a t

r-O~M(r, u) ~_

br , 0) for r > 0, a n d hence t h a t b > 0.

Suppose t h a t

lim

infr-e'M(r,

u) < br 0).

r --> co

T h e n t h e r e exists a sequence {rj} w i t h

r i ~ ~

a n d s > 0 such t h a t

rTQ1M(r j,

u) < br 0)

F o r j---- 1 , 2 , . . . a n d 0 < 0 < s . T h u s ,

-.o1~ O) b f r "-1

rj u(rj, < c(o)

for 0 < 0 < e a n d it follows t h a t

/:

r;-~lm(rj) = - rJ~(rj, O) ~ ^(1, O)dO

< --b f [ ( f c(o) r ~O (1,0)dO=b.

264 I % O I ~ A L D G A R I E I ~ Y A I q D J O H I ~ L . L E W I S

Letting j I' oQ we obtain a contradiction. Hence lim

r-~

u) = b ~ i ( 1 , 0) ~ 0

r - ~ c o

and the proof is complete in ease # : 0 with ~ : ~1.

The case

0 < # < 1. For 0 < ~ < 1 the boundary value problem

~ W + A ~ , ( a ~ l q - n - - 2 ) W = 0 on C(~) VJ= 1 on ~C(~)

has a unique symmetric solution. Choose k so t h a t the corresponding W has the value #-1 at r = 1 , 0 = 0.

Since 4 _ < h in K ( ~ , R ) it follows t h a t

M(r,u) ~h(r,O)

for 0 < r < R and hence t h a t

h(y) = #M([y l, u) < #h(]y[, O)

for

y C OK(~x) Cl B(R).

Thus, using the arguments of [7, (3.1)],

r-~'~ u) <_ r-;'~ O) ~ ~ 1R-~'~ u)

^(5.4)

for 0 < r < R . It follows that

0 < lim sup

r-~'~~

u) _~ #-1 lim

infr-~'~ u).

r ~ - o o r -e- oo

Assume t h a t lira s u p ~

r-~~

u) < m. Otherwise the proof is complete in case 0 < # < 1 with ~ = 2 ~ , .

For PR as in (5.1) we note t h a t PR, --~ PR~ in K(~, R~) whenever -~1 ~ R2.

Also, from (5.4), we have

M(r, PR) <~ h(r, O) ~_ #-~(r/R)-~~ u)

for 0 ~ r <_R. Since liminfR+~

]~-~'~M(R,

u) < 0% it follows t h a t V = limR-~ PR is harmonic in

K(a)

and

M(r, V) <_ #-~/'~

^lim

inf R-~+*M(R, u).

^(5.5)

R - ~ o o

From (5.4) and the definition of QR we have

M(R2u) -DR,

^{- -}

Pn,

^g

~-2(R1/R2)~'e*

M(R1 ' u) QR~

in

K(~, R,)

whenever R, </~2. Letting R2-+ m it follows t h a t 0 < V -- Pn, < (constant) Qn~ in K(~, R1).

Thus

V(y) =-

lim

V(x) =

#M(]y[, u) on aK(~). (5.6)

x - + y

A ]VIAXIMIY]VI : P R I N C I P L E W I T H A P P L I C A T I O N S TO S U B H A R M O N I C F U N C T I O N S I N n - S P A C E 265

F r o m (5.2) we h a v e

e~(s, o) < A(r/R)~~ ~)

for 0 < r < ^R 2 w h e r e A is a positive c o n s t a n t i n d e p e n d e n t of R. Since

lira sup R-~o~M(R, u) < ~ ,

R--> o9

it follows t h a t QR - ~ 0 u n i f o r m l y on c o m p a c t subsets o f K(~) as R - ~ ~ . Using (5.4), (5.1) a n d l e t t i n g R - - ~ m we d e d u c e t h a t M(r, u) ~_ V(r, 0) for r > 0.

This last i n e q u a l i t y , (5.5), a n d (5.6) i m p l y t h a t lim~_~ r-;~~ u) exists [7, (4.6)] a n d h e n c e t h e p r o o f is c o m p l e t e in case 0 < # < 1 w i t h ~ = ~@l.

6. Remark

W i t h

a n d

@l a n d r aS in t h e p r o o f of t h e case # = O, let u(r, O) = rQ~r O) in K(~)

u ( r , O ) = O in R ~ - K ( ~ )

T h e n u is s n b h a r m o n i c in R n a n d satisfies t h e h y p o t h e s i s o f T h e o r e m 3. I~ence

@ = @i is t h e best possible e x p o n e n t in case # --~ 0.

I n case 0 < # ~ 1, let A a n d ~ c o r r e s p o n d to # as in t h e p r o o f of T h e o r e m 3.

I t is k n o w n [7, (1.5)] t h a t ~ ~ 1 in C(a). L e t

U ( r I O) ~ ~*)'~~ O) in K(~) a n d

u(r, O) = r x~ in R ~ - K ( ~ ) .

T h e n u is s u b h a r m o n i c in R" a n d satisfies t h e h y p o t h e s e s o f T h e o r e m 3. T h u s t h e e x p o n e n t @ = ~ 1 iS best possible w h e n 0 < ~t < 1.

ReIerenees

1. BAEI~NSTEIN I I , A., A generalization of t h e cos ~@ theorem, T r a n s . . A m e r . M a t h . Soc.

(to appear).

2. --~-- Integral m e a n s of schlicht functions, Acta M a t h . (to appear).

3. C O U R A N T , R . & I h L B F ~ T , D., M e t h o d s of mathematical physics, I. Wiley-Interseienee, 1966.

4. D A I n L B E R G , B., M e a n values of s u b h a r m o n i e functions, Arlcivf. M a t . i0 (1972), 293--309.

5. --~-- G r o w t h properties of s u b h a r m o n i e functions, thesis, University of G6teborg, 1971.

6. D A V I S , B. & L E w i s , J. L., A n e x t r e m a l p r o p e r t y of s o m e eapacitary m e a s u r e s in E/, Proc. A m e r . M a t h . Soc. 39 (1973) 520--524.

266 : R O N A L D G A I ~ I E P Y A N D ; I O I t N ~L. L E W I S

7. ESS]~N, IV[. & LEWIS, J. L., The generalized Ahlfors-Heins theorem in certain d-dimensional cones, Math. Scand. (to appear).

8. FEDE~E~, I-I., Geometric measure theory, Springer-Verlag, 1969.

9. LELONG-FERRAND, J., Extension du thdoreme de Phragmen-Lindel6f-I-Ieins aux Fonetions sousharmoniques dans u n ebne ou dans tm eylindre, C. R. Acad. Sci. Paris 229 (1949), 411--413.

10. GEHRING, F., Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499-- 519.

11. tIA~DY, G., LITTLEWOO9, J., a n d POLYA, G., Inequalities, Cambridge U n i v e r s i t y Press, 1964.

12. I-IEINS, M., Selected topics in the classical theory of functions of a complex variable, I-Iolt, l~inehart, a n d Winston, 1962.

13. I-IELMS, L., Introduction to potential theory~ }Viley-Interscienee, 1969.

14. tIi~BEX, A., E i n raumliches analogon desWiman-Heinssehen Satzes, Studies in m a t h e m a t i c a l analysis a n d related topics, Stanford University Press, 1962, 152--155.

15. TALPV~, 3/i., On the sets where a subharmonic function is large, thesis, Imperial College, London, 1967.

l~eceived March 13, 1974 t~onald Gariepy a n d J o h n Lewis U n i v e r s i t y of K e n t u c k y

Lexington, K e n t u c k y 40506