A R K I V F O R M A T E M A T I K B a n d 7 n r 31
1.67 06 1 C o m m u n i c a t e d 12 April 1967 b y OTTO FttOSTtCIAlV
Smoothness of the boundary function of a holomorphic function of bounded type
By HAROLD S, SHAPIRO
I n a recent p a p e r [5] the author has proved the following theorem:
Theorem A. Let/(z) be holomorphic and o/bounded type in Iz ] < 1 and suppose its radial boundary values/(e g~ coincide almost everywhere with a/unction _F( O) o/period 27~ and class C ~ such that/or some positive A,
m a x [F (=) (0) ] <~ (An) 2~, n = 1, 2 . . . (1)
0
Then / is bounded in I z/< 1, and consequently / and all its derivatives are uni/ormly continuous in [ z [ < l . I / the right side o/ (1} were repla~ed by (An) vn /or some p > 2 , the resulting theorem would be/alse.
I n [5] this theorem was proved b y a method based on weighted polynomial approxi- mation. I t was also conjectured t h a t theorem A is true "locally", t h a t is, if in (1) the m a x is over the interval [01, 02] instead of [0, 27r], a corresponding conclusion holds for the neighbourhood of the arc joining d ~ and e i~
The main purpose of t h e present paper is to prove this conjecture. An altogether different, more direct, method is employed. Actually the relevant theorem (Theorem 1 below) is formulated for a half plane rather t h a n a d i s k - - t h e version for a disk (or indeed a n y domain having an analytic arc on its boundary) can easily be deduced from t h a t for a half plane.
We also take this opportunity to settle another point from [5]. On p. 334 we stated t h a t "it seems most plausible" an analogous theorem holds for meromorphic functions of bounded type, if (1) is replaced b y the stronger condition t h a t F belong to a D e n j o y - C a r l e m a n quasi-analytic class (that is, under such a hypothesis on F , / is bounded, and in particular is free of poles, near the boundary). This "plausible"
assertion is false, as shown in the corollary to Theorem 2 below. I n fact, no m a j o r a n t short of (An) ~, which implies analytic continuation of / across the b o u n d a r y arc in question, can guarantee the absence of poles near the boundary. (But, for a plausible conjecture, see the concluding remarks.)
Before turning to Theorem 1, we wish to recall some facts concerning functions of bounded type; we formulate t h e m for functions in a disk, the definitions a n d results for functions in a half plane are similar. F o r the notions of "inner" a n d
" o u t e r " function see I-Ioffman [2] (also [4] b u t in [4] these terms are not used).
A singular/unction is an inner function without zeros. A complete divisibility theory exists for singular functions (see [2] pp. 84-85); a n d every holomorphic function /
30:5 443
H. S. SHAPIRO, Smoothness of the boundary function
of b o u n d e d t y p e in the u n i t disk can be represented as a Blaschke p r o d u c t times a n outer function times the quotient of two relatively prime singular functions. I f t h e singular f a c t o r in the d e n o m i n a t o r reduces to a c o n s t a n t / is a Smirnov/unction.
This is equivalent to saying t h a t ] = Bg where B is a Blaschke p r o d u c t a n d f e a + z
g(z) = exp | ~ da(t), j e - z
w h e r e b y a is a measure on the circle whose positive v a r i a t i o n is absolutely contin- uous with respect to arc length. The significance of S m i r n o v functions is t h a t t h e generalized m a x i m u m principle holds, i.e. a S m i r n o v f u n c t i o n / is b o u n d e d b y t h e essential s u p r e m u m of the moduli of its b o u n d a r y values. More generally, if t h e b o u n d a r y function belongs to L v, t h e n / E H p. The corresponding results localized t o the n e i g h b o r h o o d of an arc on t h e b o u n d a r y are also true (see [4], also papers of G. Ts. T u m a r k i n referred to there).
T h e o r e m 1. L e t / ( z ) be holomorphic and o/ bounded type in the upper hall plane.
Write z = x +iy, and suppose there exists a/unction F(x) in/initely di//erentiable on an interval (x D x2) and "o/Gevrey class", that is/or some A > 0
IF(n)(x)l<(An) 2~, Xl<X<:x2, n = 0 , 1 , 2 . . . (2) such that F(x)=limu~o /(x +iy ) /or almost all x in @1, x2). Then /(z) is bounded (and there/ore /(z) and all its derivatives are uni/ormly continuous) in a neighbour- hood o/ each point x in (xx, x2). Moreover, i/ the right-hand side o/(2) were replaced by (An) vn with some p > 2, the theorem would be/alse.
L e m m a . F e r n >~ 2, ~ 2 ~ (k/n) k < 3.
Proo/o/lemma. L e t m be t h e least integer n o t less t h a n i n . The s u m on the left is less t h a n
2 k + ~ < l + n ~ < l + n t m d t < l + m + ~
k = l k = m k f f i m
< 3 .
Proo/ o/ theorem. T h e t r u t h of the last assertion in t h e theorem is established b y constructing a suitable counter-example, just as in [5]. W e therefore confine our a t t e n t i o n to t h e sufficiency of (2). The proof will be b y contradiction. I f / were u n b o u n d e d in t h e n e i g h b o u r h o o d of some b o u n d a r y point lying in (xl, x2), it would a d m i t a representation / = g U -1 where g is a S m i r n o v function, a n d U a singular function relatively prime to the singular f a c t o r of g, such t h a t the representing meas- ure of U has derivative equal to + oo at some point x a where x 1 < x a < x 2. I t t h e n follows t h a t for e v e r y ~ > 0 a n d e v e r y y ~ > 0 t h e f u n c t i o n P~(y)=lg(x3+iy)l"
I U(xa+iy)l -~ is u n b o u n d e d for 0 < y < y a . I f therefore we show t h a t for e v e r y x 3 such t h a t x 1 < x 3 < x 2 there exists a positive 8 such t h a t P~(y) is b o u n d e d for small y, the t h e o r e m will be proved.
B y trivial changes of the i n d e p e n d e n t variable we m a y arrange t h a t A = 1 in (2), x 3 = 0 ; t h e n (2) holds for Ixl < a , where a is some positive number. Now,
AliKIV FOR MATEMATIK, Bd 7 nr 31
~-~ F (k) ~0)
F ( x ) = ~ ' x k § l ( x ) § k=o k!
where R~(x) < Ixln n n ~ . <~(nelxl) n for Ix]<a.
B y hypothesis, we h a v e almost everywhere on ( - a , a), g(x) = F(x) U(x), hence a.e. on ( - a , a), ]g(x) - s n _ l ( x ) U(x)[ = I Rn(x) U(x) l < (he ]xl) ~
N o w G(z) = g(z) - S~ ~ (z) U(z)
Z n
is a S m i r n o v function in t h e u p p e r half plane whose b o u n d a r y values are b o u n d e d o n ( - a , a) b y (ne) ~, therefore in some n e i g h b o u r h o o d of 0 we h a v e I G(z)] ~<2(ne) ~.
T h u s
]g(iy) - Sn_l(iy) U(iy) l <~ 2(hey) n, 0 < y <~ yl. (3) Now, for some positive n u m b e r B, [ U(iy) l >~e -(81y). W r i t i n g 5 = 1/8B we h a v e f r o m (3),
Ig(iy) l <~ I S n _ l ( i y ) [ § 2(hey) n e ~lsy
i U(iy ) i~ . (4)
N o w (4) holds for all n. I n particular, we m a y choose n such t h a t
e 2 e 2
- - - l < n ~ < - - .
Y Y
T h e n , t h e second t e r m on the right of (4) does n o t exceed 2e-n+(l/sY)<2 if y < e - ~ - ~ . Moreover,
k=O k=O k = l
b y t h e lemma.
Thus, the left side of (4) is less t h a n 6 ~or small y, a n d t h e t h e o r e m is proved.
Theorem 2. Let Pk be any decreasing sequence o/positive numbers tending to zero.
There exists a meromorphic /unction /(z) o/bounded type in I zl < 1 whose poles cluster at every point o/ I zl = 1 and whose boundary /unction coincides a.e. with a / u n c t i o n
. ~ ( 0 ) = ~ = 1 Ck e-ikO, where ck are complex numbers such that ]ck [<<-e-k~k.
Proo/. L e t ~ be a n y complex n u m b e r of m o d u l u s one which is n o t a root of u n i t y , a n d let r n --e -1/n.. Consider/(z) = ~ = 1 a~/(z-zn) where zn =rn ~ . W e shall show t h a t positive n u m b e r s a n can be chosen so t h a t the series converges u n i f o r m l y on e v e r y c o m p a c t subset of ]z I < 1 n o t containing a n y of t h e points zn a n d represents a mere-
n. s. SI~APIRO, Smoothness of the boundary function
morphic function of bounded t y p e with the asserted b o u n d a r y behaviour. First of all, suppose ~ n2an < ~ . Then the series converges as stated, and moreover, if B(z) denotes the Blaschke product formed with zeros at {Zn} we have for ]z]< 1,
B(z)_ an
[l(z) B(z)] < ~ a. < < ~ n~a. <
Thus / B is a bounded holomorphic function in < 1, a n d / is of bounded type.
Now, it is readily verified t h a t the b o u n d a r y values of z/coincide a.e. with the func- tion
= k
~ cke -ikO,
where C k ~ a n z n .k - 1 n 1
Thus, to complete the proof it suffices to show t h a t if the a n tend to zero fast enough, we have
ane -(k/n~) <e kv~. (5)
n = l
l~ow define the function M(k) to be the greatest integer not exceeding p~89 Clearly M(k) is non-decreasing and tends to infinity with k. Now, we can choose positive numbers an so t h a t
~n~an < ~ (to satisfy the earlier requirement),
1
an< 1,
1
an<~89 -kpk, k = 1 , 2 , 3 , . . . .
n = M(k) + 1
(6)
(7) (8) Concerning the last inequality, observe t h a t since M(k) tends to infinity we h a v e for each r only a finite n u m b e r of conditions imposed on the remainder ~n%r an, therefore the inequalities (8) can be satisfied b y positive numbers an upon which we can moreover impose the conditions (6), (7). Now the left-hand side of (5) does not exceed
M(k)
(k/n~)
~ an.
an +
n = l n - M ( k ) + l
The first sum is bounded b y 8 9 2 <<.89 ~vk (by (7)), and the second b y 12e k~k, b y (8). Theorem 2 is proved.
Corollary. I] Bn are any positive numbers increasing to in/inity, we can/ind a mero- morphic /unction satis/ying the hypotheses o/ Theorem 2 and such that
m a x [ F(n)(0) ] ~< (nBn) n.
We shall not carry out the details of the proof. Since IF(n)(0)[ ~< ~ = 1 ]ck] k", it is a m a t t e r of showing t h a t if p~ tend to zero slowly enough, we have
e-kVk k ~ <~ (nBn) n
k ~ l
446
ARKIV FOR M.kTEMATIK. Bd 7 nr 31 which is straightforward b u t tedious. As the result is needed only for a counter- example we carry out the estimation only for the concrete choice Pk = 1/log (/C + 1). To estimate ~ - 1 exp ( -/c/log (/c + 1))/cn, b r e a k up the sum into ~ _ ~ a n d ~ = ~ +~ where m = re(n) is chosen as the least integer such that/c~ ~< exp (89 (/c + 1)) holds for all k > m . I t is easy to see t h a t m is around 2n(log n) 2. Then the second sum is bounded
X~m -ak~n ~ ~Oo ~-akLn
uniformly with respect to n a n d the first sum is majorized b y Lk=l e ~ Lk=l ~ where a = 1/(1 + l o g n). Now, comparing the last sum with an integral we show easily t h a t it is O(n/ea) ~+1, therefore, finally, the original sum and hence m a x lF(~)(0)l is bounded b y (An log n) n for some constant A. Thus, F belongs to a D e n j o y - C a r l e m a n quasianalytic class, which disproves the conjecture in [5].
Concluding r e m a r k s
1. We have still not been able to settle whether in Theorem A or Theorem 1 the Gevrey class can be replaced b y one of the slightly larger Carleson-Korenblyum quasi-analytic classes (see the discussion in [5], p. 333, also K o r e n b l y u m [3]). F o r example, would Theorem A be true with the right side of (1) replaced b y (An log n)2n?
2. The rather drastic smoothness imposed on F could be v e r y m u c h relaxed if we had some a priori information a b o u t the singular factor U. For instance, it is easy to see b y the method used to prove Theorem 1 t h a t if I U ( x § l>~By ~ for some positive B, (~ (here B, ~ m a y even depend on x) for x l < x < x 2 then if F(x) satisfies a Lipschitz condition of a n y positive order on [x D x~] we can conclude t h a t / is bounded (hence uniformly continuous) in a neighborhood of each point x,
Xl < X < X 2.
3. Under the hypotheses of Theorem A, / can have only finitely m a n y zeros in Izl < 1 (this follows with the help of considerations outlined in [1] p. 331). As B. I.
K o r e n b l y u m of K i e v has pointed out to the author, it can also be shown in this case t h a t / has no (non-constant) singular factor. K o r e n b l y u m has suggested to the author the interesting question whether perhaps, under the assumption t h a t / is meromorphic and of bounded t y p e and F satisfies (1), we can conclude t h a t / has no (non-constant) singular factors (either in n u m e r a t o r or denominator), i.e. F is a quotient of Blaschke products times an outer function. I f this were true, one would h a v e the interpretation t h a t the Gevrey smoothness of the b o u n d a r y function indicates the absence of singular factors, rather t h a n boundedness in the interior.
University o] Michigan and University o/Stockholm
B I B L I O G R A P H Y
I. CARLESON, L., Sets of uniqueness for functions regular in t h e u n i t circle, Acta Math. 87, 325-345 (1952).
2. HO~FMAN, KENNETH, B a n a c h spaces of Analytic Functions, Prentice-Rail, 1962.
3. KOI~EI~BLyUYl, B. I., Quasianalytic classes of functions in the circ]e, Dokl. Akad. N a u k U S S R 164, 36-39 (1965) (Russian).
4. PR~VALOV, I. I., B o u n d a r y properties of analytic functions, 2 ed., Moscow-Lgrad. 1950 (Russian) (a G e r m a n edition also exists, Dcutscher Verlag der Wissenschaften).
5. SHAeXRO, H. S., W e i g h t e d polynomial a p p r o x i m a t i o n a n d b o u n d a r y b e h a v i o u r of analytic functions, contained in " C o n t e m p o r a r y Problems of t h e Theory of Analytic F u n c t i o n s " , 1Vauka, Moscow 326--335 (1966).
T r y c k t den 8 april 1968
Uppsala 1968. Ahnqvist & WikseNs Boktryckeri AB
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