<-M, f lu(r,O)lp O<M THE POISSON INTEGRAL.

h S T U D Y I N T H E U N I Q U E N E S S O F H A R M O N I C F U N C T I O N S .

BY

FRANTISEK WOLF

i n S T o c K H O L M . 1

I n this paper I w a n t to deduce some uniqueness theorems for h a r m o n i c f u n c t i o n s w i t h assigned b o u n d a r y values in t h e u n i t circle.

I n this direction there exists a classical result for h a r m o n i c f u n c t i o n s con- t i n u o u s on the boundary, based on the fact t h a t h a r m o n i c functions t a k e t h e i r extreme values on the frontier. Here, there was understood by a b o u n d a r y value w h a t we are g o i n g to denote by uP(z) (cf. 2. I). I t was shown t h a t a f u n c t i o n , h a r m o n i c in a d o m a i n a n d such t h a t u D-~ o at all b o u n d a r y points, vanishes identically. Even discontinuous b o u n d a r y values, defined as limits along t h e radius, have been considered, especially by G. C. Evans, in his book on t h e l o g a r i t h m i c potential. The h a r m o n i c functions h a d to be restricted by one of t h e f o l l o w i n g m a j o r a n t s :

2 ~

lu0,0)l <-M, f lu(r,O)lp O<M

o

The aim of this paper is t o consider (i) more general b o u n d a r y values, such as UD, defined in 2.2 or limits uL along the radius, or even more general curves, defined in 6. o; (ii) more general m a ] o r a n t s .

Thus we prove in 7 . 4 . 6 a result which in a simplified form runs;

I f (i) u(r, 0) is h a r m o n i c in t h e u n i t circle,

(ii) at every b o u n d a r y point 0o, u (r, 0) converges to zero, if (r, 0)-~ (I, 0o) in any sector (cf. def. in I.O),

(iii) for every e > o, there is a n B < I such t h a t [u(r, 0 ) [ ~ e ~/(1-r)'' for r ~ R ,

t h e n u ~ o.

N o w : M a c a l e s t e r College, St. P a u l , M i n n . , U. S. A.

5

66 Franti~ek Wolf.

I f m--< I, t h e n we may, in (ii), r e q u i r e a t e v e r y b o u n d a r y p o i n t t h e con- vergence o n l y in a sector, h o w e v e r small, which contains t h e r a d i u s in its i n t e r i o r .

I f we define a b o u n d a r y value n o t as t h e limit in a sector, b u t t h e l i m i t a l o n g a curve, especially t h e radius, t h e n we are no m o r e able to p r o v e a u n i q u e n e s s t h e o r e m . E v e n if these b o u n d a r y values are zero, t h e r e can exist b o u n d a r y points, n e a r which t h e h a r m o n i c f u n c t i o n is u n b o u n d e d . B u t we can still prove the f o l l o w i n g t h e o r e m .

I f (i) u(,',0) is h a r m o n i c in t h e u n i t circle,

(ii) lira I (r, 0)1< for all 0,

(iii) lira u (r, 0)_< o--< lira u(r, 0) f o r a l m o s t all 0,

r ~ l r ~ l

t h e n t h e r e is a reducible set of points N, such t h a t u ( r , 0) takes c o n t i n u o u s l y t h e value zero at every b o u n d a r y p o i n t w h i c h does n o t belong to 9~. A t an isolated p o i n t of this set, the a n a l y t i c f u n c t i o n f ( z ) , f o r w h i c h

3~f(z)--u(r,O),

has a pole of finite order.

W e r e c a l l t h a t a r e d u c i b l e set o f points is such t h a t it c o n t a i n s no p a r t d e n s e in itself. Now, if we study t h e b e h a v i o u r of t h e f u n c t i o n in t h e neigh- b o u r h o o d of an i s o l a t e d s i n g u l a r p o i n t (eft section 4), we m a y hope to find the m o s t c o n v e n i e n t a n d best c o n d i t i o n s f o r the n o n - o e c u r e n e e of an i s o l a t e d s i n g u l a r b o u n d a r y point. B u t a reducible set w i t h o u t isolated points is empty. H e n c e t h e s e last c o n d i t i o n s t o g e t h e r w i t h t h e c o n d i t i o n s of t h e above t h e o r e m , give u n u m b e r of very g e n e r a l u n i q u e n e s s t h e o r e m s (eft section 8).

T h e f u n d a m e n t a l t h e o r e m , a g e n e r a l i s a t i o n of t h a t given above, is e n u n c i a t e d in 7. o. T h e importan~ n o t i o n of a r e s t r i c t e d Poisson I n t e g r a l on an arc of t h e f r o n t i e r is defined in 2.9. I t describes a, w h a t we m a y call, n o r m a l b e h a v i o u r of a h a r m o n i c f u n c t i o n in the n e i g h b o u r h o o d of t h e f r o n t i e r . I t m a k e s it possible to use f o r such f u n c t i o n s t h e t h e o r y of t h e P o i s s o n integral, of which an a c c o u n t can be f o u n d e . g . in E v a n s ' book.

]in section 2 t h e r e is a n u m b e r of r e s u l t s f r o m t h e t h e o r y o f t h e P o i s s o n i n t e g r a l By m e a n s of a c o n f o r m a l r e p r e s e n t a t i o n of a g e n e r a l d o m a i n on a u n i t circle, we define t h e P o i s s o n i n t e g r a l f o r g e n e r a l domains (eft 24). T h e m o s t i m p o r t a n t results are to be f o u n d in 2. I4, 2. I8, 2 . 1 9 a n d ~. 2o.

In section 3 we deduce some known lemmas for harmonic and analytic functions. The chief difference from the classical cases is the use of T. Carle- man's extension of LindelSf's theorem and the authors extension of the Pragmen- LindelSf's theorem.

Section 4 is devoted to the study of a harmonic function which is con- tinuously equal to zero on the boundary everywhere except at one point. We find it convenient to study the problem in a half-plane.

In section 5, we state and deduce lemmas from the theory of conformal representation.

Section 6 is devoted to lemmas and can be considered as the beginning of the proof of the main theorem. We find it stated in 7-o and proved in 7. I and 7.2. I n 7. oi and 7. o. 2 I state two particular cases which are important for the uniqueness problem of trigonometrical expansions. I n 7. o. 3 and 7 . ~ I state alternative conditions for the validity of 7. o. I n 7.4 we prove a result based on a double system of curves, which might prove useful in many cases.

We deduce from it, in 7.4.6, a very important result of which I have above stated a particular case.

Section 8 deals with conditions which make isolated singular points im- possible and lead to uniqueness theorems. The great variety of different possible uniqueness theorems has not been exhausted and the section presents rather a few examples for constructing such theorems with suitable conditions.

In 9. o we state our main theorem for general domains.

As regards the conditions in 7. o, the first four are essential. Condition (iii) may be somewhat relaxed by demanding only lira u (r, (Kr, ,,~)) > -- oo. Then we

r ~ l

have to substitute Poisson Stieltjes integrals in our resoning. I f we restrict a(~) in (iv) to be < o o, the assertion of the theorem remains the same. Otherwise we should get at the end of 7.0 u = R 1 P I c u c (cf. 2. I7) instead of u = R P I c u c . I t seems t h a t with our method conditions (v) and (vi) are essential, too. But, by using some other method, they might possibly be improved.

The first result o f this kind I have explained in a talk at Professor G. H.

Hardy's Conversation (3lass in Cambridge in December I937. In their full generality, the results have been made public when I had the honour to be invited by Mittag-Leffler's I n s t i t u t e to deliver two lectures on harmonic functions.

68 Franti~ek Wolf.

I want to express my deep gratitude to the Swedish government which, by granting me during two years a scholarship, made it possible for me to finish this paper3

I. Notation. We denote the complex variable by z ~ - x + i y ~ r d ~ or by

~ - ~ + i ~ = Q # :~. The functions f ( z ) , g(z), h(z) are supposed to be analytic in certain specified domains. Their real and imaginary parts, or harmonic functions in general, will be denoted by u (z) -~ u (x, y) -~ u (r el~ v (~, V). For functions transformed by a conformal representation ~ : ~ ( z ) , z = ~(~) we shall use the notation f(~), u(~) etc. The domain D(z) in the z-plane and the domain D(~), in the ~-plane, are connected by the conformal transformation. The letter C is reserved for the unit-circle and H for the upper half-plane. The domains will be supposed simply-connected and bounded by free rectifiable Jordan curves.

The functions a(z), b(z) will be defined only on F ( D ) , the frontier of D. I f the unit-circle C(~) is conformally represented by ~ ~(z) on the domain D(z), and if u(~) or a(~) have a certain property A, then we say t h a t u(z) or a(z)

>>have the property A in the corresponding unit-circle>>. Further, N(zo)will denote a neighbourhood of z o and e a positive number arbitrarly small. I f z o is a point of the boundary F ( D ) , t h e n we say t h a t z converges to zo in a sector, if, denoting by Q (z) the distance of z from F(D), we have lira I ~ (z) > o. The geometrical significance of this is well known. All boundary functions in C are supposed L-integrable as functions of 0. Generally, a boundary function in D will be supposed to 'be >>L-integrable in the corresponding C,>.

2. I. Definition. L e t u(z) be defined in D(z) and z 0 < F(D). Then a is a boundary value of u(z) in z 0, if there exists a sequence {zk} such t h a t zk < D, zk-~ z0 and u (zk)-~ a. All such boundary values in a point Zo are values of the function u ~ (Zo).

2.2. Definition. The number b is a boundary value in the strict sence, if there exists a sequence {zk} which converges in a sector to z o and such t h a t u (z~) -~ b. All boundary values in the strict sense at the point Zo are values of the function UD(Zo).

1 I a m also v e r y m u c h i n d e b t e d to P r o f e s s o r F. Carlson for h i s k i n d n e s s and for reading the m a n u s c r i p t a n d calling m y a t t e n t i o n to a n u m b e r of i m p e r f e c t i o n s , a n d to P r o f e s s o r T. C a r l e m a n , w h o as D i r e c t o r of Mittag-Leffler's I n s t i t u t e h a s i n v i t e d m e to lecture.

2.3- I f uD(z) is finite in a closed set of F ( D ) , t h e n uD(z)is t h e r e bounded.

The analogous resul6 for us(z) is false. 1

2.4. I

2 . 4 . D e f i n i t i o n . I f

2~

I f I - - r ~

u ( r e i ~ I - - 2 r c o s ( 0 - - ~ ) + r ~a(e'~)d~F'

0

t h e n we say t h a t u(z) is the Poisson i n t e g r a l of t h e b o u n d a r y f u n c t i o n a ( z ) . . . S h o r t l y we shall write: u ( z ) = P i c a (z). I f u (z) is defined in D (z) a n d ~ = ~ (z) represents c o n f o r m a l l y C(~) on D(z), t h e n

is equivalent w i t h

u (z) = P I ~ a (~) u (~) = P I c a (~).

2.5. I f u (z) is defined in H , t h e n t h e two assertions 2.5. I

2 . 5 . 2

are equivalent.

The c o n d i t i o n

u (z) = P I . a (z),

y f a (x') (z) = ~ (x - x') ' + v ~

--ao

a~

fla(x) ldx

J I + x ~ < o o

d :~P

is equivalent w i t h the L-integrability of a(x) in t h e corresponding C.

The relation z. 5.2 follows f r o m 2. 4. I by a conformal r e p r e s e n t a t i o n of C o n IT.

2.5. I f u (z) -~ P I p a (z), then Up (Z) = a (z) almost everywhe~'e, and f o r the upper and lower bounds we get the relations

U.B. u(z) -< U.B. a (z) L . B . u(z) >-- L . B . a (z).

t Cf. E. LIIqDELOF, Calcul des r6sidus, Coll. Borel, Paris 19o5, p. 121. From the function E/~(z) we can construct a Gegenbeispiel.

function boundary p. 633).

70 Franti~ek Wolf.

For D - ~ C this is a well known property of Poisson integrals. 1 For a general domain, it follows from the definition of the Poisson integral (cf. 2.4).

W e get the existence of up (z) only almost everywhere in the corresponding C.

Since F ( D ) is rectifiable, this is also, by a theorem of Riesz-Privaloff ~, almost everywhere on F(D). The two inequalities for the bounds are easy deductions

f r o m 2 . 4 - I.

2.7. I f u (z) is bounded, then u (z) -~ P I p UD.

By 2.4, it is sufficient to prove it for D ~-- C. But then, it is a well known result (cf. Evans, p. 52).

2. s. I f and

~ a (z) b (z)

2

almost everywhere on a rectifiable p a r t K o f F ( D ) , then ( u - V) D : 0 at all interior points o f K.

W e may again consider only the case of D = C. W e have u - - v - ~

= P I c [ a ( z ) - b(z)] and at K, which is a part of the circumference (a < 0 < fl), a ( z ) - b(z)-~ o almost everywhere. W i t h o u t changing the functions u(z), V(z) we may suppose a(z) and bIz) to be equal everywhere in (a, ~). And it is a classical result that the Poisson integral converges uniformly to its boundary in any interval which is interior to an interva.1 of continuity of the function (cf. Hobson, Theory of functions, II. Cambridge 1926,

2.9. Definition. I f there is a z o ~ F ( D ) and a neighbourhood N(zo), such that u(z) is a P I in N(zo). D, then we shall say that u ( z ) i s a restricted Poisson integral s in z0, and we shall write u(z)~-RPID(Zo).

I f D I ~ D , K ~ F ( D . D 1 ) and u ~ P / D , , then we shall say t h a t u(z) is a restricted Poisson integral on K. Shortly u ( z ) ~ - R P I D ( K ) .

2. io. I f u =- R P I ( z o ) , then u = R P I ( z ' ) f o r all z'" < F ( D ) . N(zo).

2. I I. I f u -~ t~ Pl(zo), then up is. one-valued and finite almost everywhere on

F(D).

G. C. EVANS: The Log. Potential, New York, 1927, p. 40. Corallary. Particular case of F a t o u ' s theorem in Acta math. 3 o. I9O6 , p. 345-

F. & M. RIESZ, Ueber Randwerte analytischer Funktiouen, Stockholm Congress 1916; LUSlN- PRIVALOFF, Ann. de l'Ecole Normale, T. 42, I925; t h e simplest proof F. RIESZ, Math. Zeitschrift, Bd. 18. 1923. P. 95.

a W. H. Young has introduced the notion of a , r e s t r i c t e d Fourier series~. Cf. e.g. HOBSON I I p. 6116.

I f u ~ - R P I ( K ) then Up is one-valued and finite almost everywhere on K.

This follows f r o m 2.9 a n d 2.6.

2.12. I f the values of UD(Zo) are finite, then u = RPI(zo). I f uP(z) is finite on g < F ( D ) , then u = R P I ( K ) .

This is a consequence of 2. I, 2 . 7 a n d 2 . 9 , because t h e values of u D(z) form closed set of points and if it does n o t c o n t a i n the infinite point it m u s t be bounded. F o r t h e second p a r t of t h e assertion we use the He~ne-Borel theorem.

2.13. I f u(z) is harmonic in D~, and D < D ~ , then u - ~ R P I ( K ) for a K which lies in the interior of D~.

On such a K, uD(z) is evidently bounded.

2.14. I f u(z) = PIp,, and D < DI, then we have also u(z) = PID.

By a eonformM r e p r e s e n t a t i o n we reduce the case to D = C. F u r t h e r u(z) is t h e difference of two Poisson integrals with positive b o u n d a r y functions.

Hence, w i t h o u t loss of generMity, we m a y suppose t h a t u ( z ) h a s a positive b o u n d a r y f u n c t i o n on F(D1). W e p u t

The

if UD,(Z) <-- A i f uD,(z) > A.

f u n c t i o n AU (Z)= PID,(AUD,) aS a f u n c t i o n of A is non-decreasing a n d lim Au (z) ~- u (z)

in Dl (cf. 2.4.1). I n D = C, AU(Z) is bounded. W e have therefore (of. 2.7.)

2 r r

f ^l--q*

Au (~) = P I c (Auc) = ~ i - 2 q c o s ( a - ~1 + 0'" Auc (e"p) d ~.

0

I f A - + oo, t h e limit in the first member exists. U n d e r the i n t e g r a l sign t h e r e is a non-decreasing sequence of positive functions. H e n c e lim Auc(d'P)

A ~ a o

exists a n d is L-integrable. Thus we g e t 2. I5. I f

u (~) = ~ c (uc).

(i) K < F (D. D1)

(it) u = P I p up, v = PIp, vD,

(iii) K rectifiable and u p - - v D , almost everywhere on K, then ( u - v) D = o at all inner points of K.

72 Franti~ek Wolf.

By 2.14, u---PID. D, a n d v : PII). D,. NOW, the result follows f r o m 2.8.

2.16. I f u = R H ( g , a < ~ ) ) and v = R P I ( g , a ( ~ ) ) then ( u - - v ) " = o at all interior points of K.

This is a corollary of 2.15.

2. I 7 9 Definition. I f

2 ~

I f I ~ r ~

u(z)=~--~ ~ _ 2 r c o s ( 0 _ 9 ) +

r~dU(~)

0

a n d U (9) : U1 (9) + U~ (9), where U1 is a n i n t e g r a l a n d U~ a non-decreasing function, t h e n we shall say t h a t u (z) is a lower Poisson-Stieltjes i n t e g r a l and we shall write

u(~) = l P I ~ ( v ) .

d U I t is k n o w n ~ t h a t uc exists almost everywhere a n d t h a t it is equal to d--O"

2.17. I. A suffiNent condition for u (z) to be a lower Poisson-Stieltjes integral is the existence of an A such that

u (z) >-- A , z < D . 2. I7.2. I f

u(~) = 1PX. (u), u>-- P I D ( d ~ ) = P I D u . . then

I t is sufficient to prove t h e inequality for D =- (7/. I f U is an integral, t h e n u = P!~uD. W e may, therefore, suppose t h a t U is a n o n decreasing function. T h e n u (z) >_ o.

Using F a t o u ' s t h e o r e m we deduce from

2 ~

u (z) = lira flJ- e ~ l z ~ d f O g - - 2 o r c o s (0 - - 9 ) + r ~u(ed'r)d9

0

the desired result

2 ~

I f I -- r8

u(~)_> ~ i - 2 r c o s ( O - 9 ) + r ~uc(e'~)dg"

0

x EV~.Ss, p. 4 o, Corollary.

2. I8. I f u(z) is harmonic in 1),

2. I8. I Dk < D, 2fDk + ~F(Dk) ~ C,

2. I 8 . 2 u(z) = PIDk UD~

2. I 8 . 3 [ul <_ M for z < F(Dk)D, k = I, 2, . . .

2. I 8 . 4 . there exists a boundary function a(z), equal to one of the values of UD, which is integrable in the corresponding C, then

u = P I p up.

W i t h o u t lack of g e n e r a l i t y we m a y suppose D ~ C, a n d D k - D t ~ o, k ~ 1.

I f t h e last c o n d i t i o n would n o t be satisfied, it would be easy to c o n s t r u c t new D~ s a t i s f y i n g this a n d all t h e previous conditions.

T h e set F ( C ) . F(Dk). F(D~) consists of at most t w o points. L e t t h e r e exist two p o i n t s z,, z2. T h e n we can join t h e m by two curves Kk < Dk a n d Kt ~ D,.

D1 will be c o n t a i n e d in a d o m a i n b o u n d e d by Kk a n d a c i r c u l a r arc Zl, z2, and F(D,) can have p o i n t s on z~,z2, b u t n o t on t h e c o m p l e m e n t a r y arc. Since K, separates Dk f r o m zl, z2, F(Dk) c a n n o t h a v e points on z,,z2. T h i s shows t h a t t h e r e are no more c o m m o n points on F ( C ) .

T h e points o f F(Dk) are, t h e r e f o r e (i) i n n e r p o i n t s of arcs F ( D k ) . F ( C ) , t h e n uc =-UDk; (ii) points w h i c h are n o t on F ( C ) , t h e r e [UD] g M; a n d ( i i i ) t h o s e of t h e e n u m e r a b l e set

F ( C ) - F ( D k ) . F(D~).

k,l

W e define a n d

(z) = max (a (z), ^- N) ( z ) = p r r (z).

A t a l m o s t all t h e points of the first k i n d we shall h a v e ~vuD k >--UDk, since at those points u ~- R P I ( z ) . A t t h e p o i n t s of t h e second k i n d we h a v e uDk~--_Ar, a n d t h e e n u m e r a b l e set of t h e p o i n t s of t h e t h i r d k i n d m a y be d i s r e g a r d e d . H e n c e in v i r t u e of 2. I 8 . 2 we shall have ~-u - - u --> - - h r - M i n all Dk. T h e same is clearly t r u e a t t h e p o i n t s of F(Dk). H e n c e we g e t z e u - - u - ~ l P I c . Since (lvu - - u)c ~ o, 2.17- 2. gives ~.u - - u >-- o or

u <- P l c ~uc.

74 Franti~ek Wolf.

For N - ~ r162 this becomes

u <-- P I c uc.

By means of a similar reasoning we can deduce the opposite inequality and complete the proof.

2. I9. I f (i)

(ii) (iii) then

D ~ %Dk

for a l l z < F ( D ) there is an N(z) and a k such that J O . N ( z ) < D k , and u = PIDk "Dk,

u = P I p up

We may again suppose D : C. I n every N(z), uc is almost everywhere one-valued and finite and, by (iii) and 5.5, L-integrable on F ( C ) . ~ ( z ) . By means of the Heine-Borel theorem it is easy to show t h a t uc is L-integrable on the whole T'(C).

W i t h o u t loss of generality we may suppose t h a t the 2Y(z) are all circles with centre in z. Since uc exists one-valued and finite almost everywhere, we can construct, by diminishing, circular neighbourhoods, such t h a t ue would be finite and one-valued at the points F ( N ( z ) ) . F ( C ) . Now, u ( z ) i s bounded on F ( N ( z ) ) . C. To every z ~ F ( C ) we make correspond a closed circular neighbourhood N ' (z) whose radius is half of t h a t of N(z). By the Beine-Borel theorem we may cover F ( C ) by a finite number of N'(z), say N'(zk), k = I, 2 , . . . K. Then

K

D~+I = C - y , X(zk)

k = l

is completely interior to C and there is an M such t h a t lu(z) l ~ M in it.

:Now it is easy to see t h a t the conditions of theorem 2.18 are satisfied for Dk = ~V(z~).

2.20. I f u = R P I ( z ) at all points z of l~(C), then u - ~ P I D u D .

OUt of these neighbourhoods (s. 2.9) we can choose a finite number of them which completely cover F ( D ) and such t h a t in the rest of D, u is bounded.

Then we can apply 2.19.

2.21. I f K < F ( D ) is a curve with its both eudpoints and u ( z ) = l ~ P I ( z ) f o r all z < ~K, then u = R P I ( K ) .

The proof follows from 2.9 and 2.20.

2.22. I f u = P i e , then

uniformly in O.

W e start from the expression

2 r

I lei~p-~'re i~

f ( z ) = ~ j ~ _ r e i o uc (~) d9~

0

and establish the result by the usual reasoning.

2.23. I l l ( z ) = u + i v and u -~- PIH(a(x')), where

then

unijbrmly f o r z ~ ~ in H.

t

The condition for a (x') is equivalent with the L-integrability of the boundary function in the corresponding C. We may therefore use 2.22. By

z - - i z 4 - i we represent C(~) on H(z). Hence we get

I I , ( x ' + y ' + I I'

i - l ~ p - y 2(x'+ y~+ y +,) 2y

uniformly for z-+ ~r

2 . 2 4 . I f 2 . 2 4. I

2 . 2 4 . 2

2.24.3

Now, 2.22 gives the required result.

f ( z ) = u + i v , u = P I ~ (a (x')) ]a(x'))]~<M for x' > N , then f ( z ) is bounded f o r x > N + 2, o < a < y < b.

76 Franti~ek Wolf,

W i t h o u t l a c k of g e n e r a l i t y we m a y s u p p o s e N ~ o. T h e n

w h e r e

a n d

(//)

u = ~ +

(x - x ' ) ~ + y= ^{d x P ~} ~ 1 "~- u 2

• u f dx" < M

lu, l-<-~- (~__ ,)= ^{+ y , -}

0

0 0

I a(x')[ y~dx'<

I",l-<Yz ~,+~,=+ ~ j , +~,,

- - a o - - q . :

f o r o < y < c . H e n c e u is b o u n d e d in t h e s t r i p x > N + I o < y < e , f o r a n y a r b i t r a r y p o s i t i v e c.

N o w we u s e 3. r in a w a y s i m i l a r to 3.2 a n d we deduce t h e b o u n d e d n e s s of v in

x > N + z , o < a < y < b < c .

3.0. I n t h i s s e c t i o n we shall p r o v e s o m e r e s u l t s a n a l y t i c f u n c t i o n s w h i c h we shall use later.

3. I- I f ~ u [ - - < M i n r < . R , t h e n

a b o u t h a r m o n i c a n d

F r o m

we deduce

Ou] 2M

O r ~=o, o--- e

2 ~

u (r e ~'~ ~ - _2% I

f

d - 2 ,- o cos (0 - ~) + r' u (e e i~) d ~

-

^{, ,}

0

2r

, - o o -

0

H e n c e t h e d e s i r e d i n e q u a l i t y easily follows.

3.2. I f I ~

u(z)

is h a r m o n i c in D, defined by a < a r g z <

2 ~ u ( z ) - ~ O (z e) f o r z - ~ :r u n i f o r m l y in D ; t h e n f ( z ) = u + i v = o (z~) f o r z--* ~ , u n i f o r m l y f o r a + ~ ~ a r g z ~ f l - - 8.

F o r ~ = z o.s, o < s < I we g e t

zo ~o -~o

0 0 0

where ~ OU is t h e derivative in t h e d i r e c t i o n of the n o r m a l to a r g z ~---argzo. I n o r d e r to g e t u p p e r b o u n d s f o r these derivatives, we use 3. I f o r cercles w i t h centres in ~, a n d radius e [ ~ [ . T h e rest of t h e p r o o f is s t r a i g h t f o r w a r d .

[ 1 / ]

3.3. I f l ~

f ( z ) : u + i v

is an entire f u n c t i o n , 2 ~ [ u [ - - e x p e[z[n+~ yn f o r all e > o a n d [ z [ > R(e); 3 ~

u(z)-~ o(z TM)

f o r z -~ or u n i f o r m l y in

a ~ argz<fl,

l i n t e g r a l ; t h e n

f(z)

is a p o l y n o m i a l of /-th degree.

U s i n g 3.2 f o r cercles w i t h centre at z r a d i u s

y/2

we g e t

[(

^Y

)+:/()1

^{~ Y} < exp ,

2 n+2 [ g I n+~ yn ^'/]

3.3. ~ I f ( z ) l -< exp ~ I z l + ~ 2 - for a l l e > o a n d [ z l > R ( ~ ) .

By 3.2, it follows f r o m 3 ~ t h a t

~ + ~

f o r a r g z ~ - - . 2

f(2')

⁼

0

^{(~,l 4-1}

T h e f u n c t i o n

I f g

g (,) = (z) - f ( o ) - V. f (o) . . .

(o)] /z, +,

Now, we use t h e generalized

~ + ~ satisfies 3 . 3 . 1 a n d is 0(I) on a r g z = - - .

2

t h e o r e m of P h r a g m 6 n - L i n d e l S f I to the f u n c t i o n

g(e~(~+~)/2~ ~)

in ~ ( ~ ) > o. W e g e t

f(z) -~ o (z z q 1)

u n i f o r m l y f o r all z--* or H e n c e

f(z)

m u s t be a p o l y n o m i a l of /-th degree.

3.4. I f

f(z)

is a p o l y n o m i a l of degree l, whose coefficients are real, a n d t h e r e exists a sequence of p o i n t s {ak} such t h a t I ~ ak -~ oo, 2 ~ J { f ( a k ) } --- o(ak), 3 ~ lira a r g ag ~-z~x where x is n o t a f r a c t i o n , with a d e n o m i n a t o r less t h a n or equal l, t h e n

f(z)

is a c o n s t a n t .

1 F. WOLF, An extension of the P. L. theorem, Journal of the London Math. S. I939.

p. 208.

78 Franti~ek Wolf.

If, indeed, t h e h i g h e s t p o w e r in f ( z ) is ekz k (o <-- k <-- 1), t h e n , if 3 ~ is satisfied,

~5"{f(ak)} - ek I~* I * sin (k ~r x).

F o r k > o, this would be a c o n t r a d i c t i o n to 2 ~

3.5. I f I~ f(z) is a n a l y t i c for O, <-- 0 <-- 0~, 2 ~ If(z) l _< exp It r <~176 f o r a n , > o , o ~ _ < o _ < o ,

~nd I~1 >-R(~),

3 ~ f ( z ) = o (z ~) f o r 0 --- O,

f ( ~ ) = o (,~) f o r o = o~

4 ~ (0) = (0 0 0 b + (0, O)

O~ ^-- 0 i t h e n f ( z ) = o (z'P (e)) u n i f o r m l y f o r 01 -- 0 --< 0~.

W e p r o v e it by a p p l y i n g P h r a g m 6 n - L i n d e l S f ' s t h e o r e m to f(z)/z'P (-il~ in 01 ~< 0 --< 0~.

I 0

2 0

0

3 then

4.0. I f

u (x, y) is harmonic i n H , u = R P I ( z ) f o r all z < F ( I t )

fl-.(~,o)l

- - m

oo oo

- (*, v) = ~ ( . _ ~)~ +

QO

The series ~ , an z" eonverges f o r all z and J ( a k ) = o f o r all k.

1

By 2 ~ a n d 2. II, UH iS one-valued a n d finite almost e v e r y w h e r e . Condition 3 ~ is e q u i v a l e n t with t h e i n t e g r a b i l i t y of UH in t h e c o r r e s p o n d i n g C. By 2.5, 1o (x, y) = P I H uH a n d 2 . 7 gives

(u ^- p ) ~ = o.

H e n c e s ( x , y ) is c o n t i n u o u s and e q u a l to zero on y = o. I t can be e x t e n d e d to a f u n c t i o n which is h a r m o n i c in t h e whole plane by t h e e q u a t i o n

4. o. I s (x, - - y) = - - s (x, y).

a n a n a l y t i c e n t i r e f u n c t i o n I t m a y be c o n s i d e r e d as t h e i m a g i n a r y p a r t of

co

~ a ~ , z n w h i c h is r e a l o n t h e r e a l axis.

1

4. I. C o n d i t i o n s f o r t h e n o n - e x i s t e n c e o f t h e s i n g u l a r p a r t .

4. I . I . I f the conditions o f 4. o are satisfied a n d u (z) = o (z2/y), then u = P I H UH.

B y 2 . 2 3 , we find t h a t s = o ( z ~ / y ) in H , a n d by 4. o. I also in t h e w h o l e plane. F u r t h e r 3. I gives h (z) = s (z) + i t (z) = o (z2/y) w h i c h s h o w s t h a t

3 ~ [h (z)] --- o ~ o r u = P l ~ .

4. I. 2. I f the conditions of 4. o are satisfied and (i) u(z) = o(zm), (ii) u ( z ) = o (z) or~ a sequence o f points {ak}, such that lim ] a~.l ~- r162 lira a r g ak -~ 7~ z where z is

k ]c

not a f r a c t i o n w i t h a denominator less than m, then u = P I H u~.

p > m -!

C o n d i t i o n (i) gives, in t h e s a m e w a y as above, s ( z ) = ~ / ~ , a ~ z k I . N o w ,

L I A

we a p p l y 3.4.

[ 7 ]

4. I. 3. I f the conditions o f 4. o are satisfied a n d (i) I u I ~ e x p ~ I z I "+~ y'~

f o r all e > o and I z I > iR (e)(ii) u ( z ) = o (z g+ 1 ) f o r z -+ ~ u n i f o r m l y in a < a r g z "< fl, 1 integral, (iii) there exists a sequence of p o i n t s {ak} such that I ~ l i m [ a k [ = ~ ,

k

2 ~ u ( a k ) - ~ o ( a k ) , 3 ~ lim a r g ak---zx where z is not a f r a c t i o n w i t h a denomi~ator

k

less than or equal to l, then

u = P I H U H .

(i) (ii)

W e d e d u c e t h e p r o o f in t h e u s u a l w a y f r o m 3-3 a n d 3.5.

4 . 2 . I f

u ( x , y) is harmonic in H , u = i e P I ( ~ ) fo,- I~l >- M,

- - M

(.

_ ~), + v~d~ + v(~) + J ~ a , , z ~

M 1

(iii) then

z Cf. Carleman's generalization of LindelSf's th. Acta m, 48.

80

where v(z) is harmonic in uniformly for z - * r The all k.

Similarly to 4. o

Franti~ek Wolf.

c~

series ~ a,,z" converges for all z and r o for 1

we have (U -- p)H ---- O f o r

Ixl ~-M.

H e n c e q ( x , y ) =

= u - - p = - - q ( + x , - - y ) defines a f u n c t i o n h a r m o n i c in

I~1-> M,

which m a y be considered to be the i m a g i n a r y part of a f u n c t i o n g (z) analytic in I zl ~ M.

This can be w r i t t e n g ( z ) = ~ a,~z '~, where ~(a~)----o. W e may, evidently, take a 0 = o . T h e n we get

m l

which proves our assertion.

4.3. I f (i) zo is an isolated singular point on F ( C ) , i.e. i f there is a circular neighbourhood N(zo), such that at all points z < F ( C ) . N(zo), z 4= Xo, u = R P I ( z ) (ii) u (z) = o (z -- Zo) - m in z < C. 2V(Zo),

then there are constants ck such that

4 . 3 . 1 ^{u (~) - 3 ' ~k (z - z 0 ?}

] , f

^{- ~ ~ -} 2 ~. cos (e - <f) + r" dq~

L k = 0 l F ( C ) . N ( z o)

is a harmonic function which is continuous in N(zo). C, zero on F ( C ) . N(zo) and o (z - zo).

I f we define the functions

0 Sn-I

4 . 3 . 2

$1=

^{I q-} 2 Z T n C O S 9 1 0 , S n - - O0

1

which are harmonic in C and S D = o at all points of F ( C ) except at 0 -= o, then the imaginary p a r t of the sum in 4.3. I may be substituted by a sum of the form."

4 . 3 - 3 Proof.

1d$

d~ Sk (r, 0 - - 0o), o ~ ( d D = o.

k = l

W e represent, by

. z + z o z - - z o

H ( ~ ) o n C(z) a n d apply 4. 2. W e g e t

v f

~-(~)

v "~ ~

~

- (~) = ~ (~ - ~,) +

F(H) 2r

Now, we h a v e only to p r o v e t h a t t h e i m a g i n a r y p a r t of t h e series c a n be w r i t t e n in t h e f o r m w h i c h occurs in 4-3- I. B u t this is e v i d e n t f r o m

~ = i + 2 i Z o ' - - I

z -- z o N o w we show t h a t 4 . 3 . 3 is a n a l t e r n a t i v e form.

W e h a v e f o r z 0 = I

[ ] _[

8 ~ = 3 ~ ~ + 2 ~ ' ~ = o ~ = - , : 7 - i z + [ z - - I J z - -

1

Similarly

s~ o s~

[o~]

=~o0

~7

= _ J(~).

[

~ - - - J i ~ - z . Z = 3 :=

[ (" + ')2 1 [ I '~.] I

2 2 ~ _- r

Now, we shall suppose t h a t , f o r all k g d - - I , we h a v e 4 . 3 . 4

( k -

~)!

& -Y,c~,,~gC'),

c~,~ - - 2~_ ~

a n d we shall show t h a t this is t r u e even f o r k = 1. W e have indeed S1 0 S l _ l __ I k - 1

l - - I

which is of the r e q u i r e d form a n d cz, 1 - - e t - l , l - 1 . T h e e q u a t i o n s 4 . 3 . 4 are 2

such t h a t it is clearly possible to express J ( ~ ) by m e a n s of S~. W e have, thus, p r o v e d t h a t 4 - 3 . 3 c a n be s u b s t i t u t e d in 4-3. I.

5.0. I n this section we shall f o r m u l a t e some t h e o r e m s r e p r e s e n t a t i o n .

5. o . I . O s t r o w s k i has p r o v e d t h e f o l l o w i n g t h e o r e m s : 1

on c o n f o r m M

i A L E X A N D E R O S T R O W S K I , Acta math. V. 64, I935, p. IOO, I16, 173.

6

82 Frantiiek Wolf.

L e t ~---~ q~(s) represent e o n f o r m a l l y D(~) on

D(z).

F u r t h e r

.F(D(s))

h a s at Zo a ))corner)), i . e . has two h a l f t a n g e n t s , which f o r m an i n n e r angle 7 > 0 . Similarly F(D(~)) h a s a corner a t ~o = ~ ( z o ) with a n a n g l e 7~ > o . T h e n

5 . 0 . 2.

a n d 5 . 0 . 3 .

arg

Z - - 2' 0

a r g ( z - - S o ) + c + e(Z--Zo)

a r g ~ ' ( z ) = ( Y 7 - I ) a r g ( z - - Z o ) +

c + ~ ( Z - - S o ) .

W h e r e ~, sl converge to zero, as z - ~ z 0, u n i f o r m l y in a sector.

5 . 0 . 4 . W e denote by c~, c~ two a r b i t r a r y positive constants such t h a t c 1 < c ~ . If, n o w z l ~ z ~ are two points in a sector of

s o

such t h a t

t h e n

c<[ z,-zo[<c

I Z ~ - Zol

w h e n z~, z.2 converge t o w a r d s z 0.

5-o. 5. F i n a l l y ,

log [ z -- Zo [ 7 a n d

log [ s - - Zo[ 7 ^{- - I} for z converging t o w a r d s

z o

in a sector.

5.o. 6- (Warschawskil). W e d e n o t e by s the l e n g t h of

F ( D ) f r o m

the point z o on and by

O(s)

the angle between the real axis a n d the t a n g e n t to

F(D)

•

at 8. If(i) f {

o

exists, (iii)

D(z)

on D (Z), then

cos 0(s) -- cos ~?(+ O)[ds, 6 > o converges, (ii) lim ]z(s)--Zo]

8 8

has all i n n e r angle Q at z 0 a n d (iv) ( - ~ q~(s) represents C(()

t ~Tbe r d a s R a n d v e r h a l t e n etc. M a t h . g e i t . 35 (1932) P. 427 9

lira ~ (~) - - ~ (~o)

z.--.,z, o (~' - - ~O)Zr/q

exists u n i f o r m l y for z < D (z).

5.o. 7. I t is easy to show t h a t t h e conditions are satisfied in the case

that I o ( 8 ) - o(+_ o)1 -< c l ~ l ~ ,, > o.

F o r condition (i) this is e v i d e n t and, as for (ii), we have

8 8 8

0 0 0

0

(o(~)--o(+_ o)) d s [ -~-

0(8) - o ( • o)1 d s )

= 8 + O(s~+l).

5. o. 8. (Warschawskil).

L e t

F(D)

have an arc K w i t h a c o n t i n u o u s t a n g e n t and z = lp(~), (~--= ~(z)) represent

D(z)

eonformally on C(~). W e denote by K ' an arc i n t e r i o r to K , a n d by 7' t h e corresponding arc of /;'(C).

A necessary a n d sufficient condition for t h e relations ] ~o' (z) -- 90' (z')] --< const ]z -- z' [ =, z, z' < g '

I~' (~) - ~' (~')l -< const ] ~ - ~' L ~, ~' < r',

is t h e existence of a k such t h a t

5.0.9. 1 O ( s ) - - 0 ( 8 ' ) l - < k l s - - 8 ' l ~ s , s ' < K .

5 - I . o . Definition. The curve

K(z)

is said to have t h e property E a t t h e point z0, if it is analytic in a certain n e i g h b o u r h o o d of z0, except possibly at t h e point z o itself.

I f it is n o t a n a l y t i c at z o, t h e n we suppose moreover

K(z)

to be such t h a t there exists a d o m a i n D* = D*, + D7 for which (i) K = F ( D , * ) - F ( D ~ ) (ii)F(D*) has a corner a t z0, a n d (iii) /)~*, D* are inverse to each o t h e r w i t h respect to the a n a l y t i c p a r t of

K(z).

W e see t h a t if

K(z)

is a n a l y t i c also a t

Zo,

t h e n t h e last conditions are superfluous.

5. I . I . I f ~ = ~ 0 ( z ) represents c o n f o r m a l l y H(~) on

D*(z)and

o = ~ ( z 0 ) , t h e n K(~) is, by our hypothesis, t h e u p p e r p a r t of t h e i m a g i n a r y axis. :Now

1. c. p, 447.

84 Franti~ek Wolf.,

we see easily, by 5-o. 2 a n d 5. o. 3, t h a t K ( z ) is rectifiable n e a r z0, t h a t it has~

a t a n g e n t at to, which is t h e bissectrice o f t h e c o r n e r of F(D*).

F u r t h e r , i f t~ is inverse to, z~, t h e n

lira (arg t~ + a r g t~} ~-~ z t i m a r g z

z x ~ O z ~ K , Z ~ Zo

and, by 5- o. 4,

u n i f o r m l y , i f z - ~ t o in a sector.

5-I. 2. A n i m m e d i a t e consequence o f t h e s e results is t h e following.

I f (i) D (t) has a c o r n e r a t to, (ii) K < F ( D ) h a s t h e p r o p e r t y E a t one side of Zo a n d (iii) ~---~(z) represents e o n f o r m a t t y H(~} o n D(z) so t h a t o = ~(z0);

t h e n ~----q~(z) represents also e o n f o r m a l l y H ~ ( ~ ) ( > H(~)), whose i n n e r a n g l e is, l a r g e r t h a n z, on D 1 (z) whose i n n e r angle at to is larger t h a n t h a t of D(z):

H e r e K ( t ) comes to be in a sector of D 1 (t) a n d t h e results of 5. o are. vat+ff in a n 'one-sided' sector of D(z) whose one side is K(z) itself.

5. I. 3. I f (i) D (z) has a corner at Zo, (ii) K(z) lies in a ,vector at z o a~d has the property E at that point and (iii) ~ = q~ (z) represents 1) (~) on 1) (z), th'e~ K(~,) has the property ~E at ~ o ~ (Zo).

L e t D* (z) ~-- D* (z) + D* (z) be t h e d o m a i n of 5. I . o . Since K l i e s i n a sector a t to, we m a y suppose w i t h o u t loss of g e n e r a l i t y t h a t / ) * ~ D. L e t D+(t) be represented c o n f o r m a l l y on H(w), by z ---- g) (w). Then ~=qD(Z)' trans- f o r m s D* (z) into /)+ (~) ( < D (~)) which, by ~ --- ~0 (~p (w)) is conformalty represented on H(w). H e n c e D* (~) satisfies 5. I. (iii).

I t is easy to see t h a t D* (~) satisfies also (i) a n d (ii) a n d K(~} h a s there- fore t h e p r o p e r t y E .

5.2. W e shall say t h a t K(z) has the property E* at zo if i t h a s t h e property E a n d if,

(iv) d e n o t i n g by s the l e n g t h of K(z) f r o m t h e point t o on a n d by 0 ( s ) t h e angle between the t a n g e n t a n d a fixed direction,

5.2. i. 1 0 ( s ) - 0 ( o ) l < - Cs:, = > o ,

is fulfilled, f o r t h e p o i n t s, in a 2Y(zo).

5.2.2. If(1) D(z) has a corner at zo and at each of the parts of F(D), in a certain neighbourhood of z0,

5 . :. 3. ,I,o (,s) - , o < ) I - < - C t s - s t I , - > o is satisfied,

(ii) K ( z ) has the .property :E* ~at ,z o a n d ,lies i n a sector at zo (iii) ~ = 9 9 (5)represents D ( ~ ) z o n f ~ r m a l l y on I ) ( z ) ,

,then K (~) has also the property E * ,at ~o = "9 9 (Zo).

I t is, by 5. I. 3, e ~ d e n t l y strffieient t o s h o w Chat /~7(~) satisfies 5.2. I. I t i s .easy ~o verify ~hu*, M C h o ~ h t h e eonstanCs ehange, t h e p r o p e r t y /~* is in-

v a r i a n t with the tran~rformation ~ - - , t o = ( z - z0):. 1

W e may t h e r e f o r e uuppose r Ih.e ~ o parts o f F ( D ) at Zo have a com- m o n "tangent and Chat, t h e r e f o r e , 5 . 2 . 3 :is valid n o t only separately on both s i d e s ,of Zo, bu~ in a ee.r~in #w<)-si,ded n e i g h b o u r h o o d o f Zo.

.Now, we see tha~ t h e c o n d i t i o n s o f 5 . 0 . 8 are satisfied. H e n c e

~" (z) - - 9" (so) (z - zo) ~

is ,bo,unded, if z < F ( D ) . 2V(~o). B y t h e P h r a g m 6 n - L i n d e l 6 f t h e o r e m it m u s t be so .ev.en f o r z < D . 2V(zo). A n d t h e same d e d u c t i o n can be m a d e for ~p' ( ~ ) =

=-I/,99'.(z), in the c o r r e s p o n d i n g C . N ( ~ o ) . H e n c e 99'(z) is in absolute value b e t w e e n *wo positive constants. W e have, t h e r e f o r e

a r g 99' (~) - a r g 99' (5~ ---- o ((~ - - ~o)~).

If, now, 0(a) has for K(~) the a n a l o g u o u s m e a n i n g as O(s) has f o r K ( z ) , t h e n

8 (~) - - 0~ (O) --- 0 (s) - - Os (o) + arg 99' (5) - - arg 99' (So) = o (s a) + o ((z - - Zo) ~) or, using 5. o. 7,

o

(a)

- 0~ (o) = 0 ((~ - ~o) o) = o (<C - Co) ~ = o ( 0 % Thus, w e have proved t h a t K(~) satisfies condition 5.2. I.

5 . 2 . 9 . W e have t h e f o l l o w i n g result similar to 5-I. 2.

I f (i) D ( z ) has a corner of the same kind as in 5.o. 6 a t z o (ii) K < F ( D ) has the p r o p e r t y E* at one side of z o and (iii) ~ =99(z) represents conformally H(~) on D (z) so t h a t o = q0 (So), t h e n ~ = 99 (z) r e p r e s e n t s c o n f o r m a l l y also H i (~), whose inner angle a~ ~---o is larger t h a n z, on D, (z), whose inner angle at Zo is larger t h a n t h a t of D (z).

i W A R S C H A W S K I , 1. c. p. 446.

86 Frantiw Wolf.

Further, instead for 5. o. 5, Warschawski's more precise result 5. o. 6. holds in D (z).

5.3. L e t ( = o < D ( ~ ) < C(~) and ( a < O < f l , r = I ) < F ( D ) . F ( C ) . B y z = ~ ( ~ ) , C(z) is represented eonformally on D(~), in such a way that o=~p(o).

Then -do >- I for a < O < f l .

Since ~ (~) is analytic on (a, fl), ~' (~) must exist everywhere. By the theorem of Julia-Caratheodory 1, we have, for a < ~o < fl, e~~ = ~(d~~ and z --- 9(~)

, I - -

- Igl-" > (e.o/I ! .

If we put z=~---o, we get

5.3. ~. I f a (~) is a boundary function of 1)(~), which is integrable on. (a, ~) and bounded on the rest of.F(D), then it is integrable in the eorre~lwnding C(r,O).

We have indeed

flaIo)lao= fto(,)la, la l<_ f lo(,)ld,

o ( d ~) = "

and on the rest of F ( C ) it is bounded.

5.4. For application of our uniqueness theorem to general domains we shall use a result proved by W. Seidel. =

5.4. I. If z = ~ v ( ; ) represents conformally a convex 1)(z) on C(;), then I z ~ ' (z) [ is increasing on every radius. The boundary function, thus defined, is almost everywhere finite and greater than a certain positive number.

5.4. 2. I f z = ~p (~) represents conformally a domain with bounded ~outer curvature~ 1) (z) on C (~), then ] ~0' (~)l is everywhere greater than a certain positive number.

This is a generalization of the preceding result and of another of Seidel's theorems. The outer curvature in a frontier point is the reciprocal value of the radius of the greatest circle which goes through the point and is completely

x Cf. e.g. NEVANLINNA: E i n d e u t i g e analytische F u n k t i o n e n , p. 52.

2 Ueber Ritnderzuordnung bei k o n f o r m e n Abbildungen, Math. A n n a l e n , IO4 (I931), pp. 212~

217, 222.

outside D. By 9 linear transformation, which transforms this circle into the unit circle, we reduce this theorem to the theorem of gulia-Caratheodory (cf. 5 . 3 ) .

5.4.3. I f

a(z)

is a boundary function, integrable on F(D), and D has a bounded outer curvature, then

a(z)

is integrable in the corresponding C.

The proof is identical with t h a t given at the end of 5.3. I.

5.5. I f K < F ( D ) is nnalytic and K ' < K completely interior to K, then the L-integrability of a (z) on K ' is equivalent with the L-integrability of a (z) on the arc in the corresponding C.

If ~ = ~0(z) represents conformally C(~) on D(z), then, on K ' , ]q~'(Z) l is between two positive constants. Hence the result readily follows.

5.6. Let

D(z)(< C(z))

be a domain bounded by ~ curve

K(z)

~oining the points e i~, d : , and by the circular arc between those points. At ei%

K(z)

has the property E and

D(z)

an inner angle Q. I f z < / ) ( z ) , z ~ e i~ and

; ~-~v(z)(z = ~p(~))represents H(~) on D(~), (o0= ~(e~'~)), then

I

5 . 6 . r. - o ( ~ r (F/T))

I - I ~ l

for any positive e.

If, moreover,

K(z)

has the property E* at g% then

!

~-I~1

Proof. By applying from 5. o. 5

5.6.3.

Since the

first the intermediary t r a n s f o r m a t i o n ~ = ~-1, we get

(~) - e i~ = 0

(~-~:~+~), (~' (~)-~

= o ( ~ + ~ / : ~ + q .

circular arc has also the property E* at e i~ these relations are valid uniformly not only in a sector but, by 5. I. 2, for all z < D (z).

:Now, let z - ~ r e i~, z * = e in• be two points lying on a circle with centre at d ~. Then we get

- I~--~] < 2 a r c (z, z*) -< -2 ~,oma~ IV'/C)I" I C - C'1-1 -< o (C~ '~ + " ~C/~)).

Thus 5.6. I is proved.

I n order to deduce 5.6.2 we have to use the more precise result of Warschawski (of. 5. o. 6). Hence we shall find instead for 5.6.3, t h a t

88

a n d 5 . 6 . 4

are between two as above.

Franti~ek Wolf.

I

finite positive constants. The rest of the proof is t h e same

6.0. I n the unit-circle C(r, O) we define, by the equation O = K ( r , 4), a system o f curves /s which shall satisfy the following conditions."

i ~ The function K ( r , 4) is continuous as a function of two variables, and periodic, with the period 2 ~c, in i,"

2 ~ K ( I , ~ ) ~ - ~ and (dK/dr)~=t is finite;

3 ~ The curve 0 = K ( r , ~) has the property E* (el. 5. o. I) at the point r - - ~, 0 = L 6.0. I. A consequence of t h e definition is t h e u n i f o r m convergence of K(L~) to K(40) as ~,~-~ ~o.

6 . 0 . 2 . The second condition in 2 ~ signifies geometrically t h a t no curve is t a n g e n t to t h e circumference of the unit-circle.

6. o. 3. I f we say t h a t u(z) has a p r o p e r t y in (a, fl), t h e n we m e a n t h a t it has ~hat p r o p e r t y in the domain b o u n d e d by t h e circular arc a--< 0--< fl a n d by K (a), K(~).

6. ~. Definition. W e shall say t h a t u (r, 0) satisfies c o n d i t i o n A if (i) it is h a r m o n i c in C a n d (ii) lira ]uIr , K ( r , 4))1 is finite f o r all ~ except f o r t h o s e

r ~ l

belonging to a n enumerable set ~.

6. 2. I f u(r, 0) satisfies c o n d i t i o n A, t h e n for every perfect set ~, t h e r e is a n u m b e r M o a n d a section (~. (a, b), such t h a t

lu (r, K(r,

--<Mo for X < (a, b).

Since u(r, K ( r , )~)) is a c o n t i n u o u s f u n c t i o n of r a n d ~, the set (M) = lim/~' [1 u (r, K(r, ~))l <-/If, r --< O]

is closed. F r o m (ii) it follows t h a t

lim 9A (M) > C (&) a n d in p a r t i c u l a r t

1 We put ~ ) = Z 0 n .

1

[

lira ~ ( M ) + O~ . ~ = ~ .

M ~ 1 J

Mo

Hence follows the existence of a M o such that 9A (3/o) + ~ 0~ is dense in a

1

section of ~. But, then, even ~[ (3/o) must be dense in a section ~ . (a, b). The rest follows from the closure of ~[ (2[0).

6.3. I f u (r, 0) satisfies the condition A, then the inner points of "the set of points a, for which u ~ l~t)I(et~), form an open everywhere dense s e t ~ . By ~, we shall denote the closed set complementary to ~).

If we take ~ = (a, b) in 6. 2, we see that u D is finite ~.t a partial :interval of (a, b), and since this is arbitrary it follows that u" is finite at an every- where dense set of points. By 2. I2, ~ is, therefore, also everywhere dense.

6. 4. If u(r, 0) satisfies the condition A and ~ = ~ + ~, where ~ is per- fect and ~ reducible, then, if ~ is not empty, we can find a closed interval (a, b), containing a section ~1 of ~ , and having no points of ~ in its interior.

Further, there exists an M, such that 6.4. I.

for ~ < ~31.

I n one

[ u (r, KCr, ~)) [ _< M

of the intervals complementary to ~, there exists a section of ~.

To this we apply 6.2.

7.0. Theorem. I f

(i) u (r, O) is harmonic in C,

(ii) the curve K(,~), defined in 6. o, forms with the circumference the angle fl(O)

(iii) lira l u (r, K(r, "~))l is finite for all O, except f o r the points of an enumerable set ~.

(iv) lira u (r, K(r, ,9)) --< a(,9) --< lira u (r, K(r, ,9)) where a(~) is L-integrable,

~ ' ~ 1 r---~ l

(v) for every ,~ and ~, there is a N(e~), such that

I u (r, 0)[ --< exp [e/(I--r)~/2, ~(~)] for (r, O) < N(e i a),

90 Franti~ek Wolf.

(vi) for every ,9 there exists an angle A(,9):

Cl(I--~') ~ 0 - - , 9 ~ ei~(I--r )

hating K(,9) in its interior, such that

for (r, O) < A (,9), . (,9) integral,

then u = B P I ( z ) (cf. 2. I) for all z < F ( C ) , except for the points of a reducible set ~.

R e m a r k . W i t h o u t changing the conclusions, it is possible to introduce a reducible set ~1 for the points of which no conditions need to be satisfied. We may, indeed, by the following reasoning, show t h a t in every complementary interval of ~1, there is only a reducible set of singular points. The final set of singular points is again reducible.

7. o. I. I n order to illustrate by simpler examples the meaning of the theorem, we enunciate two particular cases.

I f (i) u(r,O) is harmonic in the unit circle C(r,O),

(ii) lira l u (r, 0) 1 is finite, except at the points of an enumerable set ~,

r ~ l

(iii) lira u (r, O) _< a (0) _< lim u (r, O) where a (0) is L-integrable,

(iv) u (r, O) ~-- o u,dformly for all O,

then u = R P I ( z ) for all z < F ( C ) except for the points of a reducible set ~t.

This theorem is f u n d a m e n t a l for application to the uniqueness theory of sum- mable trigonometrical series (eft F. Wolf, On (C, k) summable trigonometrical series). 1

7. o. 2. I f (i) u (x, y) is harmonic in the upper half-plane H,

(ii) lira l u (x, Y) I is finite except at the points of an enumerable set of points gJ,

y ~ O

o o

- - f la(x) l

(iii) lira u (x, y) ~ a(x) <-- lira u (x, y), where j I + x ~ d x < ~ ,

y~O y ~ O

(iv) u ( x , y ) = o(y - ' ( x ) ) uniformly in any finite interval ( - - X , X);

then u ~ B P I ( z ) for all z < F(H) except for the points of a reducible set of points 9t.

This theorem is equally f u n d a m e n t a l for the uniqueness theory of (C,k) summable trigonometrical integrals.

I t is deduced from 7. o by the conformal representation of H(~) on C(z).

I t is easy to a e ~ n e

suitable K(`9).

Proc. London Math. Soe. S. 2, Vol, 45, 1939, P. 328.

7 . 0 . 3 . I f t h e first f o u r c o n d i t i o n s of 7. o a r e satisfied a n d O'~v(r,O)

u (% O) = Or,~O,~_,,~e

w h e r e v ( r , 0 ) is a h u r m o n i c f u n c t i o n c o n t i n u o u s in C + F ( C ) , t h e n t h e r e s u l t of 7. o h o l d s g o o d , w i t h n ( ~ ) - - n .

F r o m 3- I i t follows i n d e e d t h a t ,

' < - ~ ( i - ~ l , o l ? f I v ( * ) - ' ( ~ ~ 1 7 6 ~-~T~ol;

Iz--zol =1--1 zol s i m i l a r l y f o r h i g h e r d e r i v a t i v e s

0 n y

o~, o a . - ~ = off - I * o

I)-".

H e n c e 7 . 0 (v) a n d (vi) a r e satisfied.

7. o. 4- I f t h e first f o u r c o n d i t i o n s of 7. o are satisfied a n d

R 2 ~

(,I f ^{, , >}

0 o

t h e n t h e r e s u l t of 7. o holds good, w i t h

~, (~) = m + - - 2

P W e h a v e i n d e e d

2K,

,u o,o,l=l f u r,o do I

0

,R 2 r

o o

R 2zr 1

<- f f 'u(r,e)l'rded,']

0 o

F u r t h e r f o r an a r b i t r a r y e > o t h e r e exists a ~ > o s u c h t h a t

f f l -

r ) ~ I u (,-, e ) I ~ , - e , 9 d e <

D

if t h e a r e a of D is less t h a n 6.

92 Franti~tek Wolf.

Now, we t a k e a R such t h a t t h e a r e a of / ~ < r < I is less t h a n (I. T h e n , 2 R + I

f o r all r > , ~he circle C~ r o u n d ( r , 0 ) w i t h t h e radius 3

d o m a i n a n d we g e t

, o r

I - - r o

- - - - is i n t h e 2

1

[ + , f f ]

u(r,O)~)

~ ( i - - - - r

]u(r,O)lPrdrd8 ?

Cr

1

r :22+~p

f I+ dr dO] ?

c~

2 1

,~rt+ -- --

. 2 P ~ P

_<

2

u (r,0) = o ( I --

r) -m-~/p.

:7. i I f

+(i) :(a, fl) is an i n t e r ~ a l o f ~ (el. 6.:3), (ii)

u(r, K(r, a))

is b o u n d e d ,

~ii) :7 .such t h a t a < y < 8, a n d

~ucY,7)

is finite bounded,

a n d t h e r e f o r e u (r, K ( r , ~))

(iv) D = E [ K ( r , a ) < O < K ( r , 7), o < r < t] + / ~ [ , " < B ] ' , a n d z = 9(C) is t h e

.r, ~ r , 0

c o n f o r m a l r e p r e s e n t a t i o n of D(z) o n H(,~), ,such t h a t , , ~ = ~ ( o , ) , o = ~ (o), t h e n

7. t. ~ ~ (C) = p(C) + .8 (C)

w h e r e

7. i. 2 p (~) = p i n u u

a n d

k ~ n(~). fl(a)/n--~

7.I.3 Z ak q.

k ~ l

7 . 1 . 4 . F u r t h e r t h e r e is a n a n g l e B ( a ) : - - ~ x < ~ 9 < ~ 1 in w h i c h

p(~)=o(~)

u n i f o r m l y f o r ~-~ ~ .

R < I, but so large that D should be connected.

The f u n c t i o n

u(z)

^{is, a n}

R P I

a t the points of

K(a)

^and

K(fl)

(cf, 2, I2) except possibly a t their' e~ldpoints e i~,

el:.

F r o m the definition of ~ it follows t h a t

u = . R P I c ( e t~) a:<:c}<fl.

By 2. 9 a n d 2. I4 we have also

u--~RPID(e~).

The only possible sin~u,l~r p o i n t of u(z) on

F(D)

is, therefore, e% Condition (iv) shows t h a t 4. o (ii), is, satisfied.

The d o m a i n D satisfies the c o n d i t i o n s in 5.3 a n d b y 7, o (iv) a n d 5.3. I up, which exists a l m o s t - e v e r y w h e r e on F ( D ) ; is integrable in, t h e corresponding C.

H e n c e 4. o (iii) is sa,tisged, a n d 4. o gives, 7: I. I a n d 7.1.2'.

L e t t h e i n n e r angt, e of D a t e ~ be denoted, by e (--<~'(~), cf, 7 . 0 (ii)). No.w, w~e, use t h e second' pa.rt of 5.6 a n d we g,et: f r o m 7. o (v)

/'u,(r l! = I-(z) I -< exp [d(;z--r)~/': /~

^{- <}

7. I. 5 ~ exp [e,~g/2:~ ` sin-~J2~: ' (arg ~,)'].

exp [a~: ~: s i r e 1 a r g ~].

W e deduce, i~ a similar way from. 7'. o (vi} t h e existence of an~ a n g l e o < arg ~ < ~ i,r, which

7- I. 6 u (r = 0.(r (:~:)/@

F r o m

7.

I,. ~, 7- I. 2 a n d 2.23 it fo.l.tow, s thab 7. r. 5 and 7. I. 6 are valie~ a~so for s (~). No,w, 3- 3 gives 7. I. 3-

By 5 . 2 . 9 , ~ : ~ ( z ) represents a D ~ ( z ) > D ( z ) on D , ( ~ ) > H ( ~ ) . F u r t h e r

D~ (~) ~ H(~)

h a s a positive a , g ] e a t ~ below t h e real positive axis a n d m a y be supposed t o be so small tha~ Da (z) - -

D(z)

< A (e) (cf. 7. o (vi)). Then, by 7. o (vi) a n d 5.o. 6, there is a n angle -- d < a r g ~ < 6 in Dx(~) in which u : o(~e'*(~)/~).

I n -- ~/2 < a r g ~ <

~/2

we have, by 3 . 2 , f ( ~ ) - ~ u(~) +

iv(~)=a(~en(~)/z).

F u r t h e r , by 7. 1.3, h ( ~ ) = s(~) +

it(~)-~ o(~e '~r

a n d t h e r e f o r e also g(~)~-p(~) +

iq(~)=

= o(~e.,~(,)/~)

in the same angle.

Since we k n o w f r o m 2.24 a n d condition ( i i ) t h a t

g(~)~-o(I)for

any c o n s t a n t positive J ( z ) , we can apply 3.5- We get g ( ~ ) : 0 (~) in a certain p a r t i a l angle B ( a ) : - - 8 ~ < a r g ~ < ~ . T h u s , 7. I . 4 is proved.

7.2. P r o o f o f

7. o.

I f the conditions of 7 . 0 are satisfied, t h e n so is condition A of 6, I. I f the set of exceptional points is n o t reducible, then, by 6.4, there is a n i n t e r v a l (a, b), such t h a t t h e exceptional points in (a, b) f o r m a perfect set ~ , a n d there is a c o n s t a n t N such t h a t

7.2.1

lu(r,K(r,~))[ <~ 2(