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~)
If not, what..~eometric conditions guarantee
an___~d ~
coincide?
ri~_.~p~ ~4
?
Note that for ~ ~ ~ it becomes easier to transfer many theorems, known for the disc, on approximation by polynomials or by rational frsctions in various metrics (of. references in [4]). It is possible to obtain for such domains conditions that guarantee convergence of boundary values of Cauchy type integrals [4]. As I have proved, ~ is a rather wide class containing in particular all domains ~ bounded by curves with finite rotation (cusps are allowed) [4]. At the same time, it follows from characterizations of proved by me earlier that ~ coincides with the class of F~ber domains, introduced and used later by Dyn'kin (cf. e.g.[6],[7]) to investigate uniform approximations by polynomials and by Anderson and ~) In virtue of a well-known V.I.Smirnov theorem, the analog of this property for Cauchy integrals is always true.
315 Ganelius [8] to investigate uniform approximation by rational fract i o n s w i t h fixed poles. This fact seems to have stayed unnoticed by the authors of these papers, because they reprove for the class of Faber domains some facts established earlier by me (the fact that domains with bm~uded rotation and without cusps belong to this class, conditions on the distribution of poles guaranteeing completeness, etc.). The following question is of interest. PROBLEM 2. Suppose that the interior domain belongs to belongs to
K (=~) ~
?
6- o to {lwl> }
~+
of a curve r
. Is it true that the ` extgrior domain 6
als__._~o
(Of course, we use here a conformal mapping of
).
For ~p with p >J the positive answer to the analogous question is evident.At the same time the similar problem formulated in [9] for the class S of Smirnov domains remains still open.At last it is of interest to study the relationship between the classes S of Smirnov domains andAoof Ahlfors domains (bounded by quasicircles [SO]), on the one hand, and K and ~p (considered here) on the other. See [9] for more details on ~ a n d A o . It is known that ~ p c S , K~:=~ ([4],[11]). At the same time there exist domains with a rectifiable boundary in A o which do not belong to S (of. [ 3 ] , [ ~ ) . Simple examples of domains bounded by piecewise differentiable curves with cusp points show that K\ Ao ~% ~ . PROBLEM 3. Pind ~eometric conditions
~uaranteeing
G KD2o A°. Once these conditions are satisfied, it follows from the papers cited above and [12], ~3] that many results known for the unit disc can be generalized. One of such conditions is that ~ on and without cusps.
should be of bounded rotati-
RE FEREN CE S I,
2.
X B e ~ e a E ~ s e B.B. MeTo~ HHTerpa~OB w~na K o ~ B paspHB-HRX l~aH~l~Ix s ~ a ~ a x Teop~Gi rOJIOMOp~H2X ~ y H E L ~ O~J~io~ EOM!DIeEc-HO~ HepeMeHHO~i. "COBpeMeHS~e npo6xeM~ MaTeMaTHEH", T.7, MOOEBa, 1975, 5-162. ~ a H ~ a ~ E ~.H. Hepei~yx~H~e rpa~m~H~e s s ~ a ~ Ha IL~OCEOCT~, MOCEBa, HayEa , 1975.
316
3. D u r e n P.L., S h a p i r o H.S., S h i e 1 d s A.L. Singular measures and domains not of Smirnov type. - Duke Math. J., 1966, v. 33, N 2, 2%7-254. 4. T y M a p z ~ H r.~o P p a H H ~ e CBO~CTBa a ~ a x H T ~ e c ~ H x ~ , n p e ~ c T a B m ~ X ~HTerpa~a~m T~na E o ~ . -MaTeMoC6., I97I, 84 (126), 3 , 425--439.
5.H a a T a z B ~ x ~ B.A. 0 c m I t ~ y z ~ a p ~ HHTerpaxax Ko~m. - Coo6~. AH r p y s . c c P , I 9 6 9 , 53, ~ 3, 5 2 9 - 5 3 2 . 6.~ H H ~ K ~ H E.M. 0 paBHoMepHoM n p ~ 6 ~ e H ~ MHOrO~eHS~m B EOMr~eEcHo~ ~ O C E O C T ~ . - 8a~.Hay~H.CeMEH.~0M~, 1975, 56, 164--165, 7.~ H H ~ Z ~ H E.M. 0 paBHOMepHoM n p ~ 6 ~ e H ~ ~ B mop~aHOB~X 06~aCT~X. -- C~6.MaT.~. 1977, 18, ~ 4, 775--786. 8. A n d e r s s o n J a n - E r i k, G a n e I i u s T o r d. The degree of approximation by rational function with fixed p o l e s . - Math.Z., 1977, 153, N 2, 161-166. 9. T y M a p E ~ a r.h. I ~ a H ~ e CBO~CTBa E o H ~ p M m ~ OTO6ps~e-H ~ HeEoTopRx E~aCCOB o6~aoTe~.-c6."HeEoTopHe BonpocN CoBpeMeH-HO~ Teopm~ ~ y H E n ~ " , HOBOCH6HpcE, I976, I49-I60. I0. A x ~ ~ o p c ~. ~ e E n ~ no EBaS~EOH~OpMmm~ oTo6pa~em~M. MOCEBa, M~p , 1969. II. X a B ~ H B.H. l ~ ~ e CBO~OTBa ~HTe#ps~OB T E a K O ~ ~ IBp-Mo~eoE~ conp~l~eHs~X ~ y H E ~ B o6XaCT~X CO c n p ~ e M o ~ rpaH~~e~. -MaTeM.C6., 1965, 68 (II0), 499--517. 12. B e x N ~ B.H., M ~ E x ~ E o B B.M. HeEo~opwe CBO~CTBa E O H ~ O p ~ x ~ E B a S ~ E O H ~ O p ~ X OTo6pa~eHm~ ~ n p ~ e T e o p e ~ EOHCT-pyET~BH02 Teop~E ~ y ~ . -- HsB.AH CCCP, c e p ~ MaTeM.~I974, ~ 6, 1343--1361. 13. B e x ~ ~ B.H. E O H ~ O p M ~ e O T O 6 p a ~ e ~ ~ n p H 6 2 e H ~ e a H a J m T ~ e CKEX ~ B o6~a0T~X 0 EBaS~EOH~Op~Ho~ rpaH~e~. - MaT.c6., 1977, 102, ~ 3, 331-361. G. C. TUMARKIN
TYMAPEEH)
CCCP, 103912, MOCEBa, npocn.MapEca 18, M o C E O B O ~ reo~oropasBe~o~ ~CT~TyT
CO~NTARY A complete geometric description of the class ~p, ~/0
equals
0
IPI,~,0~ =~-4~ ~I~ =~-~/~. • be a simple closed oriented L~punov curve; ~4~..., ~
be points on
P , IPloss(IP '~K) ,~$5 )
in the space L ~ [ ~
it
be the essential norm of P
(L ~ (r, ~ k ~ on P
.asp~-o.,ed tha~"I P Io.~,.>..m.o..~~ (p,~,.~
with the weight ~ ~ T ) ~
(~ (p,.~)~-~.~i.,¢~,
~.~.~i~
deZi,,,~ ~,y (3)) • Then "i-Z[ 5] it ~pro'Ved If our Conjecture is
t'~'0 "P'IIo~. t.~= ~,~,,, I Pl ~'j'~ true thenl P l ~ = ~ (p, ~ , ) .
In conclusion we
T (~thout ,ei~t)
note that in the space
IPlo~ -IPl
~
on the circle
([3]). But i~ ~ene~l the ~o~
IP l depends on the weight and on the contour
r
(E3],[5] ).
327
REFERENCES
I. r o x d e p r H.ll., E p y ~ H ~ E H.H. 0 BopMe npeo6pasoBa~ r~depTa B npocTpaHcTBe LP . - SyH~.aaax. ~ ero n p ~ . , I968, 2, ~ 2, 91-92. 2. P i c h o r i d e s S.K. On the best values of the constants in
3.
4.
5.
6.
7.
in the theorems of M.Riesz, Zygmund and Kolmogorov. - Studia Kath., 1972, 44, N 2, 165-179. E p y n R x ~ H.~., H o x o ~ c K ~ ~ E.H. 0 sopMe onepaTopa cm~ryxspnoro ~Terp~poBan~s. - ~ y a ~ . aRax. • ero np~x., 1975, 9, 4, 73-74. B e p d ~ ~ E ~ i~ H.B. 0nem~a Hop~g~ ~ E I ~ HS IIpOOTI~OTBa X a p 2 ~epes HopMy ee BemecTBemmo~ ~ M ~ o ~ ~ac~z. - B cd."Ma~eM. ~cc~e~o~a~zs", l ~ m ~ e B , ~ ] T ~ a , 1980, J~ 54, 16-20. B e p d • ~ • ~ ~ H.B., E p y n ~ • ~ H.~. To~e EOHCTSHTI~ TeopeMax E . H . B a d e ~ o n B . B . X ~ e ~ s e od OI~paH~IeHHOCTI~ C ~ H r y ~ p moro onepaTopa. - Cood~.AH rpys.CCP, 1977, 85, ~ I, 21-24. r o x d e p ~ H.II., E p y ~ ~ n ~ H.H. B ~ e ~ e ~ e ~ T e o p ~ c~jxsp~x ~e~pax~x onepa~opoB. - E ~ m ~ e B , ~ u a , 1973. H ~ ~ o x ~ c ~ z ~ H.E. ~e~ o6 onepaTope c ~ m ~ a . M. : Hay~ , 1980.
8. H a r d y G.H., L i t t I e w o o d J.E., P ~ I y a G. Inequalities. 2nd ed. Cambridge Univ.Press, London and N e w York,
1952. I. E. VERBITSKY
(H. B. BEPBI~II~) N. Ya. ERUPNIK
(H.~. KPYnHHK)
CCCP, 277028, E m m m e ~ , H ~ C ~ T y T reoalmsmrz ~ r e o ~ o ~ AH MCCP CCCP, 277003, I ~ e B , ElmmKeBc~ rocy~apcTBe~ Ym~BepcxTeT
328 IS THIS OPERATOR I ~ R T I B L E ?
6.7.
Let ~ denote the group of increasing locally absolutely continuous homecmorphisms k of ~ onto itself such that ~t lies in the Muckenhoupt class A ~ of weights. Let Vk denote the operator defined by V ~ ( { ) - - t o k , so that V ~ is bounded on B M 0 ( ~ ) if
and only if
~ CG
(~ones [3]). Suppose that P
projection of BMO onto BMOA. Por which there exists a
for all ~
C~0
such that
k~G
is the usu~l
is it true that
II~V~(1)~M0
BMOA? Is this true for all W ~ G
~(~B~O
?
This questions asks about a quantitative version of the notion that a direction-preser~¢ing homeomorphism cannot take a function of analytic type to one of antianalytic type. For nice functions and homeomorphisms this can be proved using the argument principle, but there are examples where it fails; see Garnett-O'Farrell [2]. We should point out that, the natural predual ~ormulation of this U£#~#114 . I I14 l H'
J
"
This also has the advantage of working with analytic functions whose boundary values trace a rectifiable curve. An equivalent reformulation of the problem is to ask when
V- HV~
~4 ,,+ k
is inve~ible o~ ~o, if ,.
denotes the Hilbert
transform. This question is related to certain conformal mapping estimates; see the proof of Theorem 2 in [I]. In particular, it is shoIAr~X there that this operator is invertible if H ~ WU~M0 is small enough. REFERENCES I. D a v i d G. Courbes corde-arc et espaces de Hardy generallses. -Ann.Inst.Fouzier (Grenoble), 1982, 32, 227-239. 2. G a r n e t t J., 0 ' P a r r e 1 1 A. Sobolev approximation by a sum of subalgebras on the circle. - Pacific J.Math. 1976, 65, 55-63. 3. J o n e s P. Homeomorphisms of the line which preserve BMO, to appear in Arkiv for Natematik. STEPHEN SEMMES
Dept. of Mathematics Yale University New Haven, Connecticut 06520 USA
329
6.8.
AN ESTI~LiTE OF B~O NOR/{ IN TER~S OF AN OPERATOR NORM
Let ~ be a function in B~IO (~f~) with norm II~ I, and let K be a Calderon-Zygmund singular integrel operator acting on -----L~(~ ~) . Define
~
by
Kg(~)--g~ K(Cg~)
. The theory of weighted norm
inequalities insures that ~6 is bounded on ~ if II6 II, is small. In fact the map of 6 to ~6 is an analytic map of a neighborhood of the origin in BMO into the space of bounded operators (for instance, by the argument on p.611 of [3]). ~Iuch less is known in the opposite direction. QUESTION: Given 6~ ~
; if
I~-
K61
is small I must
The hypothesis i~ enough to insure that I16 ~ is finite but the naive estimates are in terms of ~ ] + ~ . If ~ = 4 and ~ is the Hilbert transform then the answer is yes. This follows from the careful analysis of the Helson-Szego theorem given by Cotlar, Sadosky, and Arocena (see, e.g. Corollary
( I I I . d ) of [1] ). A similar question can be asked in more general contexts, for instance with the weighted projections of [2]. In that context one would hope to estimate the operator norm of the commutator [ ~ , p ] (defined by [M~,P](~)=6P~-P(~-).,_,,~ ) in terms of the operator norm
of P -
P8 . RE~ERENCES
I. A r o c e n a R. A refinement of the Helson-Szego theorem and the determination of the extremal measures. - Studia ~¢eth, 1981, LXXI, 203-221. 2. C o i f m a n R., R o c h b e r g R. Projections in weighted spaces, skew projections, and inversion of Toeplitz operators. Integral Equations and Operator Theory, 1982, 5, 145-159. 3. ¢ o i f m a n R., R o c h b e r g R., W e i s s G. Factorization Theorems for Hardy Spaces in Several Variables. - Ann. -
Math. 1976, 103, 611-635. RICHARD ROCHBERG
Washington University Box 1146 St.Louis, MO 63130 USA
330
some OPE~ PROBnEMS CO~CER~n~G H ~
6*9. old
Am~ B M 0
I. A n int erp ola t ing BIs e chk e prod u c t is a Blsschke product having distinct zeros which lie on a n ~ ~ interpolating sequence. I s ~ ~ the un~fgrmly closed linear span of the interDolatin~ Blaschke pr0duc%s? See D ] , ~ ] -
It is
known that the interpolating Blasohke products separate the points of the maximal ideal space (Peter Jones, thesis, University of California, Los Angeles 1978), . Assume 2. Let ~ be a real locally integrable function on that for every interval
where ~I is the mean value of ~ over l , and where C is a con~tant. Does it follow that ~ = ~ ~ H~ . whoso ~ L~ snd~l -l--1, then I]~ [looand
llB-'llooshould be bounded independently of Y
Using the fact that
.
QI/~
has positive real part, i% is easy of ~ satisfy I~I ~ with ~ > 0 . We say that ~ is of P a r r e a u - W i d o m o
We first sketch some relevant results showing that such surfaces are nice. In the following, R denotes a surface of ParreauWidom type, unless Stated otherwise. (1) PARREAU [4]" (a) Every positive harmonic function on has a limit along almost every Green line issuing from any fixed point in ~ . (b) The Dirichlet problem on Green lines on ~ for any bounded measurable boundary function has a unique solution, which converges to the boundary data along almost all Green lines. (2) WIDOM ~ ] : For a hyperbolic Riemann surface ~ , it is of Parreau-Widom type if and only if the set ~@@(~,~) of all bounded holomorphic sections of any given complex flat unitary line bundle over ~ has nonzero elements. (3) HASUMI [6] : (a) Every surface of Parreau-Widom type is obtained by deleting a discrete subset from a surface of Parreau-Widom type, ~ , which is regular in the sense that ~ $ e ~ : @ ( @ , ~ ) ~ 8 is compact for any ~ 0 . (b) Brelot-Choquet's problem (cf. [7]) concerning the relation between Green lines and Martin's boundary has a completely affirmative solution for any surface of ParreauWidom type. (c) The inverse Cauchy theorem holds for ~ . In view of (3)-(a), w e a s s u m e i n what follows that ~
i s
a
r e g u 1 a r
s u r f a c e
o f
P a r-
r e a u - W i d o m t y p e. The Parreau-Widom condition stated in the definition above is then equivalent to the inequality
348
~ ~(~,i6):~ Z ( @ ) } < ~ , where Z(@) denotes the set of critical points, repeated according to multiplicity, of the function ~ ~. . we set
Moreover, let ~I be Martin's minimal boundary of ~ and ~ & the harmonic measure, carried by A I , at the point • . Look at the following STATEMENT (DCT): Let ~ such that
1'1,,,I
be a meromorphic function On
h~s a harmonic majorant on ~
. Then
A
=
, where
denotes the fine boun
W
A4
function for ~
.
(Note: DOT stands for Direct Cauchy Theorem).
(4) HAYASHI [8]: (a) (DCT) is valid for all points ~ in if it is valid for some @ . (b) (DCT) is valid if and only if each -closed ideal of ~o@(~) is generated by some (multiple-valued) inner function on ~ . (c) There exist surfaces of Parreau-Widom type for which (DCT) fails. We now mention SO~E PROBLEMS related to surfaces of ParreauWidom type. (i) Find simple sufficient conditions for a surface of Parreau-Widom t,Tpe t0 Satisfy (DCT).
Hayashi ~8~ has found a couple
of conditions equivalent to (DCT) including (4)-(b) above. But none of them are easy enough to be used as practical tests. (ii) Is there, any criterion for a surface of Parreau-Widom t.ype to satisf,7 the Corona Theorem?
Known results: there exist surfaces of Parreau-Wi-
dom type for which the Corona Theorem is false; there exist surfaces of Parreau-Widom type with infinite genus for which the Corona Theorem is valid. Hayashi asks the following: (iii) Does ~ , ~ ) for any
~
h@ve onl,y constant common inner factors?
(iv) Is a generali-
zed P, and M.Riesz theorem true f or measures on Wiener's harmonic boundar,7t which are o ~ h o ~ o ~ l rize those surfaces ~
to H ~ ? Another problem: (v) Characte-
for which ~ ( ~, ~)
ment without zero~ This
for ever7 ~
has an ele-
was once communicated from Widom and seems
to be still open. On the ether hand, plane domains of Parreau-Widom type are not very well known: (vi) Characterize closed subsets of the Riemann sphere (of. Voichick
M,
~
for which
~\ E
is of Parreau-Widom
Dol).
Finally we note that interesting observations may be found in
349 work o f Pommerenke ~1~, Stanton D2~, Pranger ~13~ and o t h e r s . REPERENCES I. H o f f m a n
K.
Banach Spaces of Analytic Functions. Prentice
-Hall, Englewood Cliffs, N.J., 1962. 2. H e 1 s o n H. Lectures on Invariant Subspaces. Academic Press, New York, 1964 . 3. G a m e 1 i n T. Uniform Algebras, Pretice-Hall. Englewood Cliffs, N.J., 1969. 4. P a r r e a u M. Th~or~me de Patou et probleme de Dirichlet pour les lignes de Green de certaines surfaces de R i e m a ~ . Ann.Acad.Sci.Penn.Ser.A. I, 1958, no.250/25, 8 pp. 5. W i d o m H. ~p sections of vector bundles over Riemann surfaces. - A n n . of N~th., 1971, 94, 304-324. 6. H a s u m i ~. Invariant r~bspaces on open Riemann surfaces. -Ann.Inst,Fourier, Grenoble,1974, 24, 4, 241-286; II, ibid. 1976, 26, 2, 273-299. 7- B r e 1 o t M. Topology of R.S. Martin and Green lines. Lectures on Functions of a Complex Variable, pp.I05-121. Univ. of Michigan Press, Ann Arbor, 1955. 8. H a y a s h i M. Invariant subspaces on Rieman~ surfaces of Parreau-Widom type. Preprint (1980). 9. V o i c h i c k ~. Extreme points of bDunded -~alytic functions on infinitely connected regions. - Proc.Amer.Math.Soc., 1966, 17, 1366-I 369. 10. N e v i 1 1 e C. Imvariant subspaces of Hardy classes on infinitely connected open surfaces. - Memoirs of the Amer.Nath.Soc.. 1975, N 160. 11. P o m m e r e n k e
Ch. On the Green's function of Fuchsian
groups. - ~ n . A c a d . S c i . F e n n . Ser. A. I, 1976, 2, 408-427. 12. S t a n t o n C. Bounded analytic functions on a class of open Riemann surfaces. - Pacific J.~ath., 1975, 59, 557-565. 13. P r a n g e r W. Riemann surfaces and bounded holomorphic functions. -Trans.Amer.Nath.Soc., 1980, 259, 393-400. MORISUKE
HASUMI
Ibaraki University, Department of Mathematics, Nito, Ibaraki, 310, Japan
35O
EDITORS' NOTE. A P a r r e a u - W i d o m surface w i t h a corona has been constructed in the paper N a k a i
M i t s u r u ,
Parreau-Widom t y p e . -
Corona problem for Riemann surfaces of
Pacif.J.Math.,
1982,
103, N I, 103-109.
351
6.19.
INTERPOLATING BLASCHKE PRODUCTS &~m
If
~-Z
is a Blaschke product, the interpo-
B:~ I~,r~l ~-~ z
lation constant of ~
, denoted
A well known result of L . Carlemon asserts that B is an interpolating Blaschke product if and only if ~(~) >0 . It is also well known that the following open problems are equivalent: PROBLEM I. Can every inner function be uniforml,y approximated b,y interpolating Blaschke products? i.e., Given an T Blaschke product B
and an
such that
~ >0
is there an interpolating Blaschke product ~I
IIB - B4 II < S
?
PROBLEM 2. Is there a function Blaschke product product
~
and a~y
~4 such that
6>0
~(6)
so that for ar4¥ finite
, there is a (finite~ B!aschke
IIB-~III
>0, (2) and
G
F=BG
where ~
is an interpolatin~ Blaschke product
is an oute r functio n satisfyin~
352
O< W IG(z)l.C(~-l~l) ~
for some
, is ~ X
c~clic?)
is the Bergman space, that is
analytic ~unctions-ll~]l¢=~ IS Jz ~ ° ° " is in the Ber~man s p a c e and if I~(~)I>
O, @ > 0
, then
S
is cyclic.
If correct this would imply an affirmative answer to QUESTION I when X is the Bergman space. The conjecture is known to be correct under mild additional assumptions (see [2], [3], [4~ ). In particular it is correct when ~ is a singular ~ n e r function. In this c a s e the condition in the hypothesis of the conjecture is equivalent to the condition that the singular measure associated with ~ has modulus of continuity 0 ( ~ @ ~ ~/i) (see [1]). CONJECTURE 2. A singular ~ e r function is c2clic in ~he Bergman
,space
if and onl.T if its @ssociated sinRular m e a d e
on an~ ~ r l e s o n set.
puts no mass
(For the definition of Carleson set see [5~,
pp. 326-327. )~) For more discussion of the cyclicity of inner functions see §6 of ~6], pages 54-58, where the possibility of an '~nner-oute~' factorization for inner functions is considered. ,
u)
i
or else
9.3
- Ed.
383 QUESTION 2. Does there exist a Banach space of anal,v~ic functions. satisfyin~ (i) and (ii). in which a function
and onl~ i f
~t ~ s no
zeroj in ~
i
is c~clic if
?
N . F ~ N i k o l s k i i h a s shown E7] t h a t no w e i g h t e d sup-norm s p a c e o f a c e r t a i n t y p e has t h i s p r o p e r t y . I f such a space X e x i s t e d t h e n t h e o p e r a t o r o f m u l t i p l i c a t i o n by ~ on X would have t h e p r o p e r t y t h a t i t s s e t o f c y c l i c v e c t o r s i s n o n - e m p t y , and i s a c l o s e d s u b s e t o f t h e s p a c e X \ {0} ( t h i s f o l l o w s s i n c e t h e l i m i t o f n o n - v a n i s h i n g a n a l y t i c f u n c t i o n s i s e i t h e r n o n - v a n i s h i n g o r i d e n t i c a l l y z e r o ) . No example o f a n o p e r a t o r w i t h t h i s p r o p e r t y i s known. ( T h i s ~ y no l o n g e r be c o r r e c t ; P e r E n f l o h a s announced a n example o f a n o p e r a t o r on a Ban~ch s p a c e w i t h no i n v a r i a n t s u b s p a c e s ; t h a t i s , e v e r y n o n - z e r o v e c t o r i s c y c l i c . The c o n s t r u c t i o n i s a p p a r e n t l y e x c e e d i n g l y d i f f i c u l t . ) H ° S . S h a p i r o h a s shown t h a t f o r a n y o p e r a t o r t h e s e t o f c y c l i c v e c t o r s i s always a ~ s e t ( s e e [ 8 ] , §11, P r o p o s i t i o n 40, p ° 1 1 0 ) . F o r a d i s c u s s i o n o f some o f t h e s e q u e s t i o n s from t h e p o i n t o f view o f w e i g h t e d s h i f t o p e r a t o r s , s e e [ 8 ~ , ~ 1 1 , 12. QUESTION 3. Let_ X
c
clic.
be a s before, and l e t #
,
~, ~ e X
wish
clic?
This question has a trivial affirmative answer in spaces like the Bergman space, since bounded analytic functions multiply the space into itself. It is unknown for the Dirichlet space (that is, the space of functions with ~l~'I~< eo ); the special case ~ = constant is established in [9] •
REFERENCES
1. S h a p i r o
H a r o I d S. Weakly invertible elements in certain function spaces, and generators in ~ . - Mich.Math.Je,
1964, 11, 161-165. 2. S h a p i r o H a r o i d S. Weighted polynomial approxlwation and boundary behaviour of holomorphic functions. - B ~ . :
CoBpeMeH~e npo6xe~ Teop~ aRa~zT~ec~x ~ m ~ ,
M., Hay~a,
1966, 326-335. 3. ~ a n ~ p o r.
He~oTopHe saMe~zam~t,,~ o Becoso~ no,~z~o~,ma~z,Ho~ annpo~czMam~ rO~OMOp~HX ~ y m ~ . -MaTeM.cS., 1967, 73, 320-330.
4o A h a r o n o v
D., S h a p i r o
HoS.,
S h i e I d s
A,L~
Weakly invertible elements in the space ef squ~re-summ~ble holo-
384 morphlc functions. - J.London Nath.Soc.~ 1974, 9, 183-192. 5. C a r I e s o n L. Sets of uniqueness for functions regular in the unit circle. - Acta Nath.~ 1952, 87, 325-345. 6. D u r e n P.L., R 0 m b e r g B.W., S h i e I d s A.L. Linear functionals on ~ ~ spaces with 0< ~~ I . - J.fur reine und angew.Math.~1969,
238, 32-60.
7. H ~ E o x ~ c ~ z ~ H.E. C n e E T p a x ~ m ~ C~mTes ~ sa~a~a mecoBo~ a n n p o ~ c ~ m m m B upocTpaRcTBaX asa~mTE~ec~x ~ y ~ r ~ . - Hs~.AH Ap~. CCP. Cep.MaTe~., I97I, 6, ~ 5, 345-867. 8. S h i e 1 d s A I I e n L. Weighted shift operators and analytic function theory. - In: Topics in operator theory, ~ath.Surveys N 13, 49-128; Providence, Amer.Math.Soc., 1974. 9, S h i e 1 d s A 1 I e n L. Cyclic vectors in some spaces of analytic functions. - Proc.Royml Irish Acad.~1974, 74, Section A, 293-296.
AT,TRN L. SHIELDS
Department of Nathematics University of Michigan Ann Arbor, Michigan 48109 U.S.A.
O0~ENTARY QUESTION 1 has been answered in the negative by Shamoyan
• and l e t
X~
ITZ~l
d,eno¢e the spa,oe of ~11 f u n c t i o n s ~
= 0 ( 0 ~(~)) f~r i~[/ ~. everywhere in D a~d the PRO~gSITION for the Smirnov dual pair ( ~ H ~, ~ N. 1 using P+ C.Q-L t )), (~ ,p+~) d~ I # ~ 4 ~ , we can prove the following THEOREM. Let X
S*×c X • Suppose
Let g
be a Hausdorff topological vector space, X:-~,
that X
has the following oroDerty:
be an invariant subspace of S*: X--~X
,
E aH
~I
(H)
for some inner f~otion I . ~he~ E=I~(X). CORO~Y
E ~ N,
I. l.~f E
, then E
c
is an invariant subs~aoe of S :N, --~ N,
I*(N.)
, where
I=I E
395 COROLLARY 2. The jclosure of the linear span of the famil[
[S
does not coincide with
a ~seu~ocant, i n ~ t , i o n
N,
(see [1] f o r the e e f ~ n i t i o n ) .
4. DISTRIBUTION 0F VALUES 0F FUNCTIONS IN (N,) De~lote by ~W:(0,~)--~ the decreasing rearrangement of l~I , where ~ is a measurable function on ~ . In [4] (see also [5]) it is proved that if ~ E ~(N,) and ~A~ ~*($)' ~ ~ 0 , then #=--0 •
an,d
O't=O,
i;-'O then
~ -----0-
•
Results of [4],[8] imply
t-~-O t-~O
t-~0 RE~ERENCES
I. 2.
3.
4.
H ~ E o a ~ c E ~ ~ H.K. ~e~m~H o6 onepaTope c~B~ea. M., Hsy~a, I980. A x e E c a H ~ p O B A.B. A n n p o E c m ~ pan~oHax~m ~a~~ aHs~or T e o p e ~ M.PHcca o c o n p ~ e E m ~ x ~ y s z T ~ x ~j~ n p o c ~ a ~ c T B Ip c p ~ ( 0 , 1 ) . -~mTeM.c60pH.,I978, I07, ~ I, 3-I9. A a e E c a H ~ p 0 B A.B. ~ a p ~ a ~ T ~ e no~npocTpa~cTBa onepaTopa o6paTHoro c ~ E r a B npocTpaHc~me HP(p¢(0, 0) . - 8an~cE~ H a ~ . c e ~ L ~ . ~ 0 ~ , I979, 92, 7-29. A a e E c a H ~ p o B A.B. 06 ~-m~Terp~pyeMocmE r p a ~ r ~ x 3Ha~e~ £apMo~ecEzx ~y~. - ~TeM.sa~eTEE, I98I, 30, ~ 1,59-72.
5. A 1 e k s a n d r o v
A.B.
Essays on non locally convex Hardy
classes. - L e c t . N o t e s Math., 1981, 864, 1-89. 6. A a e E c a H ~ p O B A.B. 2L~BapEa~T~e no%upocTpa~cTBa onepaTOpOB c 2 m v a o A ~ C ~ O M a T z x e c ~ no~xoA. - 3an~cEH H a y ~ . c e ~ . ~ 0 ~ , I98I, II3, 7-26. 7. c o i f m a n R.R. A real variable characterization of H P . Studia Math., 1974, 51, N 3, 269-274. 8. H r u s c e v S.V., V i n o g r a d o v S.A. Free interpolation in the space of unifoz~ly convergent Taylor series. - Lect.Notes Math. 1981, v.864, 171-213. A. B. ALEKSANDROV CCCP, I98904, ~ e H E H ~ p ~ , He~po~Bopen (A.B.A~EECAH~POB) ~ 6 X ~ o T e ~ a ~ ~ . 2 . ~aTeMaTEo-Mexa~m~ecEE~ ~ a x y ~ T e T ~I~
396 7.12.
DIVISIBILITY PROBLEMS IN A(~) AND H~(~).
We offer tWO problems on divisibility, one in the disc algebra A(D), the other in H~(D). The first is this. ~or which X C D t ~
, do we h a v e
Ax=N A
A×
where
is the rink of fr~ctions
{~/~ : I , ~ A ( ~ ) , and
~
{~}
~vanishes nowhere in X}
is the local ,tin6 of,,,,fractions ,,
We might point out that (1) holds if ~((IT is closed. (PROOF. Put Y = X ~ T , ~= ~ , and let ~ ~ the right side of (1). Then
~.~N I t i s p r o v e d i n [1] t h a t
A~.
(1) holds i f
X
T-- F/G where F,G ~ A ~ ) ~nd G ~ n i ~ e ,
is closed in
~
~owhere i n
¥
; hence
. ~,et ~A ; then ~=~/~ where ~,~ ~ A ( ~ ) , . We have F~ ={~ ; t h ~ o(~,~}./0
?
be a prime ideal in
~oo(~),
Q is finit~l,y ~enerated, Do,,we then have
= where ~ 6 ~
Q
(Yes i f
--.ol ~ i s maximal [2].) In A(~)
we have the following.
~e~ P and Q PC Q C ~
A(D)
bsp~meid~lsin
where ~ e T
-moaule
O/P
Q ~ P
( P~ =
A C91 ~ h
P~Q
and
the right side of (2)). Then the
is not finitely generated, i.e.
a finitely generated ideal in A (D).
398
This is a corollary to Nakayama's lemma. It suggests that the answer to Problem 2 is yes, but then the proof would have to be different, RE~RENCES I. ~ o r e I i i F. A note on ideals in the disc algebra. - Proc. Amer.Math.Soc. 1982, 84, 389-392. 2. T o m a s s i n i G. A remark on the Banach algebra ~ H ~ [ ~N)'-~ -Boll.Un.Mat.Ital. 1969, 2, 202-204. FRANK FOREI~I
Department of Mathematics University of Wisconsin-Madison Madison, Wisconsin 53706 USA
399
A REFINEMENT O~ THE CORONA THEOREM
7.13.
The usual methods for proving corona type theorems [I ,2] use existence of solutions with bounded radial limits for equations ~=~
when I~IiI~I~,~U
, or something similar to I ~ I I ~ , U
, is
a Carleson measure. Our problem is a variant of the corona theorem for which apparently no Carleson measure is in sight.
~o~. ~or ~
Z~
~,pose {,I~,t~ H~ with I.¢c")l~lI~Cz'~i+lT~cz)l t3 ~ ~ho~e ~o ~,,~ H ~ --{~
The answer is known to be yes if the exponent 2 is replaced by 3 (or 2 +~ if I has no zeroes) and no if 2 is replaced by I (or 2 ~ ). See [2]. The answer is also yes if ~ , ~ are only required to be in H i . This is by a ~ argument using the estimate
-
~TJ
0 ,
O(~(6~t ~ I )
that the zeros of B
~k ) • A computation shows that this implies
have finite degree of contact at ~°°(I)
.
405 The last assertion follows from the closed graph theorem. To prove sufficiency in(b), let O0
J={5
Then ~(~) =
Y
}
C, A : ~(J) =
, one concludes
~(_I)
(4/B) I
and (~/~)I C J
o Applying (a) to
is closed. To prove necessity in (b),
let
( Again,
one can
=~(K)=~(1)n SD (~/B)I~
ignore singular inner factors. ) Then ~ ~)~~nd by the closed ideal structure theorem
K. Thus B K c l ~
o@
OA
; and so, applying (a) to ~
,the
zeros of B have finite degree of contact at ~ K ) = ~ I ) n SD. • Let us consider problem (I) in the more general case where ~ I ) caD but ~°C)=~ ~ ( I ) in t h e l i g h t o f t h e a b o v e r e s u l t s . From t h e c o m p u t a t i o n r e f e r r e d t o i n t h e p r o o f o f s u f f i c i e n c y i n THEOREM (a), it is clear that if the zeros of B have finite degree of contact at ~ ( I ) , thenBlcC? ; however, it is not difficult t o construct ewamples t o show that this condition is not necessary. On the other hand, THEOREM (a) along with the closed ideal structure theorem implies that a necessary condition for B l C O A is that the zeros o f ~ have finite degree of contact with ~o(i) ; however, this condition is clearly not sufficient. It appears that the sets ~(I) , O< ~ 0 then all closed shaped and W ( ) is ideals are standard. This is in contrast to the fact that star-shaped weights can support non-standard closed subalgebras. Apparently ~ilov first posed the problem whethe{r or not there exists any radical algebra weight W such that ~ (W(~)) contains a non-standard ide: al. The answer is affirmative [6, Theorem] for certain seml-multipli ., i,
i
i i
.
L
*) This is also a part of Lemma I in H X E o x ~ c ~ H.K.,H3BeCT~ AH CCCP, c e p ~ ~aTe~., I968, 32, II23-I137. - Ed. **) This is also a part of Theorem2in H x I ~ o ~ c ~ H.K., BeCTHN~ c e p ~ NaTeH.Nex. ~ aCTpOH., 1988, ~ 7, 68-77. - Ed.
424 cative weights [5, Definition 2.1]. These are weights where W ( M % ~ ) actually equals W ( ~ ) W ( ~ ) for many vslues of ~ , ~ in Z + • Hence we propose the following PROBLEM 2. Characteriz,e,,,,,,,,,,,,,the radical al~ebra weiKhts that
~(W("~ -
W
su~ch
has non-s,t,andard ideals.
Even substantial necessary conditions on the weight W for the existence of a non-standard ideal would be welcome. Finally we remark that one can consider related radical algebras ~t(Q÷, W ) built upon Q+ rather than E + (we again Tequire (I)(3) for ~ , ~ in ~ ) . Define for ~ non-zero in ~I(Q+, W )
Also define
~4 We pose the final problem. PROBLEM 3- Does there exist some element ~
~(~)=0
~ (Q+, W )
containir~ am
, such,,,that the closed ',ideal ~enerated b[ 0~ ins
properly contained in M
?
Preliminary results on this problem can be found in [2].
RE~RENCES I. G r a b i n e r S. Weighted shifts and Banach algebras of power series. -American J.Math., 1975, 97, 16-42. 2. G r o n b a e k N. Weighted discrete convolution algebras. "Radical Banach Algebras and Automatic Continuity", Proceedings, Long Beach, 1981, Lect.Notes Math., N 975. 3. S ~ d e r b e r g D. Generators in radical weighted ^ ~ , Uppsala University Department of Nathematics Report 1981:9. 4. T h omt ~ a s M.P. Approximation in the radical algebra ^~4(W~) when ~ W ~ is star-shaped. "Radical Banach Algebras and Automatic Continuity", Proceedings, Long Beach, Lect.Notes in Math., N 975.
425
5. T h o m a s
M.P.
A non-standard
Banach algebra of power series. 6. T h o m a s nach algebra
MARC THOMAS
M.P.
A non-standard
of power series,
closed subalgebra
- J.London Math.Soc.,
of a radical to appear.
closed ideal of a radical Ba-
submitted
to Bull.Amer.Math,So¢.
Mathematics California
Department State College
at Bakersfield 9001 Stockdale Bakersfield, USA
Hwy.
CA 93309
426
7.22. old
HARMONIC SYNTHESIS AND COMPOSITIONS
L e t ~ ~ be the algebra of all absolutely convergent Fourier series on the circle ~ :
We say ~ admits t h e h a r m o n i c s y n t h e s i s ( ~ - h . ~ ) i f t h e r e i s a sequence I ~ , } c ~ ' such t h a t n~-~I{¢ , 0 and I~(O)C I~$ ~(0), ~=J,~, . . . . The algebra ~CI contains functions not admitting h.s. though every sufficiently smooth function admits h.s. QUESTION I. Let ons ~
~
admit h~s~ Is it ~ossible to choose functi-
in the ~bove derlnit~on so that
~ = ~o ~
, ~. bein~ so~e
Zunctlons on ~-1,1] ? Denote by [~] the set of all functions ~ on [-1,4]such that ~°~6~' . This set is a BAn~ch algebra with the n o ~ U ~ {{[~] = ={{~@ ~, . It contains the identity function ~ ( X ) ~ we can reformulate our question, QUESTION 2. Let ~ ~(X) ~
X
po±nt, × = 0
X
. NOW
admit hls , Is it possible to approximate
in the algebra
[~ ] b~ functions vanlshin~ ~ a r the
?
I f I f ] c CI [ - ~ , J ] ( t h i s embedding h a s t o be c o n t i n u o u s by t h e Banach t h e o r e m ) , t h e n t h e f u n c t i o n a l ~[1 ~ )' ~ ~ O) s e p a r a t e s X from f u n c t i o n s i n [ ~3 v a n i s h i n g a t a v i c i n i t y o f z e r o . So the following question is a particular case of o u r problem.
QuEstioN 3. ~et ~ admit h,s. Is
it ~ossible that [~]cC'[-tI] ? so
we have a further specialization. QUESTION 4. Is it possible to construct a function ~ G ~ admittin~ h~ s. with
f
427
Now (1983) very little is known about the structure of the ring [~]. The theorems of Wlener-Le~ type (~]ah.Vl [2]) give some sufficient conditions for the inclusion ~ c [~] , but these conditions are much stronger than C1-smoothness of ~ . On the other hand, let the function $ - - ~ ( ~ ) be even on ~ , ~ ] and strictly monotone on [0,~S . Thus any even function on [-~,~] has a form~@# and our Question 1 has the affirmative answer. Hence If] ~ 0! and all known theorems of Wiener-Levy type are a priori too mough for this ~ . Kahane [3] has constructed examples of functions with [~]~ O ~ 4 , J ] . Thus, the ring [~] is quite mysterious. A possible way to answer our questions is the following. If C then the functional ~ : F ~ ~ O) is well-defined on [~] and generates a functional I ~ ) on the subalgebra[[#]]=~ ~ o ~ : ~ [ # ] } c ~¢' I f there is an extension of this functional to ~ with < I ~ ) , ~ > = 0 for ~(0) c I~$ ~-' (0) , then I cannot admit h.s. It is in this way that Malliavin's lemma on the absence of h.s. has been proved. Namely,
then Malliavin's functional @O
gives the desired extension of ~t(~) The author thanks professor Y.Domar for a helpful discussion in 1978 in Leningrad.
REFERENCES 1. K a h a n e J.-P. S~ries de Pourier absolument convergentes, Springer, Berlin , 1970. 2. ~ w H ~ E ~ H E.M. TeopeMH T~na B~epa-~eB~ ~ o~eEE~ ~ onepaTopOB B ~ e p a - X o n ~ o - MaTeMaTH~ec~e ~ccae~oBs2~, 1973, 8, ~ 3, 14-25.
428
3. K a h a n e J.-P. Une nouvelle reclproque du theoreme de Wiener - L~vy. - C.R.Aead°Soi.Paris, 1967, 264, 104-106. E.MoDYN'K!N
CCCP, 197022, ~ e R ~ r p s ~ yx. npo~.HonoBa, 5 ~eHEH2pa~cE~ SxeE~poTexH~ecE~ ~HCTETyT HM.B.H.Y~B~HOBa (~eH~a)
429 7.23. old
DEUX PROBT,~RS CONCERNANT LES S~RIES TRIGONO~TRIQUEH I. Soit ~
une s~rie trigon~m~trique dont les coeffi-
~ $
cients tendent vers 0 et dont lee sommes partielles tendent vers sur un ensemble ferm6 ~ c T :
N-~
~=-~
~ Soit ~
0
~+(~)
~---- 0
une mesure positive por%ee
par Y ,
~
^
telle que
A-t~on nec,essazrement °
Une r~ponse positive (dont je doute) donnerait une nouvelle preuve de l'existence d'ensembles ~(5) de Zygmund de mesure pleine. ,
2. solt S=LPcT) dans LP(~)
,
^
~~ ~ ~(~
~,~,~'~
. Pout-on =~roo~er t
par d~s pol,ynomes tri~onom~triques 9 = ~ A
~(~)~
A
t e l e que ~ ( 1 4 ¢ ) : ~(~)---> P ( ~ I ~ ) : P(~) ? La question a et~ posse par W.Rudin [I] pour ~ : 1 (la r~ponf . # # se est alors negatzve [ 2 ] ) . Pour p : ~ , la reponse positive est evidents. Pour p~- oo , la question n'a d'int~r~t que si on suppose continue (la r~ponse est negative). La question est ouverte p o u r ~ < ~ BIBLIOGRAPHIE I. R u d i n oe, 1962.
W.
Fourier analysis on groups. N.Y., Interscien-
430
2. K a h a n e J.-P. Idempotents and closed subalgebras of ~(~) - In: l~nct, algebras, 198-207, ed.T.Birtel, Proc.Intern.Symp. Tulane Univ., 1965, Chicago, Scott-Forestmann, 1966. J.-P.~AWANE
Universit~ de Paris-Sud, Nathematique, B~timent 425, Centre d'0rsay 91405, 0rsay Cedex, France
COMMENTAIRE La reponse au second probl~me est negative (voir [3] pour et 4,5] pour ~ < p < O 0 ).
BIBLIOGRAPHIE 3. R i d e r D. Closed subalgebras of C(T) . - Duke ~ t h . J . , 1969, 36, N I, 105-115. 4. 0 b e r I i n D.M. An approximation problem in ~P[ 0 , ~ S • < p/ 4/W(~) ~ -..e. ~-+'I Sometimes equality holds a.e. on an arc ~
of ~ :
WCQ)
~--4
e.g. if $ ~ ~ is in [i(~) , then (4) holds with verifies tha% if W does not vanish a.e. on ~ and then H~(~) cannot split.
•~ o ~ = 2. D] Suppose that ~ ~
P~T
. One if (4) holds,
W A ~ ~.-~
~,a
r, 4 ~ , one can construct W , W > 0 aoe° with for ~ = ~ l l ~nd all ~ [3]. ~hus (4) can fail even if (5) holds.
~,I~ >-~
QUESTION 2. Assm~e that %he inte~r~l in that &
is ~iven b x (6). o,r that
E ,EcT
se~
~---~
(~) is finite! or even
. Is ~here a measurable
,:',.~.h
where %h e first summand consists of " ~ l y % i c " an E
C,o~%ain an,y arc on which ~ ( ~, ~)
functions? ~i~h% such
(or a suitable an%lo~ue)
tends %o zero sufficien%l y slowly as I--~ 0 ? If ,~here is no such F
with M E
> 0 , ~x~ctl~ how can the variou s conclusions of Theo-
rem 2 fa,,ilt,if indeed ~he,y can? QUESTION 3. Let W(~) Assumin~ the integral in ~ )
be smooth with a single zero at ~-----0 . is f tni%¢, describe the ,invarian% sub-
spaces of the operator "multiplication b~ • " on H~(~) of the rates of decrease of W(~)
near 0 and
~(~)
in t e ~ near I.
Perhaps more complete results can be obtained than in the similar situation discussed in [8]. Finally we mention that the study of other special classes may be fruitful. Recently A.L.Volberg has communicated interesting related results for measures 9 t W ~ ~ where ~ is supported on a radial line segment. (See [I0], [13] in the reference list after Commentary. - Edo )
442 REPERENCES
I. C I a r y
S. Quasi-similarity and subnormal operators. - Doct. Thesis, Univ.Michigan, 1973. 2. H a s t i n g s W. A construction of Hilbert spaces of analytic functions. - Proc.Amer.r~ath.Soc., 1979, 74, N 2,2295-298. 3. K r i e t e T. On the structure of certain H (~) spaces. Indiana Univ.Math.J., 1979, 28, N 5, 757-773. 4. B r e n n a n J.E. Approximation in the mean by polynomials on non-Caratheodory domains. - Ark.Nat. 1977, 15, 117-168. 5. M e p r e x ~ H C.H. 0 n o ~ o T e CzCTeM a ~ a ~ z m ~ e c E H x S y H ~ . Ycnex~ MaTeM.RayK, I958, 8, ~ 4, 3--68. 6. K r i e t e T., T r e n t T. Growth near the boundary in M~) spaces. - Proc.Amer.Math.Soc. 1977, 62, 83-88. 7. T r e n t T. ~(~) spaces and bounded evaluations. Doer. Thesis, Univ.Virginia, 1977. 8. K r i e t e T., T r u t t D. On the Cesaro operator. Indiana Univ.Nath.J. 1974, 24, 197-214. THOMAS KRIETE
Department of Math. University of Virginia Charlottesville, Virginia 22903,
USA
COMMENTARY THEOREM (A.L.VoI'Berg) such that
m2(~)
There
exists W
splits even for ~ ~ ~
, W>0
T
a,e, on
.
The theorem gives an affirmative answer to QUESTION I. It may be seen from the proof that ~ (~,S) tends to zero rather rapidly for every ~ . The proof follows an idea of N.K.Nikolskii [9], p.243. PROOF. It is sufficient to construct a function W , ~ > 0 a.e. on T and a sequence of polynomials {P~ } ~ 4 such that 2
I$
(P.IT, Let
{~}~4
ID) =C0,
in
the Hilbert space IZ(WcI,I'II,)~I~L(~
be any sequence of positive numbers satisfying
443 Z S. 0 corresponds to the problem of approximation by functions admitting a quasi-conformal continuation. One of possible ways to solve problem I consists in the construction of a "Swiss cheese" satisfying ~(K)~C(K)~ ~(K) = 0 ( K ) , These problems were posed for the first time at the International Conference on Approximation Theory (Varna, 1981). REFERENCES I. L e h t o 0., V i r t a n e n K.I. Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin. Heidelberg. New-York,1973. 2. B E T y m E E R A.r. A R a a E T ~ e c ~ a s ~ O C T ~ MRo~ecTB B 3a~s~ax TeopzE npEdJm~eRE~. -Ycnexa MaTeM. sayE,1967,22,.~iS, 141--199. 3. Z a i c m a n L. Analytic Capacity and Rational Approximation. Lect.Notes in Math., 1968, 50.
V. I. BELYI
CCCP, 340048, ~osenH 48, YHEBepcETeTCEa2 77, 14RCTMTyT llpEEaa~o~ MaTeMaT~E~ E MexsaHE~
453
TANGENTIAL APPROXIMATION
8.7.
Let F be a closed subset of the complex plane C and let and G be two spaces of functions on F . The set F is said to boa set of tangential a p p r o x i m a t i o n of functions in the class ~ by functions in the class ~ if for each function ~ ~ ~ and each positive continuaus ~ on F , there i s a function ~ G with
Carleman's theorem [I] states that the real axis is a set of tangential approximation of continu@us functions by entire functions Hence, tangential approximation is sometime called Carleman approximation
PROBLEM: For given classes of functions
~
and
G
, characte-
rize the sets of tangential approximation. Of course, this problem is of interest only for certain classes and ~ . We shall use the following notations: H(~)
: entire functions
MF(C) :meromorphic functions on C H(F) U(F)
having no poles on
F
•
: functions holomorphic on (some neighbourhood of) F . :uniform limits on F , of functions in H(F) .
A(F) : ~ u n c t i o ~ continuous on F and holomorphlc on F ° . C(F) :continuous complex-valued functions on F . Each of these classes is included in the one below it. We consider each problem of tangential approximation which results by choosing ~ as one of the first three classes and choosing ~ as one of the last three. Thus each square in the following table corresponds to a problem
U(F) H((:)
A(F)
C(F)
[~]
[9]
MF(£)
[3]
H(F)
[3]
454
The blank squares correspond t o open problems. For partial results on the central square, see [5]. In [3], the conditions stated characterize those sets of tangential approximation for the classes ~=
C(F)
and G = ~ F ~ C )
. One easily checks that these conditions
are also necessary and sufficient for the case ~ = C ( F )
and ~ = H ( F )
The first column was suggested to us by T.W.Gamelin and T.J.Lyons. One can formulate similar problems for harmonic approximation. The most general harmonic function in a neighbourhood of a n isolated singularity at a point
~
~n
, ~$~
can be written in the form
=po.
+
where
KCO ) =
and
Pk,~K
k>~O
g-~
are homogeneous harmonic polynomials of degree
. The sin~alarity of ~
t i a I
if p k = O
m o n i c
, k>~ k o
f u n c t i o n
which is harmonic in ~ ties.
is said to be
. An
h a r-
is a function
except possibly for non-essential singulari-
: functions harmonic on
,
~>~
. We introduce the fol-
~t
~ F ( ~ ) : essentially harmonic functions on ~ singularities on F.
having no
~(F)
:functions harmonic on (some neighbourhood of)
~(F)
: uniform limits on
~(F)
:functions continuous on
~
, of functions in
F.
~(F) . @
c~F)
,
n o n - e s s e n -
e s s e n t i a 1 1 y
on an open set ~ c - ~ n
Let F be a closed set in lowing notations: ~(~)
k
F
and harmonic on
continuous real-valued functions on
F
.
As in the complex case, we have a table of problems.
~.
455
c(p) [s]
mFc ,CF) RE~RENCES I. C a r 1 e m a n T. Sur un th~oreme de Weierstrass. - Ark.Mat. Astronem.Fys. 1927, 20B, 4, I-5. 2. K e a ~ ~ m M, B. ~ a B p e H T B e B M.A. 06 O~OR ssaa-e EapaeMaHa. - ~oEa.AH CCCP, I989, 28, • 8, 746-748. M e p r e a a H C.H. PaBEoMepm~e np~Oa~zeHm~ ~ y s x m ~ EOMEae~cHo-rO nepeMem~oro. - YcnexE MaTeM.HayE, I952, 7, B~n.2 (48), 31-123. (English. Translations Amer.~ath.Soc. 1962, 3, 294-391). A p a E e a a H H.Y. PaBHoMepm~e z ~acaTea~m~e np~OazxeH~a saax~T~ec~mm ~z~m. - ~sB.AH ApM.CCP, cep.MaTeM., 1968, 3,
J~ 4-5, 273-286. 3. H e p c e c a ~ A.A. 0 yaBHoMey~o~ ~ ~aca~ea~Ho~ a u n p o ~ c m ~ a ~ MepOMOp~ ~ . -- HsB.AH ApM.CCP, cep,MaTeM., I972, 7, 6, 405-412. R o t h A. Meromorphe Approximationen. - Comment.Math.Helv. 1973, 48, 151-176. R o t h A. Uniform and tangential approximations by meromorphic functions on closed sets. - Canad.J.Math.1976, 28, 104-111. 4. H e p c e c a H A.A. 0 ~Ho~ecTBax Kapae~saa. - MsB.AH ApM.CCCP, cep.MaTeM., 1971, 6, ~ 6, 465-471. 5. B o i v i n A. On Carleman approximation by meromorphic functions. -Proceedings 8th Conference on Analytic ~unctions,Blazejewko, August 1982, Ed.J.Lawrynowicz (to appear). 6. ~ a ~ ~ ~ a H annpoEc~Mmm~
A.A. 0 paBHoMepHo~ ~ EacaTex~HO~ ~apMo~m~ecEo~ H e n p e p ~ m m ~ S y 2 ~ m ~ Ha ~po~sBOa~SHX COBOEyrK~OCT~X. --MaTeM.ssaeTE~ 1971, 9, Bm~.2, ISI-142. (English: Mat.Notes 1971, 9, pp. 78-84).
7. G a u t h i e r P.M. Carleman approximation on unbounded sets by harmonic functions with Newtonian singularities. - Proceedings 8th Conference on Analytic 2unctions, Blazejewko, August 1982, Ed.J.Imwrynowicz (to appear).
456
8. L a b r ~ c h e M. De l'approximation harmonique uniforme. Doctoral Dissertation Unlverslte de Montr6al, 1982. •
~D~
BOIV~
.
J
•
•
o
PAUL M. GAUTHIER
o
Departement de Mathematlques et de Statistique Unlverslte de Montreal C.P. 6128, Succursale "A" Montreal, Quebec H3C 3J7 CANADA •
•
457 8.8. old
THE INTEGRABILITY 0P THE DERIVATIVE 0F A CONPORNAL NAPPING
Let ~ be a simply connected domain having at least two boundary points in the extended complex plane and let ~ be a cenfermal mapping of ~- onto the open unit disk ~ . In this" note we pose the following QUESTION: Per which numbers p is
J'I/L
?
For p--~ the integral is equal to the area of the disk and is therefore finite. In general, it is known to converge for $/J 0
and one on
be removed?
(2) I s the theorem true for all ~, ~ p < o o
?
If W(~) } 0 in a sufficiently regular fashion then the answer to both questions is yes. For example, if W(~) ~ o -~($~ a n d ~ ( ~ ) ~ + 6° as ~ ~ 0 the divergence of the integral ~ . * ~ ( ~ ) ~ is sufficip v o ant tO guarantee that Hr(..Q.,'MF~Q~I,J,)=LIo. ' for all p:, '~ p < o o , even ~ e n ~ ( ~ L ~ ) ~ O. I n o r d e r to give an i n d i c a t i o n of hew the hypotheses are used, here i s a b r i e f o u t l i n e of the p r o o f of Theorem 1. For each ~ > ¢ we shall denote by 05 the capacity naturally associated with the Sobolev space W I'$ and A~ will stand for ~-dimensional Hausdorff me&sure. A comparison of these two set functions together with their f@rmal definitions can be found in the survey article of ~az' ja and Havin [5] • Let ~ be any function in ~$(O W~X~) , ~ p/(~-~), with the property that ~ Q ~ , ~ # , ~ = 0 for all pol~onials Q and form the Cauchy integral
~L
~-~
Evidently, ~ v~nishes identically in ~ and so, by "continuity", ~ 0 a.e. -C$ on ~lao . To establish the completeness of the polynomials we have only to prove that 5-----0 a.e. -C$ on the rest of the boundary as well, this approach having first been suggested by Havin E6] (cf.also
E3] and E4] ). The argument is then carried out
463
in two stages; one verifies that: STEP 1. ~ 0 a.e. with respect to harmonic m e a s ~ e on ~ l STEP 2, ~--0 a.e. with respect to the capacity C~ on ~ Horeover, in the process it will be convenient to transfer problem from ~ to ~ by means of confoz~nal mapping. With ~ = let ~ = ~(~ . ~or each ~ > 0 let A ~ = { ~ :I ~o¢~1 < 4 - ~. } and put
{~(~)= Thus,
~
I
~(~ ~# ~-r~
a n d ~a
are
; . the ~-~
a n d ~6 ~-- "~6 (~) •
both defined
on ~
a n d ~8
is analytic
near
3D. STEP 1. By c h o o s i n g ~ ~ 0 a n d s u f f i c i e n t l y s m a l l we c a n f i n d a corresponding ~> 0 such that the following series of implications are valid for any Borel set E ~- ~ll :
0
(1.1)
Here $ ~ p/(p-J) and p < ~ + ~ ° The first implication (i) is essentially due to Frostman K7~- Although he considered only N e ~ o n i a n capacity, his ar6ument readily extends to the nonlinear capacities which enter into the completeness problem (cf.~az'ja-Havin [5~ ). The second assertion (ii) is a consequence of a very deep theorem of Carleson [8]. Because I = 0 a.e. -05 on ~ o o and ~(~-oo) > 0 it follows that ~ 0 on some boundary set of positive harmonic meas~LTe. Consequently, taking radial limits, ~-- 0 on a set of positive are length on S~ . We may now suppose that W ( ~ ) ~ 6-~($) , where ~(~)~ + co
a s t , i o . Then, n s i ~
the f a c t t ~ t
~l~'l~&~
< ~
forp 5 . If ~£~ , let
]~, (~)~ I~ ~ ~ : I I~- ~I< ~ }. If als On
Zf
k ~ {4,~}
L~ t ~ ? ~ )
is fixed, define the (positive) function
k'
aS %he inverse Pourier transform of
= (i i" i~I ~ )- I~
~
, and for each
A C~
,f)1,(~) = Jwl~ [ II~ II Lp,(~) Q
ta~t C >0 for
All functions w i l l be real-valued.
t°O, 4} ,Ulet ~ "denote the vector space Of all polynomi~ which are homogeneous of degree ~ , with inner product
define the Bessel capacity
if
kp' < M,
~0
On
, t h e r e e= s s a co
-
C -~ .< ~Kp (B~t°))/~IP'-k " 0 , stud each function ~ ~ LP(X) which is harmcnlo on the interior ~ A , there exists a harmonic function ~ on an open
neighborhood of X THEOREM 1. I f
X
such that any
l[ ~- { II J(X) < ~
"
one of the f c l l o w i n ~ conditions holds, the~
~ s ~he LP h..,,p. a) ( [ 8 ] , [ z ] ) b) ([8])
p'>~v,
e'< ~
and there exists a constant
~4,p,(B~tm)\X)>~!2~ ~tP'-4 c) ( h i , [51 ) ~or each
a~t~o~ is .et: (±) kp' > v~ se~ E k with
~K,p' (Ek) : 0
me~X
if
k ~ {i,~}
~'0
such that
and 0~ or (ii) kp1% • and there exists a set E k with ~K,p'CEk) =O such that
,{01 0 and
:aSX\E k.
If ~ has the ~ h.a.p., then ~ has property ~*) ; this follows from Theorem 2, (I) and the Kellogg property ~5, Theorem 2]. Our question is whether the converse holds: if ~ has Rro~erty (~) , does it have the L e h.a,p.? We remark that if this question were answered affirmatively, then part c) of Theorem I would follow by use of (I).
469
2. If h.a~.
p=~
follows
and
~5
, a different criterion for the
~P
from work of Saak [9]. What is the relation between
saak,s work and Theorem 2?
3. If H ~ 0 U~ ~{0} and ~ is an open set. then o~j~(', ,~') is an increasing set function defined on the subsets . Can one characterize the compact sets ~ v i n ~
the ~
h.a.p.
b,~ means of inoreasim~ set functions which are countabl,~ subadditive a_nd have the propert,y that all Bore! sets are oapacitable? that a set
~
is capacitable with respec~ to a set function
eans t h a t
open, ~ D E
(To say
K
~. ;
See
[3],[~].
compact,
°
Per the case
p=~
, see [9 , Le.~ma
2]. REFERENCES I. B a g b y T. Approximation in the mean by solutions of elliptic equations. - Trans.Amer.~ath.Soc.
2. B y p e H ~ O B B.~. 0 n p ~ 6 m m ~ H ~ ~yH~m~ ~3 npocTpa~CTBa W~ (i~I ~z~ma~ ~ys=~ ~ npo~sBox~oro o ~ H ~ o ~ ~O~ . -- Tpy~N MaTeM.~--Ta NM.B.A.CTeF~oBa AH CCCP,1974, ISI, 51-63. 3. c h o q u e t G. Porme abstraite du theoreme de capacitabiliZeoTBa
re. -Ann.Inst.Pourier (Grenoble), 1959, 9, 83-89. 4. H e d b e r g L.I. Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. - Aota Math., 1981, 147, 237-264. 5. H e d b e r g L.I., W o 1 f f T.H. Thin sets in nonlinear potential theory.Stockholm, 1982. (Rep.Dept.of Math.Univ. of Stockholm, Sweden, ISSN 0348v7652, N 2~. 6. L i n d b e r g P. A constructive method for ~? -approximation by analytic functions. - Ark.for Mat., 1982, 20, 61-68. 7. M e y • r s N.G. A theory of capacities for functions in Lebesgue c l a s s e s . - Math.Scand., 1970, 26, 255-292. 8. P o 1 k i n g J.C. Approximation in ~? by solutions of elliptic partial differential equations. - Amer. J.Math., 1972, 94, 1231-1244.
9. C a a E
%.M. ~ o c ~
Ee~ ~ p ~ x x e ~
Ep~Tep~ ~
odxacT~ c yc~o~mBO~ sa~a-
SJ~nT~qecENx y p a B H e H ~ BNC~HX ~ o p ~ E o B . -- Ma-
470
~em.c6., 1876, 100(142), ~ 2 (6), 201--208o I0. B H T y m E ~ H A.Y. AHaJn~T~ecEa~ eMEOCT~ MHO~eOTB B s~ a~ ax Teop~ ~p~0~eH~. - Ycnex~MaTe~.~ayE, I967, 22, B~n.6, 14I--
-199. THOMAS BAGBY
Indiana University Department of Mmthmmatics Bloomington, Indiana 47405, USA
EDITORS' NOTE. Many years before the appearance of C6S the constructive techniques of Vitushkin was applied to the LP -approximation by analytic functions by S.O.Sinanyan (CMHaH~H C.0. ANHpOECHMa~HS aHa~HTMqecEH~ ~tHE~H~MH H HO~MHOMaMH B cpe~HeM no n ~ o ~ H . Ma~eM.c0., I966, 69, ~ 4, 546-578. )See also the survey MexbHHEOB M.C., C w a H ~ C.0. BonpocH TeopHH rrpH6~ixeH~i~ S y H ~ O~HOI'O Komn~exCHOre nepeMeHHoPo. - B KH: "I~TOrH HayKH ~ TeXHHF~". CoBpemeHHme npo_ 6~eM~ MaTemaTMKH, T.4, MOcKBa, 1975, HS~-BO B~T~, 143-250.
471 RATIONAL A P P R O ~ T I O N
8.11.
0F ANALYTIC P~CTIONS
old
1.
Local approximations. Let
l¢)O
and let ment ~
~ be a complete analytic function corresponding to the ele, For any ~ ~ define ~ ( ~ ) = s~p{~(~-~): ~ ~
where ~ ($) is the multiplicity of the zero of ~ at ~ is the set of all rational functions of degree at most ~. Por any ~ there exists a unique function 9 ~ , ~ ~ such that ~ n ( ~ ) = ~ (I-~) " It is called the ~ - t h diagonal Pad~ approximant to the series (1). Let ~ @ > ~ be an arbitrary fixed number and let I . I = @ -%(') ; then ~ is the function of the best approximation to ~ in ~ w i t h respect to the metric: ~ ( ~ ) = detailed discussion on the Pad~ approximants (the definition in slightl~ differs from the one given above). For any power series (I) we have
A
being an infinite subset of ~
depending on
[2]
~.
A functional analogue of the well-known Thue-Siegel-Roth theorem (see [3], Theorem 2, (i)) can be formulated in our case as follows: is ~ is an element of an algebraic nonrational function ~;hen for any ~ 8 e ~ ~ , the inequality ~ (~) > ~8~ holds on2y for a finite number of indices ~ , ~rom this it follows easily that in 01.L~ e a s e -4
C3) Apparently, this theorem is true for more general classes of analytic functions. CONJECTURE I. If
f un~ction ~
~
is an element of a multi-valued analytic
with a finite set of singular points then (3) is valid.
472 In connection with CONJECTURE 1 we note that if
~-47~(I) = + ~
(4)
A,A>0
then ~ is a single-valued analytic function; but for any the inequality ~ - 4 9~(~) > A is compatible with the fact that ~ is multi-valued (the first assertion is contained essentially in [4],[5], the second follows from the results of Polya [6]). Everything stated above can be reformulated in terms of sequences of normal indices of the diagonal Pad6 approximations (see[7], [I] ). In essence the question is about possible lacunae in the sequence of the Hankel determinants
5~.
Thus (3) means that the sequence { ~ } has no "Hadamard lacunae" and (4) means that {F~} has "Ostrowski lacunae" (in the terminology of [8]), Apparently many results on lactmary power series (see [8]) W e their analogues for diagonal Pade approximations. 2. Uniformapproximation, We restrict ourselves by the corresponding approximation problems on discs centered at infinity for the functions satisfying (1). Let ~ > ~ l , ( is ho-
} ".
lomorphic on E ) and ~R = {E: the best approximation of ~ ~(~)=H no:l~ on
2, ~) on E by the elements of ~ n : "
{II~-%~E: % ~ ~
~,
Denote by U'~
is the sup-
E e
Let ~ be t h e s e t of a l l compact8, F , F C ~)R (with t h e conn e c t e d complement) such that ~ admits a holomorphic (single-valued) continuation on C \ F , Denote by C D (F) the Green capacity of F with respect to ~ (the capacity of the condenser (E, F) ) and define •
473 For every
This inequality follows from the results of Walsoh ([9], oh.VIII). CONJECTURE 2. For ar4y
ct7/ Inequalities (5), (6) are similar to inequalities (2). To clarify (here and further) the analogy with the local case one should pass in section I from 9~ to the best approximations k ~ . In particular, equality (3) will be written as
i
CONJECTURE 3. I_~f $ which
is an element of an analytic f ~ q t i o B
D
has a finite set of sin~u,lar points then
(7) |
If tmder the hypothesis of this conjecture $ is a single-valued analytic function, both parts of (7) are obviously equal to zero. CONJECTURE 3 can be proved for the case when all singular points of 4 lie on ~ ( f o r the case of two singular points see [10]). In contradistinction to the local case the question of validity of (7) remains open for the algebraic functions also.
REFERENCES 1. P e r r o n
O.
Die Lehre von den Kettenbx4/chen, II, Stuttgart,
1957. 2. B a k e r 1975.
G.A. Essentials of Pad~ Approximant, New-York, "AP",
474
3. U c h i y a m a S. Rational approximations to algebraic functions. - Jornal of the Faculty of Sciences Hokkaido University, Serol, 1961, vol.XV, N 3,4, 173-192.
4. r o H ~ a p A.A° ~OEa~BHOe yC~OB~e O~HO3Ha~HOCTE aHaJn~T~eCEEX ~y~. -MaTeM.C6., 1972, 89, 148-164. 5. r o H ~ a p A.A. 0 C X O ~ M O C T H 2n~ROECEMsrU~ ~ e . - ~2TeM.C6., 1973, 92, 152-164. 6. P e I y a G. Untersuchungen uber Lucken und Singularitaten yon Petenzreihen.- Math.Z., 1929, 29, 549-640. 7.
r o H ~ a p A.A. 0 C ~ O ~ M O C T H a n n p o E c H M a ~ Ha~e ~ HeEoTopHx ExaCCOB MepoMop~m~X $ ~ . -- MaTeM.Cd., I975, 97, 605 - 627. 8. B i e b e r b a c h L. Analytische Fortsetzung. Berlin - Heidelberg, Springer-¥erlag, 1955. 9. W a I s h J.L. Interpolation and approximation by rational functions in the cemplex domain. AMS Coll.F~Bl., 20, Sec.e~.1960. I0. r o H ~ a p A.A. 0 cEopocTH pan~oHax~HO~ annpoEc~Mau~ EeEoTop~x aHaJL~TEeCE~X ~ y 2 E ~ . -MaTeM.C6., I978, I05, I47-I88. A,A. GONC~AR
CCCP, 117966, Mocxma
(A.A.r0RNAP)
yx.Bamzxoma 42, CCCP.
475 8.12.
A CONVERGENCE PROBLEM ON RATIONAL APPROXIMATION
old
IN SEVERAL VARIABLES
1. The one-variable case, ~ e 6 . Let me first give the background in the one-variable case. Let ~(~)~ ~ C ~ ~ , R e 6 , be a formal power series and P/Q , Q ~ 0 , a rational function in one variable ~ of type (~,~) , i.e. P is a polynomial of degree ~ ~ and Q of degree ~ 9 . It is in general not possible to determine P / Q so that it interpolates to ~ of order at least ~* 9 @ I at the origin (i.e. having the same Taylor polynomial of degree ~ v 9 as ~ ). However, given ~ and ~ , we can always find a unique rational function P / ~ of type (~,~) such that ? interpolates to ~Q of order at least ~ + ~ + ~ at the origin, i.e. ( ~ - P ) ( ~ ) ~ 0 ( ~ + $ + 1 ) . This function P/~ , the[~P a d e a p p r o x i m a n t to ~ , was first studied systematically by Pade in 1892; see [I]. In 1902 Montessus de Ballore [2] proved the following theorem which generalizes the well-known result on the circle of convergence for Taylor series. 1
THEOREM. Suppose ~ Rhic in ~ I < ~ Then the [I,,#3 formly to ~
with ~
is holomorphlc at the origin and meromorpoles (counted with their multiplicities),
-Pade approximant to
~ , ~ / ~
, conver~es uni-
, with ~eometric de~ree of convergence I in those com-
pact subsets of I~I < ~
which do not contain ar47 poles of ~ .
With the assumption in the theorem it can also be proved that P /Q diverges outsiae I I= if is chosen as large as possible [3, p.2693 and that the poles of Pw / ~ w converge to the poles of { in l~l < ~ • Furthermore, when ~ is sufficientS,y large, ~ / Q ~ is the -n~que rational function of type (~, ~) which interpolates to ~ at the origin of order at least I,+9 + I. Montessus de Ballore's original proof used Hadamard's theory of polar singularities (see [4]). Today, several other, easier proofs are known; see for instance [51,[6] ,[7] and [8]. Pad6 approximants have been used in a variety of problems in numerical analysis and theoretical physics, for instance in the numerical evaluation of functions and in order to locate singularities of functions (see [I] ). One reason for this is, of course, the fact that the Pad6 approximants of ~ are easy to calculate from the power series expansion of ~ . In recent years there has been an increasing interest in using analogous interpolation procedures to apprexi-
476
mate functions of several variables (see E9~). I propose the problem to investigate in which sense it is possible to generalize Montessus de Ballore's theorem to several variables. 2. The two-variable case, ~ ( ~ i , ~ ) ; ~,~e~. We first generalize the definition of Pad6 approximants to the twovariable case. Let ~ ( ~ ) ~ ~ G~K~~ ~ be a formal power series and let ~/~ , ~0 , be a rational function in two variables ~I and ~ of type (~,$) , i.e. P is a polynomial in ~I and ~ of de~ree ~< ~ and ~ of degree ~ ~ . By counting the number of coefficients in P and ~ we see that it is always possible to determine P ~K~-O
and 0
so that, i f ( ~ f - P ) ( ~ ) ~
for (~,K)6 ~
, where ~
chosen subset of ~ x ~
K
~
, then
, the interpolation set, is a
with ~(We~)(~+$)+~(9+~)(9+~)-
elements. There is no natural unique way to choose ~ but it seems reasonable to assume thatI(~,K):~+K~W} c ~ and t h a t ( ~ , K ) ~ =>(~,W%) 6 ~ if ~ ~ } and ~ K . In this way we get a r a t i o n a i a p p r o x i m a n t P/@ of type (w, 9) to
corresponding
to
. With a s
table choice of
, P/Q
is unique [7 , Theorem 1.I~. The definition, elementary properties, and some convergence results have been considered for these and similar approximants in [9], ~0] and [7]. The possibility to generalize Montessus de Ballore's theorem has been discussed in [6],[73 and E11] but the results are far from being complete. PROBLEM I. In what sense can Montessus de Ballore's theorem b@ ~eneraliz~d to several variables? I% is not clear what class of functions ~ one should use. We consider the following concrete situstion. Let ~ = ~ / 6 , where is holomorphlc in the p o l y d i s c ~ = ( ~ , ~ ) : l ~ I < ~ , ~=~,~ ~ and & is a polynomial of degree 9 , ~(0) ~ 0 • By the method described above we obtain for every ~ a rational approxin~nt P W / Q w of type (~,~) to # corresponding to some chosen interpolation set ~ = ~w" In what region of ~ does P w / ~ converge to ~ ? Partial answers %o this problem are given in [7] and [11] (in the latter with a somewhat different definition of the approximants). If @ = ~ , explicit calculations are possible and sharp results are easy to obtain [7 , Section 4~. These show that in general we do not have convergence in{~: I ~ l < ~ , $=~,~\{~:e(~)~---O} . This proves that the general Analogue of the Montessus de Ballore's theorem is not true. It may be
477 added, that it is easy to prove - by just using Cauchy's estimates that there exist ration~l functions ~ $ of type (~,~) interpolating to ~ at the origin of order at least ~ + I and converging ifo=ay,
as
, to
in compact
subsets
A disadvantage, however, of ~ $ compared to the rational approximants defined above is that % ~ is not possible to compute from the Taylor series expansion of ~ (see ~ , Theorem 3.3~). In the one-variable case the proof of Montessus de Ballore's theorem is essentially finished when you have proved that the poles of the Pad~ approximants converge to the poles of ~ . In the several-variable case, on the other hand, there are examples ~ , Section 4, Counterexample 2];when the rational approximants P~/~ ~ do not converge in the whole region ~ ~: ~ ( , ) = 0 } in spite of fact that the singularities of P l ~ / ~ converge to the singularities of ~ / ~ . This motivates:
\|
PROB~
2. Und.er what c ondi,t.!ons does Q~
conver~e to ~
?
The choice @f the interpolation set ~t~ is important for the convergence. For instance, if $-----J and ~i_~_~_~_oo , we get convergence in ~ \ ~ ~: ~ ( ~ ) = 0 } with a suitable choice of ~ ET, Section 4~. On the other hand if we change just one point in ~ without violating the reasonable choices of ~ indicated in the definition of the rational approximants - we get examples [7, Section 4, Cotuuterexample 1],where we do not have convergence in any polydisc around ~ 0 . PROBLE~ 3. How is the convergence the choice of the interpolation s~t ~
P~/Q. ......~.. ~
influenced b~
.~
Since we do not get a complete generalization of Montessus de Ballore's theorem it is also natural to ask: PROBLEM 4. If, the sequence .o,frat!ona,l,,,approximants aces not conver~e t is there a subs equence t ~ t
c,onv~rges _t0 ~ ?
(Compare ~7, Theorem 3.4] ) l~inally, I want to propose the following conjecture.
and the interpolation set Corollary above. )
~
2~ and the case 9 =
is suitably chosen. (Compare Ell, ~
referred to just after PROBLE~ 2
478
REFERENCES I. B a k e r
C.A.
Academic Press, 2. d e
Essentials of Pad6 Approximants.
N e w York,
1975.
M o n t e s s u s
d e
B a I I o r e
fractions continues alg~briques.
R.
Sur les
- Bull. Soc•Math. France ~ 1902~
30, 28-36. 3- P e r r o n
O.
Die Lehre yon den Kettenbr~chen. Band II.
Stuttgart, Teubner, 4. G r a g g
1957.
W.B. On Hadamard's theory of polar singularities.
In: Pad6 approximants and their applications
-
(Graves-Morris,
P.R., e.d.), London, Academic Press, 1973, 117-123. 5. S a f f
E.B.
A n extension of Montessus de Ballore's theorem
on the convergence of interpolation rational functions. - J. Approx.T., 1972, 6, 63-68. 6. C h i s h o 1 m J.S.R., G r a v e s - M o r r i s
P.R. Gene-
ralization of the theorem of de Montessus to two-variable approximants. - P r o c . R o y ~ l Soc.Ser.A., 1975, 342, 341-372, 7- K a r 1 s s o n J., W a 1 1 i n H. Rational approximation by an interpolation procedure in several variables.rational approximation York, Academic Press,
8.
In: Pad~ and
(Saff, E.B. and Varga, R.S., eds.), New 1977, 83-100.
r o H q a p A.A. 0 CXO~HMOCT~ O606~eHR~X annpo~c~Many~ ~a~e Mepo~opSHax ~ y H E ~ . -~4aTez.cS., 1975, 98, 4, 563-577.
9. C h i s h o 1 m
J.S.R.
N
-variable rational approximants.
-
In: P a d 6 and rational approximation (Saff, E.B. and Varga, R.S., eds.), N e w York, Academic Press,
1977, 23-42.
IO. F o H ~ a p A.A. JIOEaKBHOe yC~OB~e O~O3HaqHOCTE aHa~I~T~lqeoK~X ~ y ~ Hec~o~X nepeMeHHax. - ~Te~.C6., 1974, 93, ~ 2,
296-313. 11. G r a v e s - M o r r i s
P.R.
Generalizations of the theorem
of de Montessus using Canterbury approximant. tional approximation Academic Press, HANS W A L L S
- In: Pad6 and ra-
(Saff, E.B. and Varga, R.S., eds.), N e w York,
1977, 73-82. Ume~
U n i v e r s i t y
S-90187 Ume~, Sweden
479
COMMENTARY BY THE AUTHOR In a recent paper A.Cuyt (A Montessus de Ballore theorem for multivariate Pad~ approximants, Dept. of Math., Univ. of Antwerp,,_ Belgium, 1983) considers a multivariate rational approximant ~/ to ~ where ~ and Q are polynomials of degree ~ ÷ ~ and ~ $ ~ , respectively, such that all the terms of P and Q of degree less t h a n ~ $ vanish. It is then possible to determine P and Q so that ~Qhas a power series expansion where the terms of degree ~+~+~ are all zero. Por this approximant P / Q she proves the following theorem where P / Q - P~ / Q ~ and ~ and ~ have no common non-constant factor: Let ]=F/6 where F i , holomorphic in the polydisc {Z:IZ~I(R~} and ~ is a polynomial of degree ~ , ~(0) ~ 0 , and asstm~e that ~ ( 0 ) ~ 0 for infinitely many ~ . Then there exists a polynomial Q(Z} of degree $
uch that converges ,~niformly to ~
u oo uo oo of{P,/Q4
on compact subsets of
[;~.lzi, I
0
=E UE
{~:t~-~l. o • However every subset F ~ of is irregular and hence satisfies C ~ ( F ) = 0
F with H 4(F) < . This shows we cannot
easily reduce the problem to compact sets E with H~(E) < ~ . If ~(E)> 0 , one possible approach to prove CR(E)>o
E -1-T-z
is
E,0(e(2
tO consider the set A point ~ is not in ~ if and only if the line passing through ~ and whose distance to the origin is I~I , misses the set . It is not hard to see C R(E)> o i f i f E has positive area. Uy [ 24] *) has recently shown that a set F has positive area if Bad only if there is a Lipschitz continuous function which is analytic on ~ \ F . so one might try to construct such a function for the set ~ , A related question was asbed by A.Beurling. He asked, if ~ and i f E has no removable points, then must the part of the boundary of the normal fundamental domain (for the universal covering map) on the unit circle have positive length? This was shown to fail in [26]. Finally, I would like to mention that I see no reason why C ~ ( E ) is not comparable to analytic capacity. In other words, does there exist a constant K With 4/K "C~(E)~ ~ ( E ) ~ ~ C ~ ( E ) ? If this were true, it would have application to other problems. ~or example-, it would prove that analytic capacity is semi-subadditive.
E
(E)>o
REFERENCES I. A h 1 f o r s L.V., B e u r 1 i n g A. Comformal invariants and f~uction-theoretic null sets. - Acta Math., 1950, 83, 101-129 2. P a i n 1 e v ~ P. Sur les lignes singuli~res des fonctions analytiques. -Ann,Fac.Sci. Toulouse, 1888, 2. 3, A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J., 1947, 14, 1-11. 4.
B m T y m ~ ~ H
A . F . A ~ J ~ Z T m ~ e c ~ e eM~CTI, MHoxec~'B B s s ~ a x Teolm~ ~ p m 6 ~ z e ~ . - Ycuex~ ~mTeu.HsyK,I967,22,~, I4I-I99. 5. z a 1 c m a n L. Analytic capacity and Rational Approximation - Lect.Notes Math., N 50, Berlin, Springer, 1968. 6. C o 1 1 i n g w o o d E.P., L o h w a t e r A.J. The Theory
of Cluster Sets. Cambridge, Cambridge U.P., 1966~ 7. B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points. - Proc.London Math. Soc°, 1931, 32, N 2, I-9. See [27] for a short proof. - Ed.
489 8. B ]~ T y m E B H
A.r.
lIpilMep MEtoxecT~t ZZO~ZZZTe~Z~Ot ~ m ~ ,
~ro
Hy~eBoM SN~Jt~TB~eoEo~ eNEOOTg.-~0~.AH CCCP, 1959, 127, 246-249. 9. G a r n e t t
J, Positive length but zero analytic capacity
-
Proc.Amer.Math. Soc., 1970, 24, 696-699.
l O . H B a ]K o B
~.~.
11. O r o f t o n
BS1SI~aI~MB MHozecTB lit ~ y H l ~ ] ~ .
M.,
"Hayza",I975.
M.W, On the theory of Local Probability.
-Philos.
Trans.Roy.Soc., 1968, 177, 181-199. 12. S y 1 v e s t e r J,J. On a funicular solution of Buffon's "Problem of the needle" in its most general form~ - Acts Math , 1891, 14, 185-205. 13. M a r s t r a n d
J°M. Fundamental geometrical properties of
plane sets of fractional dimensions.
- Proc.London Math.Soc.,
1954, 4, 257-302. 14. D e n j o y A° Sur lea fonctions analytiques uniformes ~ singularit~s discontinues. 15. X a B M H C O H
-CoR,Acad.Sci,Pmris,
C.H. 06 ~ a a ~ T M ~ e c E o ~
1909, 149, 258-260,
e~EOCT~ MHoxecTB, CoBue-
O~Ol) H e T p M B a a ~ O O T ~ ImSJta~KUX ~laccoB aHaJt~T~qecz~x ~ y m ~ c ~ ~ e ~ e ~Bap~a B npoMsBo~m~x 06~aOTSX. - MaTes.c6., 1961, 54,
J
~ I , 3-50. 16. I~ B a H o B
~.~. 06 aHa~Tsqecxo~ eKI~OOT~ ~J~He~h~x IE~O~eOTB.
--Ycnexa MaTea.Hay~, I962, I7, I43-I44. A.M. Analytic capacity and approximation problems. 17. D a v i e Trans.Amer.Math.Soc., 1972, 171, 409-444. 18. C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and related operators. - Proc.Nat,Acad.Sci, USA, 1977, 74, 1324-1327 P.R, Schwarz's lemma and the Szeg'o kernel 19. G a r a b e d i a n function. -Trans.Amer.~lath,Soc=, 1949, 67, 1-35. 20. X a B ~ H B.H. l~aHa~Hue CB0~OTB8 asTerpa~oB Tana Koma a rapao-
~ecEa
Conp~eHHuX ~ m U ~ a ~ B 06~a0TSX C0 c~pe~seao~ rpaHa~e~. -~aTe~.c6., 1965, 68, 499-517. 21. X a B z H B.H., X a B ~ H c o H C.~. HeEOTOI~e o~eH~a aHa~ a ~ e c ~ o ~ eaEoc~z. - ~ o ~ . A H CCCP, 1961, 138, 789-792. 22. B e s i c o v i t c h A. On the fundamental geometrical properties of linearly measurable plane sets of points I. - Math.Ann.,
23.
1927, 98, 422-464. II: Math.Ann., 1938, 115, 296-329. B e s i c o v i t c h A. On the fundamental geometrical properties of linearly measurable plane sets of points I I I . -
Math.Ann.,
1939, 116, 349-357. 24. U y N. Removable sets of analytic functions satisfying a Lipschitz condition.
25.
e d e r e r
-Ark.Mat.,
1979, 17, 19-27.
H. Geometric measure theory. Springer-Verlag,
Bar-
490 lin, 1969. 26° N a r s h a 1 1 D°E. Painlev~ null sets, Colloq, d'Analyse Harmonique et Complexe. Ed,: G.Detraz, L,Gruman, J.-P.Rosay. Univ. Aix-Marseill I, Marseill, 1977. C.B. HpOcTOe ~oz,asa~em~cTBo ~eopeuu o0 y C T I m ~ M ~ X ooo6e~ooT~ a E a ~ e c z ~ x ~ m o ~ , ~o~.~e~Bop.mo~x yO~OB~ ~Lmn=az~a. - 8 a ~ . H a ~ . o e a . ~ 0 M N , I 9 8 I , I I 3 , I 9 9 - 2 0 3 .
27. X p ~ ~ e B
DONALD E. MARSHALL
Department of MathematicspUniversity of WashingtonjSeattle, Washington 98195 USA
Research supported in part by National Foundation Gramt No MCS 77-01873
491
8.16. old
ON PAINLEV~ NULL SETS Suppose that
E
neighbourhood of s e t
is a compact plane set and that
~
. i set is called a
~
is an open
P a i n 1 e v e
n u 1 1
(or 2.N. set) if every function regular and bounded in ~\ E
can be analytically continued onto that
E
has
z e r o
E
. In this case we also say
a n a 1 y t i c
c a p a c i t y.
The problem of the structure of P.N. sets has a long history. Painlev~ proved that if dorff) measure
E
has linear (i.e.
zero, then E
~ - dimensional Haus-
is a P.N. set,
this result was first published by Zoretti
though it seems that Painleve' " s theorem
[I].
has been rediscovered by various people including Besicovitch who proved that if outside
E
, and if
~
is continuous on E
E
[2]
, as well as regular
has finite linear measure,
analytically continued onto
then
~
can be
. Denjoy [3] conjectured that if
lies on a rectifiable curve, then E
E
E
E
is a P.N. set if and only if
has linear measure zero. He proved this result for linear sets.
Ahlfors and Beurling
[~
proved Denjoy's conjecture for sets on ana-
lytic curves and Ivanov [ 4
for
Sets on sufficiently smooth curves.
Davie [6] has shown that it is sufficient to prove Denjoy's conjecture for
C 4 curves. On the other hand Havin and Havinson [ ~
and Ha-
vin [8] showed that D e n j o y ' s c o n j e c t u r e follows if the Cauchy integral operator is bounded on ~ for ~ curves. This latter result has now been
proved by Calder6n
[gB so that Denjoy's conjecture is
true. I am grateful to D.E.Narshall [10] for informing me about the above results. Besicovitch
[11] proved
that every
compact set
linear measure is the union of two subsets
~I
'
~
~
of finite
. The subset E~
lies on the union of a finite or countable number of rectifiable Jordan arcs. It follows from the above result that set unless
~4 has linear measure zero. The set
~4
~
is not a P.N.
on the other
hand meets every rectifiable curve in a set of measure zero, has projection zero in almost all directions and has a linear density at almost none of its points. The sets tively
r e g u 1 a r
and
E4 and
~
i r r e g u 1 a r
were called respecby
Besicovitch
[11]. Since irregular sets behave in some respects like sets of measure zero I have tentatively conjectured
[12, p.231]
be P.N. sets. Vitushkin
[14] have given examples
[13] and Garnett
that they might
of irregular sets which are indeed P.N. sets, but the complete conjecture is still open. A more comprehensive conjecture is due to Vitushkin
[15 , p . 1 4 ~
492 He conjectures that
E
is a P.N.
set if and only if
~
has zero
projection in almost all directions. It is not difficult to see that a compact set
~
is a P.N. set
if and only if for every bounded complex measure distributed on E
,
the function
(I)
E is unbounded outside
E ~). Thus
E
is certainly not a P.N. set on E
if there exists a positive unit measure
such that
f t~-gl E
is bounded outside
E
i.e. if
E
has positive linear capacity [16,
p.73]. This is certainly the case if respect to some Hausdorff function
E
has positive measure with , such that
9,-----i-~ < oo 0
([17]). Thus in particular
E
is not a P.N. set if
E
has Haus-
dcrff dimension greater than one. While a full geometrical characterization of P.N. sets is likely to be difficult there still seems plenty of scope for further work on this intriguing class of sets.
RE FEREN CE S I. Z o r e t t i
L.
Sur les fonctions analytiques uniformes qui
possedent un ensemble parfait discontinu de points singuliers. J.~th.Pures
Appl.,
-
1905, 6, N I, 1-51.
2. B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behavior of analytic functions in the neighborhood of non-isolated singular points. - Proc.London ~ath. Scc°, 1931, 32, N 2, I-9. 3. D e n j o y ° ~
gularltes
A.
Sur les fcnctions analytiques uniformes a sin-
discontinues.
- C.R. Acad. Sci.Paris,
1909, 149, 258-
-260. ~)
See E d . n o t e a t
t h e end o f t h e s e c t i o n .
- Ed.
493 0d aHS~HT~eoEo~ eMEOCTM MHOXeOTB, COBpasm~x IuIaOOOB a ~ T H ~ e c E ~ X ~yHEI~ aeMMe IUBap~a B ~ p o H s B O ~ X 06zaoT~x. - NmTeM.c6., 1961, 54, I, 3-50. 5. H B a H 0 B ~.~. 0 lauoTese ~aHxya. - Ycnex~ MaTeM.HayK, 1964, 18, 147--149.
4. X a B H H C o H
C.~.
MeoTHO~ H e T p H B H ~ H O O T I
6. D a v i e
A.M.
Analytic
capacity and approximation
problems.
1972, 171, 409-444. X a B ~ H C O H C.H. He~oTop~e o~e~Fa a~a- ~ o F ~ . A H CCCP, 1961, 138, 789-792.
-Trans.Amer.Math.Soc.,
7. X a B ~ H ~T~ecEo~
e~OCTH.
8. X a B H H
B.H.
HEec~
B.H.,
F p ~ e
conp~eB~x
M~TeM.C6., I965, 9. C a I d e r ~ n
B o6~aCT2X CO c n p ~ e M o ~
A.P.
related operators.
Cauchy integrals
on Lipschitz
-
curves and
USA, 1977, 74, 1324-1327. Preprint, 1977.
- Proc.Natl.Acad.Sci. D.E.
The Denjoy Conjecture.
11. B e s i c o v i t c h perties
rp~e~.
68, 499-517.
10. M a r s h a 1 1
Ann.,
CBO~CTBa HHTeI~S2IOB T~na Ko,~ ~ rapMo-
~
A.
On the fundamental
of linearly measurable
geometrical
pro-
plane sets of points I. - Math.
1927, 98, 422-464. II: Math .Ann., 1938, 115, 296-329.
12. H a y m a n
W.K.,
K e n n e d y
Vol. 1. London - N.Y., Academic
13. B H T y m E ~ H
A.~.
~ep
HyxeBO~ a ~ a ~ T ~ e c E o ~
-249. 14. G a r n e t t
Positive
TeOp~ np~6~eH~.
16. C a r 1 e s o n
--~oEx.AH
~ , Ho 1969, 127, 246capacity.
-
AHSJH~TEeCEB~ eMNOCTB MHo~eCTB B s8~aqax - Ycnex~ MaTeM.HayE, I967, 22, ~ 6, 14I-I99.
L.
Selected
problems •
0.
A.F.
on exceptional
N 13, Toronto,
Potentiel
Medded.Lunds~Univ.~t.S~.,
E ~ H
CCCP,
length but zero analytic
.
•
1935,
sets.
-
1967.
Van Nostrand, °
d'equ~l~bre
sembles avec quelques applications
18. B ~ T y m
1976.
Press,
A.r.
Van Nostrand Math.stud.,
17. E r o s t m a n
Functions
1970, 24, 696-699.
Proc.Amer.~'2th.Soc.,
15. B ~ T y m E Z H
Subharmonic
~Ho~ecTBa H o ~ O ~ T e ~ H o ~
eMEOCT~.
J.
P.B.
•
et capac~te
des en-
a la theor~e des fonctions.
-
3, 1-118.
06 O~HO~ Bs~a~e ~ a ~ y a .
- HsB.AH CCCP,
cep.MaTeM., 1964, 28, ~ 4, 745--756. 19. B a ~ ~ c E ~ ~ P.8. HecEox~EO s a M e ~ a ~ o6 o ~ p a H ~ x e ~ x aHa~ecE~x ~y~n~x, npe~cTaB~M~x ~HTe~pa~oM T~na Kom~-CT~T~eca. -C~6.MaTeM.~., I966, 7, ~ 2, 252--260. W.K.HAY~,~N
Imperial
College, Department
Mathematics,
of
South Kensington,
L o n d o n SW7 England
494
EDITORS' NOTE. As far as we know the representability of a 1 1 functions bounded and analytic off E and vanishing at infinity by "Cauchy potentials" (I) is guaranteed when E has finite Painlev~'s length whereas examples show that this is no longer true for an arbitrary E (~18],[19]). We think THE QUESTION £f existence of potentials (I) bounded in
~ \~
(provided
E
is no t a P.N. set) i_~s
one more interestin ~ problem (see also § 5 of [ 4]).
495 8.17.
ANALYTIC CAPACITY AND RATIONAL APPROXIMATION
old
Let
all
E
be a bounded subset of C ^and functions ~ in ~ analytic on ~ \ m
I~1 < 4 on~O 3
.
Put
A(E,O
B(E,~)
= { ~ ~ BQE,~) "
be the set of
and with ~(co)= 0 , ~ is continuous
. The number
is called the analytic capacity of
E
. The number
~A(E,O ~--~ is called the analytic C-capacity of E • The analytic capacity has been introduced by Ahlfors [I] in connection with the Painlev~ problem to describe sets of removable sin~alarities of bounded analytic functions. Ahlfors [I] has proved that these sets are characterized by ~(E) = 0 . However, it would be desirable to describe removable sets in metric terms. CONJECTURE I. A compact set E capacity iff the projection of E
,
E cC
, has zero anal~tic
onto almost evelV direction has
zero length ("almost every" means "a.e. with respect to the linear measure on the unit circle). Such an E is called irregular provided its linear Hausdcrff measure is positive. If the linear Hausdor~f measure of E is finite and ~(~) = 0 then the average of the measures of the pr@jections of E is zero. This follows from the Calderon's result [2] and the well-known theorems about irregular sets (see [ 3], p.341-348). The connection between the capacity and measures is described in detail in [4]. The capacitary characteristics are most efficient in the approximation theory [ ~ , [6] ,[7], [8S. A number of approximation problems leads %o an unsolved question of the semiadditivity of the analytic capacity:
%(EUF) ~ c [ ~ ( E ) + ~ ( F ) ]
,
496 where C is an absolute constant and E, F are arbitrary disjoint compact sets. Let A (K) denote the algebra of all functions continuous on a compact set K , K:-C , and analytic in its interior. Let ~ ( K ) denote the uniform closure of rational functions with poles off and, finally, let ~o~ be the inner boundary of ~ , i.e. the set of boundary points of ~ not belonging to the boundary of a component of C \ K . Sets K satisfying A ( ~ ) = ~ ( ~ ) were characterized in terms of the analytic capacity [6]. To obtain geometrical conditions of the approximability a further study of capacities is needed. CONJECTURE 2. If ~ ( ~ ° ~ ) = 0
then
A(~) =~(K)
.
The affirmative answer to the question of semiaddivity would yield a proof of this conjecture. Since ~ ( [ ) = 0 provided ~ is of finite linear Hausdorff measure this would also lead to the proof of the following statement. CONJECTURE 3. If the linear Hausdorff measure of ~o~
( K
b e i n g a compact ..sub.s.et o f
~.,
) then
is zero
A(K] --C(K). f
The last equality is not proved even for K s with ~°K of zero linear Hausdorff measure. It is possible however that the semiadditivity problem can be avoided in the proof of CONJECTURE 3. The semiadditivity of the capacity has been proved only in some special cases ([9], [10-13] ), e.g. for sets ~ and ~ separated by a straight line. For a detailed discussion of this and some other relevaal% problems see [14] o
REFERENCES I. A h 1 f o r s L.V. Bounded analytic functions. - Duke Math.J., 1947, 14, 1-11. 2. C a 1 d e r ~ n A.P. Cauchy integrals on Lipschitz curves and related o p e r a t o r s . - Proc.Nat.Acad.Sci., USA, 1977, 74, 1324-1327. 3. H B a H 0 B ~.~. B a p ~ a m ~ MHo~eCTB ~ ~ y H E m ~ , M., "HayEa", 1975. 4. G a r n e t t J. Analytic capacity and measure. - Lect.Notes Math., 297, Berlin, Springer, 1972. 5. B ~ T y m E ~ ~ A.r. A H a X ~ T ~ e c E a ~ eMEOOT~ ~ o ~ e c T B B sa~avax
497
Teop~npEdx~meH~. 6. M e ~ ~ H ~ E 0 B
- Ycn~xHMaTem.HayE, 1967, 22, ~ 6, 141-199. M.C., C ~ H aH ~ ~ C.0. Bonpoc~ T e o p ~
np~6~m~eH~a ~ y ~ z ~ O~HOrO K O M ~ e E C H O P O HepeMeHHOrO. - B EH.: CoBpeMem~ge n p o 6 x e M H M a T e M a T ~ T.4, MOCKBa, BHHETH, I975, I43-250, 7. z a I c m a n L. Analytic capacity and Ration~l Approximation. - Lect.Notes Math., 50, Berlin, Springer, 1968. 8. G a m e I i n T.W.. Uniform algebras, N.J., Prentice-Hall, Inc. 1969. 9. D a v i e A.M. Analytic capacity and approximation problems. - Trans.Amer.Math.Soc., 1972, 171, 409-444. IO. M e x ~ H ~ E 0 B M.C. 0~eHEa ~HTerpaxa K o ~ no a~a~TH~ecEo~ EIDEBO~. -- MaTeM.Cd., 1966, 71, ~ 4, 503--514. II. B ~ T y m E ~ H A.F. 0 ~ e ~ a ~ T e ~ a ~ a E o m ~ . - MaTeM.c6., 1966, 71, ~ 4, 515--534. 12. ~ ~ p 0 E 0 B H.A. 06 O~HOM CB0~CTBe a H a ~ T E e c E o ~ eMEOCTE. -BeCTH~E ~IY,cep.MaTeM., ~ex., aCTpOH., 1971, 19, 75-82. 13. ~ ~ p 0 E O B H.A. HeEoTop~e 0BO~CTBa aHa~IETEeCEo~ eMEOCT~. -BeCTH~E~I~J, cep.MaTeM., MeX., aCTpOH., 1972, I, 77--86. 14@ B e s i c o v i t c h A. On sufficient conditions for a function to be analytic and on behaviour of analytic functions in the neighbourhood of non-isolated singular points. - Proc.London Math.Sec., 1931, 32, N 2, I-9. A. G. V I T U S H K I N
(A.r.BHTY~Gm)
M. S.MEL 'NIKOV
(M.C.MF/6m~0B)
CCCP, 117966, MOCKBa, y x . B a B ~ o B a , 42, MEAH CCCP CCCP, 117234, MOCEBa, Mexa~aEo-MaTeMaT~ecE~ (~m~ym_~TeT MOCEOBCEOrO yH~BepC~TeTa
498 8.18. old
ON SETS OF
ANALYTIC CAPACITY ZERO
Let K be a compact plane set and Aoo (K) the space of all functions analytic and bounded outside K endowed with the sup-norm. Define a linear functional ~ on ~ o o ( K ) by the formula
I~-$ with
~
>
ttM3/J6 {1~ I : ~ ~ K ]
The norm of
L
is called
t h
e
a n a 1 y t i c c a p a c i t y o f k . We denote it by ~(K). The function ~ is invariant under isometries of C . Therefore it would be desirable to have a method to compute it in terms of Euclidean distance. E.P.Dolzenko has found a simple solution of a similar question related to the so-called %-capacity, C1]. But for ~ the answer is far from being clear. I would like to draw attention to three conjectures. CONJECTURE 1. There exists a positive number a%7 compact set
where
~(K,~)
the line through
such that for
K
T denotes the l e n g t h o f the p r o ~ e c t i o n o f K
0
and ~ ~ T
CONJECTURE 2. There an,y compact set
C
onto
•
exists a positive number C
such that for
K
Y(K)~c I ~ (K,~)#~(~). T These CONJECTURES are in agreement with known facts about analytic capacity. For example, it follows immediately from CONJECTURE I that ~ ( K ) > 0 if ~ lies on a continuum of finite length and has positive Hausdorff length. In t~rn, CONJECTURE 2 implies that ~(K)=0 provided the ~avard length of K equals to zero. At last, let be a set of positive Hausdorff M-measure (a surveE of literature < co on the Hausdorff measures can be found in ~3] ). If ~ ~(~2~"
~t
O
then the Favard length of
K
is positive. This ensures the existence
499
K~cK
of a compact K4 , function
~,
~(K0>O
, such that
~ ( ~ ) = I. - ~
, is continuous on
the Hausdorff ~-mea~ure. Hence ~(K) >~ ~ (KO > 0 easily follows from CONJECTURE 2. CONJECTURE 3. Pot an E increasing function with
I ~(~)/~ ~----oo
and
0
there Qxists a set
K
and the ,~ being which also
~ :(0,+°°)--~(0,+°°) satisfyin~ ~ ( K ) > 0
.
To corroborate this CONJECTURE I shall construct a function
~nd a set E
such that
~(~=0,
~(E)>O
but ~ ( E ) - O •
Assign to any sequence 6 ~ { 8 . } , ~ev~O (~), a compact set E(8). • Namely, let ~a(8) [0,4J . If ~. is the union of disjoint segments , of length^j ~.(8) then ~.+~C~) is the union of ~n sets ~L\ ~n ^j , A n being the interval of length I
~(6)~-8~)
and let ~
concentric with the segment
be a constant ~e~uenoe, ( ~ ) ,
=C
~
. Put
. ~inally let
E=
= E(~°). Tt is known (see [~]) that ~(E) = 0 existence of a function ~ such that
~ (~(13)=0 and t-,-o
. This ~mplies the
~'(E(6))<Sv(t)
for any sequence 8 satisfying ~4 < ~ . Then ~(8) has the desired properties for properly chosen 8 , as will be shown later. To choose ~ pick numbers ~4 ~ 0 and ~ 4 ~ such that ~Co~i) < V~ ~(~(6&~)) 0 . ~ will be used as a generic notation for compact subsets of ~ . For any locally integrable (complex-v~lued) function ~ on ~ we denote by
its mean value over the disc ~ ( ~ , , ) = ~ : ~ will stand for the class of all functions integrable to the power ~ and satisfy
~ ~
on
I~-~I~0 , and ~ , ~ > 0 ,
503
where the infimum is taken over all sequences of sets M ~ , M ~ c with ~ M ~ 0.
(3)
N~K>~4
Es ware i~teressant, folgende Annahme zu rechtfertigen @der zu widerlegen~ HYPOTHESE 3. Die Bedingun~ (3) ist notwendig Wir wenden uns zum Schlu~ nichtlinearen elliptischen Gleichungen zweiter Ordnung zu. Wie in [ 7] gezeigt wurde, ist der Punkt 0 regular fEr die Gleichung ~IZ ( I ~ t&lP'~? ~ ~) = O, 1/C ~-K(~-~) gilt. Deshalb ist f~r N>~
N~K~4
Polglida divergiert die Reihe von Wiener f{Lr das betrachtete Gebiet, aber die Bedingung (3) ist nicht erfullt. Wit zeigen, da~ trotzdem eine beliebige in ~ harmonische Punktion, deren Randwerte der Holder-Bedingung im Punkt 0 genugen, ebenfalls der H~lder-Bedingung im l>~ukt 0 genugt~ Es sei IA Losung des Dirichletproblems /ki& ~ 0 in ~ , i&-~=0 auf 8 ~ , wobei ~ eine stetige Funktion ist, die tier Bedingung ~(~C) ~ 0 ( I ~ l ~) , ~ > 0 , gen~gt. Man kann annehmen, da~ 4>~(~)> 0 ist. Wit bezeichnen mit ~ einen beliebigen Punkt des Gebiets ~ und mit $ eine Zahl, f{~r die f~-4 ~ i~l >i ~ ist. Es sei ~----O au~erhalb von C ~ O S ( ~ _ 4 ) und ~q-----~ ~-4 Ferher sei i&~ eine harmonische ~ku%ktion in ~ , die auf ~ mit ~ ubereinstimmt. Wegen 0~ ~ C~_! auf ~ ist
510
c harmonisohe Funktien
ofze b r
O
~
~j ~ Wir f~hren eine in der Kugel ein, die auf 8 ~ 2 gleich ~ @
~_~ ist.
k,- ~ beruht, positiv mit dem Gewicht ~($C-~) zu sein, wobei ~ die Pundamentallosung ist, alle Dimensionen mit Ausnahme der folgenden drei zu opfern: ~ = ~ , ~ Y ~ + ~ , 2 ~ + 2 • Als fragw~trdigen Ausgleich gestattet uns dies, die folgenden beiden Hypothesen zu f o ~ i e r e n , die sich an die HYPOTHESEN I und 2 anschlie~en~ ~ ) ~merhung bei der Korrektur: I.W.Skrypnik teilte soeben auf der Tagung "Nichtlineare Probleme der Mathematischen Physik" ( 13 April, L0~I, Leningrad) mit, da~ er die Notwendigkeit der Bedingung (4) fur ~>~ bewiesen hat. Damit ist die HYPOTHESE 3 teilweise gestutzt.
511
HYPOTHESE I'. ~ r
E
~ >2, ~2,$+5
ist die G l e i c h ~
2K(n,-2m,)
K)4
wobei
cllp~
die m-harmonisohe Kapazitat ist, hinreichend fur die
Steti~keit der LBstulg des Diri0hletproblems mit NullraudbedinKunKen der Glelohun~
Q-A) "~4~= ~ ~ e C i ( n ) i m
HYPOTHESE 2'. FS_r ~ > 2 , tion
G~
des Operators
C
F~ilt frar die Greemache F~nk-
(-/~)"~die Abschatzun ~
i C-,,,(=,o)l wobei di9" Konstante
~2~+S
Ptmkt 0.
~
cl :- l
yon ~ und ~ a b e r nicht vom Geblet 'abh~n~t.
In der letzten Zeit wurden neue Erkenntnisse ~ber das Verhalten der LSsung der ersten Randwertaufgabe f~r stark elliptische Gleichungen der 0rdnung 2 m in tier Nahe konischer Pumkte erhalten. Im allgemeinen (s. [14] ) haben die Hauptglieder tier Asymptotik solcher LSsungen in der Umgebung des Eckpunktes des Konus die Gestalt N
cl l ZK=O ( f lt, l l)
(x/i O
Dabei ist ~ Eigenwert des Dirichletproblems fSr einen gewissen polynomial vom Spektralparameter abh~ngigen elliptischen Operator in dem Gebiet, das durch den Konus auf der Einheitssph~re ausgesohnitten wird. Die Flmktion (5) hat genau dann ein endliches Dirichlet-Integral, wenn ~e ~ > ~ - ~/2 . sie ist des weiteren stetig und gen'ugt sogar einer H~ider-Bedingung, falls R e ~ > 0 ist. Wenn im Band 0 > B e ~ > ~ - ~/2 Eigenwerte des genannten Operators existieren, dann besitzt die Ausgangsrandwertaufgabe verallgemeinerte L~sungen, die in einer beliebigen Umgebung des Eckpunktes des Konus unbeschr~akt sind, und yon einer Regularitat nach Wiener kann man selbst . bei einem konisohen Pumkt mioht reden. Es zeigt sioh ([15], [ 1 6 ] ) , ~ wir auf solche unerwarteten Erscheinungen schon bei stark elliptischen Gleichungen zweiter Ordnung mlt konstanten Koeffizienten =--a ~ -
512 stolen, falls nicht alle Koeffizienten reell sind; In [16] (eine ausfS/L~liche Darstellun~ erscheint evtl- in ,,MaTe~aTN~eoE~ O6Op~NE") wird das homogene Dirichletproblem au~erhalb eines d'6nnen Konus l'~'~=(t~ Z-'~E~ ~: ~,,.)' O~ U.,~I~ q~untersuoht, wobei ~ ein kleiner _ F,, (. )~' iiJ. rTM ~-i ~-I "J ~ ~rv-, farame~er, CLT~---- ~.~ e IK : ~ ~ e u I i ~nd w ein Gebiet im IK ist.
~s wir~ ~ie ~,~su~ ~(~)---I~IX(~)~(£,~/I~I)
aes sta~k
elliptischen Systems Qz~ ( ~ ) ~ (6~ ~C) = o betrachtet, wobei ~ ( ~ ) eine ~atrix mit homogenen Polynomen der Ordnung ~ als Elementen und ~ (6) ----0 (~) f'/r ~-~+ 0 ist. Hauptergebnis ist eine asymptotische ~ormel fur den Eigenwert k (£) , welche f't~r den eimfachsten ~all der Gleichung (6) die Gestalt •
X (~)= £n-s{ _ _ .,- z I ~
ca,pp.(D~t,O)
.n
.(~-a)/~
]
÷O(1)
(,~) (d,et; I~iKII~,K-,)
]
( (~ei: II ~, ~eX > (2-~)/2
so ~ m e n , ~a9 ~ e U~leio~u~
erfullt ist. Im Fall
h(2)= t21 ~o~21)-'(~ o0)) +
f~
~=3
gilt
~ - - +0.
Folglich erf~It j ede verallgemeinerte LBsung die H~Ider-Bedimgung, falls der 5ffnungswinkel des Konus K~ genugend klein ist. Es ist nicht ausgeschlossen, da~ die Forderum~ nach einem kleinen ~ffnungswinkel unwesentlich ist. Dies ist gleichbedeutend mit folgendem Satz HYPOTKESE 5. F ~
~=3
b_i~en elliptisehen Operator gular math Wiener.,
ist ein konischer Punkt fur einen belie-
P~(D,) nit,komplexen
Koeffiziente ~ re-
513
F~tr den biharmonischen Operator im ~ und fur die Systeme yon Lam~ und Stokes im --]R~wurden derartige Ergebnisse in "-[17J, "-[18] erhalten,
REFERENCES
I. W i e n e r N. The Dirichlet p r o b l e m - J,Math. and Phys~ 1924, 3, 127-146~ 2. W i e n e r N~ Certain notions in potential theory - J~Math and Phys. 1924, 3, 2 4 - 5 1 3. ~I a H ~ ~ c E.M. YpaBHe~S BTO10OrO I I O 1 0 S ~ ~ J U ~ n T ~ e c ~ O ~ O ~ na-
pa6o~m~eci¢oro T~a, M., Hayz,a, 197I. 4. M a 3 ~ ~ B.r. 0 n O B e ~ e H ~ B 6 ~ S Z r p S H ~ H pemeH~R saXaqR ~ p s x Jle ~ 6 R P s p M o ~ , e o E o r o oIIepSTOpS. - ~oEJI.AH CCCP, 1977, I8, ~ 4,
15-I9. 5. M a 2 ~ s B.r. 0 !DSI~JI~pHOOTm H8 I~H~I~8 IDSmSHR~ @JI~IITBRS01~X ylmBHe~mi~ ~ KoH~olxmoro OTO6paxe~ms. - ~or~I.AH CCCP, I963, 152, 6, I29V-I300.
6. M a s ~ s B . r . 0 ~oBe~eH~ ~ 6 ~ s rlmHs~u peme~s sa~a~s ~ p s x ae ; ~ s ~ z ~ n ~ a ~ e o ~ o ~ o ypa~se~as B~OpO~O nops;m~a B ~ B e p r e H T S O ~ ~OpMe. - ~ a ~ e a . s a a e ~ F m , 1967, ~ 2, 209-220. 7. M a s z s B . r . 0 ~ e n p e p u z ~ o c ~ z B r l m S a ~ o ~ ~ o ~ e pemeHz~ r m a s a J I ~ H e ~ X @ ~ a i i T ~ e c F ~ x ypaBHeHZ~. - BeOTH.~[rY, 1970, 25, 42-55 (nonlmBF~: BeO~H.~rY 1972, I, 158). B. G a r i e p y R., Z i e m e r W,P. A regularity condition at the boundary for solutions of quasilinear elliptic equations. Arch,Rat.Mech.Anal., 1977, 67, N I, 25-39. 9. E p o ~ ~ ~.H., M a ~ ~ s B.r. 06 O T O y T O T ~ ~enpepB~ooT~ Henl~elmBHOCTZ uo re~epy pemeHz~ K B S S ~ H e ~ x ~m~Tz~ec~mx ypB~H e H ~ B6JL~3~ HepSryJlSpHOH TOV~ER. - T I ~ MOCE.MaTBM.O--BS, 1972,26,
73-93. I0. H e d b e r g L. Non-linear potentials and approximation in the mean by analytic functions. - Math,Z,, 1972, 129, 299-319, II.A d a m s D.R., M e y e r s N. Thinness and Wiener criteria for non-linear potentials. - indiana Univ,Math.J., 1972, 22, 169197. I2. M a s ~ s B.T., ~ o H ~ e B T. 0 p e r w ~ q p H o o ~ no B ~ e p y rlmKZ~HO~ TOPAZ ~ S u o ~ r a l X ~ O H ~ e o x o r o onepaTopa. - ~o~.Bo;az,.AH,
514
1983, 36, 2 2. I3.
m a z ' y a V~G. Behaviour of solutions to the Dirichlet problem for the biharmonic operator at the boundary point, Equadiff IV, Lect.Notes Math , 1979, 703, p 250-262
14. E o H ~ p a T ~ e B B.A. ElmeB~e s a ~ s ~ ~ smmn~ecF~x YlmB-HOHI~ S 06JIaCTSX C EOH~0CF~M~ ~Jl~ yraOBHMH TO~I~SMH. T I ~ MotE. aa~eM.o-Ba, 1967, I 6 , 209-292. 15. M a s ~ s B.T., H s 3 8 p 0 B C.A., H a a M e H e B C ~ m ~ B.A. 0TCyTCTB~e T e o p e ~ TmnS de ~ o p ~ s ~as C~a~HO s a a ~ T ~ e c z ~ x y p S B H e H ~ O KOMII~IeEoHHM~ l ~ o s ~ u a e H T a ~ . - 3aTr.Hayq.OeM~H.ZOMM, I982, IIS, 156-I68. 16. M a s ~ s B.F., H a s a p o B C.A., I I a a ~ e H e B C ] ~ ~ B.A. 06 o/o~opo/~ux p e m e H ~ x sa~a~z Jl~p~x~e Bo B~e=HOOT~ TOH~O~O ~¢o]¢~0a. -/~oI~a.A~ CCCP, 1982, 266, • 2, 281-284. 17. M s s ~ B.F., H a a ue H e Be ~ ~ ~.A. 0 n l ~ a ~ n e ~ m E csw~Ma ~ 6aral~OHa~eCl~OrO ypaBHeHas B 06Y~aGTB C EOHJ~NOCI~M~ TO-~ . - 14SB.BY3oB, 1981, ~ 2, 52-59. 18. M a s ~ s B.L, H a a m e H e B C ~ a ~ B.A. 0 CBO2Cr~ax l~m e H ~ TpeXMepH~X sa~sq T e O p ~ ynlmjrocT~ ~ r ~ p O J ! ~ H ~ K ~ B O6aSCTSX C ~SOaSpOBaHHHM~ OC06eHHOOTm~S. -- B 06. : ~ H S S ~ E 8 O]UtOmHO~ c p e ~ , HoBoc~6spc~, 1981, BUn.50, 99-121.
V. G.MAZ'YA
(B.r.MAS~)
CCCP, I98904, JIelmHrlm~, He TpO~mOl~ea, ~ e H g H r l ~ o ~ m ~ ~ooy~81oc TBSHHR~ ~H~BepC~TeT,
MaTeMaTaXo-aexa~eo~caJl
~a~vev
515
THE EXCEPTIONAL SETS ASSOCIATED WITH THE BESOV SPACES
8.21.
For ~ real and 0 < p , ~ < o o , we will use Stein's notation P~ for the familiar Besov spaces of distributions on ; see IF] and [S~ for details. The purpose of this note is to generally survey and point out open questions concerning the general problem of determining all the inclusion relations between the classes ~ ~, $ ~>0~ of exceptional sets naturally associated with the spaces A~ for various choices of the parameters &,p )~ ; c.f.[A~S], These exceptional sets can be described as sets of Besov capacity ze-
AA
on
~)~
l~l~p¢
some fixed smooth dense class in the spaces
the n o = (q~si-no=) of ~
is extended to all subsets of
E~B~p~
iff A ~ p ~ < E )
bitrar~ compact ~rite A ~ s
set
K
such
, hold. Now when 4 ~ p,~ < OO , there is quite a bit that can be said about this problem. First of all, one can restrict attention to ~ ~ 4 ~/& . Functions in A ~ for p > ~ / A are all equivalent to continuous functions and hence AA, p, $ (E) > 0 iff E ~ . Continuity also OCCURS, f o r example, when ~ - I'l,/& and ~= 4 . Secondly, i n t h e range 4 < p .< ~/~., ~< ~ co , there appear to be presently four methods for obtaining inclusion relations. They are: I. If
~
C ~j~
( continuous embedding), then clearly
. Such e~Bbeddings but not ve~ypoften. However, since & . ~ >0 , with ~ denoting the usual class of Bessel potentials of ~P functions on ~ (see [ $] ), and since the inclusion relations for the exceptional sets associated with the Bessel potentials are all known [AM], it is easy to see that A>,%, s co)
~(~) >0 ; 2) the integrals
Ak=kl g
are finite. Putting
, consider entire functions of
A o=
- 2
Z
:
(.-kI 0
00
~k
and
wC ®) e
~et, finny, {~k}4
(o < l~kl .
(it is assumed that
~C~) ~
~lw ~ ( ~ ) %~4-0
at
~ 67~
E
).
The structure of ~(~) -sets is well understood for many important classes ~ (see[l] ,[2]; [3] contains a short survey). The same cannot be said about the family ~(~A) , ~ being the space of all functions analytic in ~ with finite Dirichlet integral The description of
0 is a ~ K , L =) -set (though we know it is when C= 0 or when E satisfies (c)). All this is closely connected with our PROBLE~ (or better to say with its slight modification). DEFINITION. I) A Lebesgue measurable function q on the line is said to be ~-s t a t i o n a r y on the set E~ E c~, if there are functions ~4 ~ . . - , ~ ~ W ; (~) (i.e ~ ( ~ ) , absolutely continuous and with the L¢(~) -derivative) such that
~ I E =~41E, ~, 2) A set t y
S~
~IE, .... ~-~1
E, (or E
is said
E(S~)
~ H~(IR),
~IE have
IE
O.
t
r o p e r -
) if ~ ~-stationary on
E ~
~-=-0.
It is not hard to see that
E ~ C~s E, E e t C ) ,
l~es E > O ~
Ee(S~),
~=~,2,...
co
and that if
E ~ ~-i S t
then
E
is a (k, ~ k )
-set for eve-
ry semirational k [3]. moreover if there exists a ~ H 2 ( ~ ) ~ ~ @ 0, stationary on the set E , then E is not a (K,~k) -set for a semirational k (which may be even chosen so that k agrees with a linear fmnction
on (-co,O)
).
Another circle of problems where ~ -stationary analytic functions emerge in a compulsory way is connected with
539
3. Jordan operators. We are 6oing to discuss Jordan operators (J.o.) ~of
the form
~+~ where ~ is unitary, ~ =O,~G-Q~ (in this case we s a y T is of order ~ ). It is well known that the spectrum of any s u c h ~ lies on ~ so that ~ is invertible Denote by ~ ( ~ ) the weakly closed operator algebra spanned by ~ and the identity I We are interested in conditions ensuring the inclusion
T-' ~ P,, (T).
(**)
EXAMPLE. Let E be a Lebesgue measurable subset of T and be the direct sum of (~+I) copies of L ~ ( T \ E ) The operator
I = Y( E,~)defined by the
I
( ~ 4 ) × ( ~ * {)
H
-matrix
~. I
.
( ~ being the operator of multiplication by the complex variable is a J.o. of order ~ . It is proved in [3] that
J-'e g(l)
< :- E ~(S;).
Here ( ~ ) denotes the class of subsets of T defined exactly as C8~) in section 2 but with ~ replaced by the class of all functions absolutely continuous o n ~ . The special operator I - I c E ~ ) is of importance for the investigation of J.o. in general, Namely [3],if ~ is our J.o. (~) of order ~ and ~ stands for the spectral measure of ~ then
•herefore i~ E e ~=,ACS'~)
( ~ particular ~ ~eS E >0 ~d
E ~ CC) ) then (**) holds whenever ~ U ( E ) 0 Recall that for a unitary operator T (i.e. w h e n T = ~ Q =
0
540 in (*)) the inclusion (~*) is equivalent to the van lshing of ~ on a set of positive length A deep approximation theorem by Sarason [61 yields spectral criteria of (**) for a normal T . Our questions concer~ing sets with the property ~ and analogous questions on classes ( ~ ) , ( ~ ) are related to the following difficult PROBLEM: which spectral condltions ensure (**~ for T ~ ~ ~ ~ is normal and .....~ is a nilpotent com~ntin~ with ~ ?
where
REPERENCES 1. X p y m e B C.B. Hpo62eMa O~HoBpeMeHHO~ annpoEc~Ma~EE z cT~paH~e Oco6eHHOCTe~ ~Hwerpa~oB T~na Kom~. - T p y ~ MaTeM.EH--Ta AH CCCP, I978, IS0, I24-195. 2. E p ~ K E e B., X a B ~ ~ B.H. H p ~ R ~ Heonpe~e~H~ocT~ onepaTopoB, nepecTaHOBO~HRX CO C~BErOM I. -- 8anEcE~ Hay~H.Ce~H. ~0M~, I979, 92, I34-170; H. - ibid., I98I, II3, 97-I34. S. M a E a p o B H.F. 0 CTa~zoHapH~x ~yR~IE~X. - BeCTH~K ZIY (to be published). 4. H a v i n V.P., J o r i c k e B. O n a class of uniqueness theorems for convolutions. Lect,Notes in Math , 1981, 864, 143170. 5. X a B E H B.H. H p z H ~ H Heonpe~e2~HHoc~z ~x~ O~HoMepHax H o T e ~ a 2OB M.P~cca. - ~ o E ~ . A H CCCP, 1982, 264, ~ 8, 559-568. 6. S a r a s o n D. Weak-star density of polynomials. - J.reine umd a~ew.Math., 1972, 252, ~-15.
V. P. H A V I N
(B.H.XABHII)
CCCP, 198904, ZeH~Hrlm~, IleTpo~mope~, JleR~HrpajIcE~ r o c y ~ a p C T B e H ~ yH~Bep-CE TeT M a T e M a T ~ K o - M e x a H ~ e c E ~ ~BEyJIBTe T
B. JSRICKE
Akademie der Wissenschaften der DDR Zentralinstitut f~ur Mathematik und Mec~ DDR, 108, Berlin Mohrenstra~e 39
N. G. M A K A R O V
CCCP, 198904, JleHEHrpa~, HeTpo~Bopen, JIeHEHrpaJIc~zi~ rocy~apc TBeHHNI~ yH~BepOZTeT MaTeMaTI~Eo-~exaH~ eCKEI~ ~Ey2BTe T
(H.LMAKAPOB)
541
PROBLEM IN THE T ~ O R Y
9.5.
OF FUNCTIONS
In 1966 I published the following theorem: There exists a constant @ • 0 nomials Q
such that a%7 collection of poly-
of the form
k
)
with
is a norma ! famil,y in the complex plane. See Acta Math., 116 (1966), pp.224-277; the theorem is on page 273. This result can easily be made to apply to collections of polynomials of more general form provided that the sum from I to oo in its statement is replaced by one over all the non-zero integers. One peculiarity is that the constant @ > 0 r e a I I y m u s t b e t a k e n quite small for the asserted normality to hold. If ~ is I a r g e e n o u g h, the theorem is f a I s e. The results's proof is close to 40 pages long, and I t h ~ k very few people have been through i%. Canoone find a shorte r and clearer proof?
This is my question.
Let me explain what I am thinking of. Take amy fixed ~ , 0 < 2 < < ~
and let ~
be the sllt domain
CbO
C\ U If
~
i s any p o l y n o m i a l , w ~ t e
By d i r e c t harmonic e s t ~ t i o n much t r o u b l e t t ~ t
in
~3
one can f i n d w i t h o u t too
542
_~
' 7 + ~ ~'
where K~ (~) depends only on ~ and ] . (This is proved in the first part of the paper cited above. ) A natural idea is to try to obtain the theorem by making ~ ~ 0 in the above formula. This, however, cannot work because Kf (~) tends to oo as ~ - ~ 0 whenever $ is not an integer. The latter must happen since the set of integers has logarithmic capacity zero. For polynomials, the estimate provided by the formula is too crude, The formula is valid if, in it, we replace ~ I ~ ( ~ ) [ by an~ function subharmonic in ~ having sufficiently slow growth at ~ and some mild regularity near the slits K ~-~, ~ ÷ ~ ] • P o 1 y n o m i a 1 s, however, are s i n g 1 e - v a 1 u • d in ~ . This single-valuedness imposes c o n s t r a i n t s on the subharmonio function ~ [ ~(~)I which somehow work %o dimlniah ~8(~) to something bounded° (for each fixed • ) as ~--~ 0, provided that the sum figuring in the formula is sufficiently small. The PROBLE~ here is to See quantitativel,7 how the constraimts cause this d~m~n~shin~ t o take place. The phenomenon just described can be easily observed in one simple situation. Suppose that U(~) is subharmonic in ~ ,%hst ~(~)~< ~[~l there, and that E~(~)] + is (say) continuous up to the slits ~ - ~ , ~ + ~ . If U(x)~< ~ on each of the intervals El,!,- ~ , 'H, * jo]
, then
This estimate is best possible, and the quantity on the right blows up as ~ ~ 0 . However, if U ( ~ ) = ~ i ~(~)l where ~(~) is a s i n g i e v a I u e d entire function o f exponential type A < ~ , we have the better estimate
M with
a constant
CA
cA • A i
n d e p e n d e n t
of
~
. The improved
543
result follows from the theorem of Duffin and Schaeffer. It is no longer true when with L ~ oo •
A ~ ~
--
consider the functions
~(~)~~
The whole idea here is to see how harmonic estimation for functions analytic in multiply connected domains can be improved by taking Into account those functions' simgle-valueduess. PAUL KOOSIS
Institu% Mittag-Leffler, Sweden McGill University, Montreal, Canada UCLA, Los Angeles, USA
544 9.6. old
PEAK SETS FOR LIPSCHITZ CLASSES The Lipschitz class
analytic In D
(~ s~bo~
, consists of all functions
, continuous on ~ D
I{(~,) { ( ~ ) I . < ~ I ~ A closed set
A A , 0< % ~ ~
ql ~
q
and such that
(~)
~T
E , E c T , is called p e a k s e % f o r A~ E ~ % ). i~ the~ e~s~s a ~unction ~, I ~ A~
(the so called
p e a k
f u n c % i o n) such that
I{1/J , and if $ 2 ~ is uniformly continuous, then IT IS CONJECTURED that no nonzero function exists in the domain of ~ ~ishes
which
in a set of positive measure. K~FEI~NCE
I. d e B r a n g e s L° Espaces Hilbertiens de Fonotions Entieres., Paris, Masson, 1972o L.DE •RANGES
Purdue University Department of Nath. Lafayette, Indiana 47907
USA
553 FROM THE AUTHOR'S SUPPLEMENT,
1983
The problem originates in a theorem on quasi-analyticity due to Levinson [2~ . This theorem states that ~ cannot vanish in an interval without vanishing identically if it is in the domain of ~ where K is sufficiently large and smooth. L a r g e means that the integral in (*) is infinite, s m a I 1 that it is finite. The smoothI is non-decreaness condition assumed by Levinson was that ~ K sing, but it is more natural E3~ to assume that ~ I K I is uniformly continuous (or satisfies the Lipschitz condition). A stronger conclusion was obtained by Beurling E4~ under the A Levinson hypothesis. A function ~ in the domain of ~ cannot vanish in a set of positive measure unless it vanishes identically. The Beurling argument pursues a construction of Levinson and Carleman which is distinct from the methods based on the operational calculus concerned with the concept of a local operator~ The Beurling theorem can be read as the assertion that certain operators are local with a trivial domain. It would be interesting to obtain the Beurling theorem as a corollary of properties of local operators with nont~ivial domain~ . . . ~ . The author thAn~S Professor Sergei Khrushchev for informing him that a counter-example to our locality conjecture has been obtained by Kargaev. REPERENCES 2. L • v i n s o n
N. Gap and Density theorems
Providence, 1940. 3. d e B r a n g e s
- Amer.Math.Soc.,
L. Local operators on Pourier transforms
Duke Math.J., 1958, 25, 143-153. 4. B e u r 1 i n g A. Quasianalyticity and generalized distributions, unpublished manuscript, 1961.
CO~AENTARY
P.P.Kargaev has DISPROVED the PIRST and the LAST CONJECTURES. As to the first, he has constructed an entire function k not only of minimal exponential type, but of z e r o o r d e r such that ~ is not local. Moreover, the following is true.
-
554
THEOREM (Kargaev). Let
be a positive function decreasi~
to zero on [0 ,+@@) . Then there exist a function ("a divisor"),
a set
e c
, and a function
~ t h the
I~(~)
followin~ properties:
i~e~e >O, ~
* ~
is bounded away from zero on
eooo~ n ~(,)=th(~+~)] but
-4
e
, where
.o see k~ ~o o~ ~oro order,
K~ "s ot l o c a l . The LAST CONJECTURE i s disproved by the f a c t (also found by
Kargaev ) t h a t there e x i s t r e a l f i n i t e
v e ~ large lacunae in {(~
~
~-~
))~=4
~p~
Borel meam~es ~
Then h k
with
(i.e. there is a sequence
of intervals free of
I~I
, ~
< ~
,~=~,~,"-
tending to infinity as rapidly as we please) and with
vanishing on a set of positive length. Take
where
onn ~
~
b ~- ~ q
,
is a suitable mollifier and k : ~ C ~ ( ~ , ~ p ~ ) ) is a Lipschitz f~ction and
z~ ~=+~,
if • grows rapidly enough. Then the inverse Fourier transform vanishes on a set of positive length and belongs to the domain of ~ . Kargaev's results A l l soon be published. The THIRD CONJECTURE is true and follows from the Beurling-~alliavin multiplier theorem (this fact was overlooked both by the author and by the editors). Here is THE PROOF: there exists an entire function ~ of exponenti~ t~e ~ ~ ~0 satis~yin~ t lk I ~ on ~ . Then ~ ' ' ~ ~[X) is in the domain o f K •
555
NON-SPANNING SEQUENCES 0F EXPONENTIALS ON RECTIFIABLE PLANE ARCS
9.10.
Let A : (~) be an increasing sequence of positive numbers with a finite upper density and let ~ be a rectifiable arc in ~ . Let C (~) denote the Banach space of continuous functions on with the usual sup-morm. If the relation of order on ~ s d e n o t e d b y < and if Eo and Z 4 are two points on ~ such that ~ 0 < Z! we set
The following theorem due to P.Malliavin and J.A,~Siddiqi [ 7]gives be a necessary condition in order that the sequence (eXZ)~A.~
~on-spanning in C(y; THEOREM. If the class
c°° (Mr.,,
i,,s non-empty .for some
Zo,ZIC
M~= ~p_ ~ ~hen
(eX~) ~ eA
?
, where
~cA
is ~on-spa~in~ in C (y).
It had been proved earlier by P.~alliavin and J.A.Sidaiqi [6] that if ~ is a piecewise analytic arc then the hypothesis of the above theorem is equivalent to the :5/Itz condition ~ ~-~ < c o In connection Wlth the above theorem the following problem remains open. PROBL~
I~ Given any non-quasi-analytic class o.f functions.....on
in the sense .of Den,~o~-Carleman , to f..ind a non-zero
function. ~e.l.on-
~im~ to that class and hay in ~ zeros of infinite order at two peint.s
ix }
With certain restrictions on the growth of the sequence partial solutions of the above problem were obtained by T.Erkamma [3] and subsequently by R.Couture [2],J.Korevaar and M.Dixon [4] &nd
556
M. Ltundin ~ 5 ]. Under the hypothesis of the above theorem, A.Baillette and J,A. Siddiqi [1] proved that ( e A~ ) A e A is not only non-spanning but also topologically linearly independent by effectively constructim~ the associated biorthogonal sequence.
I n this connection the following
problem similar to one solved by L.Schwartz
[8] in the case of linear
segments remains open. PROBLEM 2, To characterize the closed linear span of
(eA~)AeA
i_.nn C (V) when it is non-spanning. @
REFERENCES I. B a i 1 1 e t t e
A.,
S i d d i q i
ctions par des sommes d'exponentielles
J,A. Approximation de fonsur u n arc rectifiable
-
J.d'Analyse Math., 1981, 40, 263-26B. 2~ C o u t u r e R. U n th~or~me de Denjoy-Carleman sur une courbe du plan complexe. 3. E r k a m m a
- Proc,Amer,Math,Soc.,
d'approximation de Muntz.
- C,R,Acad.Sc. Paris,
4, K o r e v a a r J,, D i x o n
et le th~or~me
1976, 283, 595-597.
M. Non-spanning sets of exponenti-
als on curves. Acta Math.Acad.Sci.Hungar, 5~ L u n d i n
1982, 85, 401-406.
T. Classes non-quasi-analytiques
1979, 33, 89-100.
M. A new proof of a ~t[utz-type Theorem of Korevaar
and Dixon. Preprint NO 1979-7, Chalmers University of Technology and The University of Goteborg, 6. M a 1 1 i a v i n P°, S i d d i q i J.A. Approximation polynSmiale sur un arc analytique dans le plan complexe. C.R.Acad. Sc. Paris, 1971, 273, 105-108. 7. M a 1 1 i a v i n P., S i d d i q i
J.A. Classes de fonctions
monog&nes et approximation par des sommes d'exponentielles sur u n arc rectifiable de ~ , ibid., 1976, 282, 1091-1094. 8~ S c h w a r t z L. Etudes des sommes d'exponentielles. Paris,
Hermann,
1958.
J.A°SIDDIQI
I
Department de Mathematlques Universit~ Laval Quebec, Canada, GIK 7P4
557
T It is a well-known fact of Nevanlinna theory that the inequality in the title holds for boundary values of non-zero holomorphlc functions which belong to the Nevanlinna class in the unit disc. But what can be said about s!,m~ble functions S with non-zero Riesz projection~_~=~= 0 ? Here~_~ %e~ E ~ ( - ~ ) ~ , l ~ J ~ 1 • ~>Oc Given a positive sequence ~ M ~ } ~>~ 0 dsfine
It is assumed that
a)
M~q, j=~,...,,},
~D(o,c) - uz(.) ~ Co,c).
j~ , ~ ~HC~(_Q,O))
is continuous on the set
~cm)= ,T. ~.J~Cm), is well-defined in ~ proved in [I]. If there exist
C
Define
~ CO ,c),
~D( ~ , c ) =e~Uz(~' ~)~ CO,c) , ~ppose t ~
.
n :{~n:
~
the restriction ( n , O)
. Then
m~}, ~ ~=~~'...'d~
. The following uniqueness theorem has been C = ( C 4,C 2 ~ , , . , C ~ ) ~
~+
and functions
a~t(=, an)> nclt=~c~.c~+ ...+c,~ }.
Note that the theorem is important for studying homogeneous convolution equations in domains of real ( R ~) or complex (C") spaces (see [I] , [2] , [3] ).0ne might think that,=--0 on ~ , as it occurs in the one-dimensional case. However there exists an example (see [I] )~ vhere all conditions of the uniqueness theorem are satisfied, but
562
~0 in ~ (for sufficiently large IICII ).Hence the appearance of the set ~C is therefore inevitable although ~ C does not seem to be the largest set where ~ = 0 • PROBLEM. Pind the maximal open subset of the domain ~
where
REFERENCES
I.
2.
3.
H a ~ a ~ E o B B.B. 06 o ~ o ~ TeopeMe e~HCTBeHHOCT~ B Teop~ ~m~ ~mi~x EO~,~eEcHHX uepeMeHHHX X o~opo~mHe ypsmHeH~ TZna c~epT~Z ~ Tpy6~aT~X 06AaCT~X ~ -- HSB.AH CCCP, cep.~aTeM. 1976, 40, ~ !, 115-132. HauaaEoB B.B. 0 ~ o p o ~ e CHCTeMH ypasRem~ T ~ a cBepT-EH Ka B~ny~HX O6maCT~X ~. -~o~x.AH CCCP, 1974, 219, ~ 4, 804-80V. Ha~ax~oB B.B. 0 pemem~x ypa~Rem~ 6ec~oHe~oro ~op~•a B ~e~CTB~TeX~O~ C4AaCTH. -- MaTeM.c6., 1977, 102, ~ 4, 499--510. V. V. NAPALKOV
(B.B.HAIIA~OB)
CCCP, 45O057, Y#a yx. Ty~aeBa, 50 F m m . l p c ~ @ ~ a x AH CCCP C e ~ o p MaTeMaTNEM
CHAPTER
10
INTERPOLATION, BASES, MULTIPLIERS
We discuss in this introduction only one of various aspects of interpolation, namely the
f r • •
(or Carleson) interpolation by
analytic functions. Let X
be a class of functions analytic in the open u ~ t
. We say that the interpolation by elements of X is free if the set XIE
(of all restrictions ~I E
disc
on a set E c
,Se X ) can be desc-
ribed in terms not involving the complex structure inherited from ~ . So, for example, if ~ (see formula (C) ments of H ~
satisfies the well-known
in Problem 10.3 below), the interpolation by ele-
on E
rich, bounded on ~
is free in the following sense: , belongs to H ~ I ~
lation for many other classes that the space ~ I ~ =>~I
Carleson condition
X
a n y
lunc-
. The freedom of interpo-
means (as in the above example)
is ideal ( i . e . ~ X I ~ , ] ~ l ~ I ~ l
on ~ >
~ )" Sometimes the freedom means something else, as is the ca-
se with classes
~
of analytic functions enjoying certain smooth-
ness at the boundary (see Problem 10.4), or wlth the Her~ite interpolation with unbounded multiplicities of knots (this theme is treated in the book H.E.H~KO~BCE~, ~eE~N~ 06 oHepaTope C~BEIB, MOCKBa, HayEa , 1980, English translation, Springer-Verlag, 1984; see also the article B~HoPpa~oB C.A., P y E ~ H C.E., 8an~cEE Hay~H~X CeMzHapoB ~ 0 ~ , I982, I07, 36--45).
564
Problems I0.1-I0.5 below deal wit~ free interpolation which is also the theme (main or peripheral) of Problems 4.10, 6.9, 6.19, 9°2, 11.6. But the imfoxlmation, contained in the volume, does not exhaust the subject, and we recommend the survey BNHOPpa~OB C.A., XS3~H B.~., 8an~cF~ Hay~H.ce~HapoB ~0MM, 1974, 47, 15-54;1976, 56, 12-58, the book Garnett J., "Bounded analytic functions" and the recent doctoral thesis of S.A.Vinogradov "Free interpolation in spaces of analytic functions", Leningrad, 1982. There exists a simple but important connection of interpolation (or, in other words, of the moment problem) with the study and classification of biorthogonal expansions (bases). This fact was (at last) widely realized during the past I~-20 years, though it was explicitly used already by S.Banaoh and T. Carleman~ Namely, every pair
of biorthogo~l
, ~'/" { ~ A' } l ~
families ~-----{~I}AG ~
tors in the space V , ~
( &A are vec-
belong to the dual space, < ~ ' ~ > = ~ A ~
)
generates the following interpolation problem: %o describe the coefficient space ~ V (~S ~
~,~>}Ae~
)
of formal Fourier
expansions ~
~ ~, ~> ~ . There are also continual analoA gues of this connection which are of importance for the spectral theory. "~reedom" of this kind of interpolation (or, to be more precise, the ideal character of the space ~ V
) means that ~
is an
unconditional basis in its closed linear hull. This observation plays now a significant role in the interaction of interpolation methods with the spectral theory, the latter being the principal supplier of concrete biorthogonal families. These families usually consist of eigen- or root-vectors of an operator ~ is oftem
differentiation or the backward shift, the two being iso-
morphlc) : T ~ l = ~-~ unt of
(in Function Theory
( ~
. Thus the properties of the equation
is the given function defined on 6" ) depend on the amo-
m u 1%
sending ~A
~I A , ~ E g
i p 1 i e r s
to ~(~) ~A
, where ~
of ~
, i~e. of operators ~ - ~
denotes a function C--~ ~ or the
565 multiplier itself). These multipliers ~ of ~ ~ - ~ C T ) ) (given ~
may turn out to be functions
and then we come to another interpolation problem
, find ~
). The solution of this "multiplier" interpola-
tion problem often leads to the solution of the initial problem ~
~ . Interpolation and multipliers are related approximately in this
way in Problem 10.3,whereas Problem 10.8 deals with Fourier multipliers in their own right. These occur, as is well-known, in numerous problems of Analysis, but in the present context the amount of multipliers determines the convergence (summability) properties of standard Pourier expansions in the given function space. (By the way, the word "interpolation" in the title of Problem 10.8 has almost nothing to do with the same term in the Chapter title, and means the interpolation
o f
o p e r a t o r s . We say "almost" because
the latter is often and successfully used in free interpolation). We cannot enter here into more details or enlist the literature and refer the reader to the mentioned book by Nikol'skii and to the article H1~a~6$v S.V. ,Nikol'skii N.K., Parlor B.S. in Lecture Notes in Math. 864, 1981. Problem 10.6 concerns biorthogonal expansions of analytic functions. The theme of bases is discussed also in 10.2 and in 1.7, 1 10, 1.12. Problem
10.7
represents
an
interesting
and
vast
aspect
of interpolation, namely, its "real" aspect. We mean here extensio~ theorems b. la Whitney tending t o the constructive description of traces of function classes determined by global conditions. Free interpolation by analytic functions in ~ functions in ~
(and by harmonic
) is a fascinating area (see, e.g., Preface t o Gar-
nett's book). It is almost unexplored, not counting classical results on extensions from complex submanifolds and their refinements. Free interpolation in $ ~ is discussed in Problem 10.5.
566
10.1. old
NECESSARY CONDITIONS POR INTERPOLATION BY ENTIRE ~UNCTIONS
Let ~ be a subharmonic function on C such that~(¢+Izl)= ~(~(~)) and let A# denote the algebra of entire functions such that I ~(~)I~ < A for some A 2 > 0 Let V denote a discrete sequence of points { a%l of C together with a sequence of positive integers ~p~] (the multiplicities of {~n] )" If~A#, ~#~ 0 , then V(~) denotes the sequence ~a~} of zeros of and ~% is the order of zero of ~ at @ ~ . In this situation, there are THREE NATURAL PROBLemS to study. i. Zero set problem. Given ~ , describe the sets V(~) , ~ A 3 . Ii. Interpolation oroblem. If { ~ , ~ - V c V(~) for some ~, ~ A# , describe all sequences { ~I$,K } which are of the form Kr ~cK)(~)
'
O~< J
0 & rcE)
:
.
I~1~1~1
The conditions ~( ~ ) > 0 and ~ (E) < + ~ are important for the problems of interpolation theory in ~P as well as in other spaCeSo [3]. Everything said above makes plausible the following conjecture. *) The{ "(C) does not i m p l y ~ E ( ~ ) = proved with help of [3].
~ ( E ) , ~B E ~
N
NPA
where
stands for the B l a s c ~ e product ~enerated
byE. CONJECTURE 1 follows from CONJECTURE 2. To see this it is sufficient to apply the Earl theorem [5] about the interpolation by Blaschke products. It is not hard to show that the zero set F of the corresponding Blaschke product can be chosen in this case satisfying 6"(E)P0, ~ ( E ) < ~ (see [6], §4 for details), ~p~°° ) that every follows fr= = + ~=i~ inner function I *~ ~ satisfies
M (v
for ~ es, ts [8], [91 ead to the lowing question. QUESTION. Is there a sin6~lar inner L function ~n U ~ I~ ? ,, REFERENCES 9. B ~ H O r p a ~ o B C.A. My~TZna~xaT~Bm~e CBO~CTBa CTeHeHEaX p~OB C noc~e~oBaTe~HocT~ XO~m~eHTOB ~8 ~P -. ~O~a.AH CCCP, 1980, 254, ~ 6, 1301-1306. (Sov.Math.Dokl., 1980, 22, N 2~ 560-565) I0. B e p d z ~ E z i~ I~.3. 0 ~yx~TZn~EEaTopax IIpOCTpaHCTB ~ ~rH~.aHa~ms z e r o npEJI., 1980, 14, BI~II.3, 67-68.
575 10.4.
FREE INTERPOLATION IN REGULAR CLASSES
Let ~ denote the open unit disc in C and let X be a closed subset of ~ . For 0 < % < I , let ~ & denote the algebra of holomorphic functions in ~ satisfying a Lipschitz condition of order . The set X is called an interpolation set for ~ & if the restriction map
A~
~,,,L~p(~,X)
is onto. The interpolation sets for ~ & , 0< & 0.
In the limit case ~ =~ there are (at least) three different of posing the problem: I. We can simply ask when the restriction map A4-~ Lip ('[, X) is onto, A~ being the class of holomorphic functions in ~) satisfying a Lipschitz condition of order I . 2. We can also consider the class
ways
A~ = H(~) a c ~ (D) and call ~ an interpolation set for (the space of Whitney jets) such that A with , on X .
A~ if for all ~ C ~(X) ~ 0 there exists { in
576
3. Finally one can consider the Zygmund class version of the problem. Let A , denote the class of holomorphic functions in h having continuous boundary values belonging to the Zygmund class of . We say that ~ is an interpolation set for ~, if for any ~= in the Zygmund class of ~ there exists ~ in ~ such that
on X In [I] and [2], it has been shown that Dyn'kin's theorem also holds for ~ -interpolation sets. For A 4 interpolation sets the Carleson condition must be replaced by (2C) ~ N ~
is a union of two Carleson sequences.
Our PROBLE~ is the following: which are the interpolatiom sets for the Zygmund class? Considering the special nature of the Zygmund class, I am not sure whether the condition describing the interpolatinn sets for A~ (one can simply think about the boundary interpolation, i.e. ~ C ~ ) should be different or not from condition (K). Recently I became aware of the paper [4], where a description of the trace of Zygmund class (of ~ ) on any compact set and a theorem of Whitney type are given. These are two important technical steps in the proofs of the results quoted above and so it seems possible to apply the same techniques. REFERENCES I. B r u n a J. Boundary interpolation sets for holomorphic functions smooth to the boundary and B~i0. - Trans.Amer.Eath.Soc.~1981, 264, N 2, 393-409~ 2. B r u n a J., T u g o r e s F. Free interpolation for holomorphic functions regular up te the boundary.-to appear in Pacific J. Math°
3. ~ U H ~ ~ ~ ~ E.M. ~[~o~ecTBa C B O 6 0 ~ O ~ ~ e p n ~ ~z~ F~ac-COB r ~ e p a . - ~aTeM.c6opH., 1979, 109 (151), ~ I, 107-128 (Math.USSR Sbornik, 1980, 37, 97-117). 4. J o n s s o n A., W a 1 1 i n H. The trace to closed sets of functions in ~ with second difference of order 0(~). - J. Approx.theory~ 1979, 26, 159-184. J.BRUNA
Universitat autBnoma de Barcelona Secci~ matematiques. Bellaterra (Barcelona) Espa~a
577
old Let ~N b e t h e u n i t b a l l o f ~N ( ~ > ~ ) a n d d e n o t e b y I~ H~(]~ N) the algebra of all bounded holomoz~hic functions i n ]~ , An a i ~ a l y t i c s u b s e t ~ o f ]~1t is said to be a z e r o - s e t
f o r H ~ ( R N) (in symbois, E~EH'(R ~) ) if there exists a non-zero function # in H ' ( B N) with E=~-~(O) ; E i s said to be a n i n t e r p o I a t i o n s e t f o r ~o(~N) (in symbols: E ~ I ~ ( ~ N) ) if for any bounded holomorphic function ~ on E there exists a function ~ in H °°(~N) with ~IE = ~ . The problem to describe the sets of classes ~ H ~ ( B N) and I ~ao(~N) proves now to be very difficult. I would like to propose some partial questions concerning this problem; the answers could probably suggest conjectures in the general case. Let A be a countable subset of ~ . Set
1,.What are the sets
~
such that
~& ~ H ~ ( ~
N)
?
PROBLEM 2. What are the sets
A
such that
T& E l Ho~(~N)
?
~ 0 ~
It follows easily from results of G.N.Henkin [I] and classical results concerning the unit disc that the following two conditions
are ~ecessary
~e~
for
7~ ~ ZH®r.~BN)-
'"' ) 0} be the biorthogonal family to [ ~ : k(~)> O} , ooe
L;( )
~-~-e,_ O.,t, , IfflO. 0
Condition (2) implies that qX are analytic outside ~ , continuous up to the boundary and bounded (by the constants, which may depend on ~ ). Let B(~) be the class of all functions ~ analytic in int ~ , continuous in ~ and such that
0 Putting
C =~,
~K,X----~A ,
4 k(~)>0
in (I), associate with
581
every function ~ ~0 ~>~o
(~e
E)
582
It was shown in [2] and [3] how the general case (i.e. the case of an arbitrary ~ analytic in int ~ ) can be be reduced to the case ~ ~(~0) • PROBLEM 3. Show that for any domain int tion
~
with the properties
~
there exists a func-
(1), a), b), c).
REFERENCE S
I.
~ e o H T ~ e B
2.
.[ e o H T ~ e B A.~. K Bonpocy 0 npe~cTa2~e~E aHaJn~THqeoENX ~yHELG~ B 5ecEoHe~HO~ B~LUyF~O~ odaaCTH p ~ a M ~ ~ p H x ~ e . - ~OE~.
A.$.
P~j~ SECnOHeHT. M., Hs~Ea, 1976.
AH CCCP, 1975, 225, ~ 5, I013-i015. 3.
~ e o H T B e B A.~. 06 O~HOM r@e~CTaB~eHH~ a H ~ T ~ e c K o ~ # y ~ E u ~ B 6ecEoHeqHo2 B ~ m y ~ o ~ o6aacT~. -- Anal.Math.s 1976, 2, 125-148. A. F. LEONT iEV
(A.$.~EOHTBEB)
CCCP, 450057, Y~a y~. TyEaeBa, 50 ~Hpcz~ ~s~ AH CCCP
583
10.7.
RESTRICTIONS OF THE LIPSCHITZ SPACES TO CLOSED SETS K
The Lipschitz space of the semi-norm
A~(~) ~
K
is defined by the finiteness
~Cl~l) k
Here as usual A~ = ( ~ - ~) and %~ #(G)) = ~(SC+~). The majorant : ~÷ --~ ~ + is non-decreasing and~,GJ~+0) = 0 . Without loss of generality one can suppose that ~0($~$K is non-increasing. Let X ~ ) be the closure of the set Co°° in A w ( ~ ) . This n o t e d e a l s w i t h some problems c o n n e c t e d w i t h t h e space o f t~oes A t (E) -- At, (RT I~ and with its separable subspace ~,(F) -~, where~F~ %" is an arbitrary closed set. Among ~ spaees~u~der consideration there are well-known classes C ,C and A ~ + 4 whose importanCen ~is~indubitable" Recall that C & @ ~-th derivatives satisconsists of all functions ~ U with fying HSlder comdition of order @ . Replacing here H'~Ider condition by Zygmund condition K) we obtain the definition of the class
A g+t CONJECTURE 1. There exists a linear continuous extension M
tor
~'
op~ra-
k
A~(F)-~A~(~ ).
A nonlinear operator of this type exists by Michael's theorem of continuous selection [I]. The lineartiy requirement complicates the matter considerably. Let us review results confirming our Conjecture I. Existence of a linear extension operator for the space of jets ~ , o L ( F ) connected with ~ -CL,~(-'I~ ~) is proved in the classical_ ___W~hitney th\eorem --[2J . But the method of Whitney does not work for A~+~(~) ° Recently the author and P.A.Shwartzman have found a new extension process proving Conjecture I for ~ = ~ (the case ~-----~ is well-known, see for example E3 ] ). The method is closely connected with the ideology of the local approximation theory [4] ). The following version of Conjecture I is intresting in connection with the problem of interpolation of operators in Lipschitz spaces.
~ction
$
satisfiesZ y ~ d
conditionif I A ~ I = 0 0 ~ I )
584
CONJECTURE 2. Let c0~,
~ =t,~
, be majerants. There exists a K
linear exstension operator K
~ : C C F ~ C C ~ ~) mapping
~C
F)
A~C~), ~=~,~.
into
The above mentioned extension operators do not possess the required property. If Conjecture 2 turns out to be right we would be able to reduce the problem of calculation of interpolation spaces K for the pair A ~ ( F ) , $ : ~, ~ , to a similar problem for ~ . PROBLEM. Pind condition necessary and sufficient for a ~iven function
~
# ~CCK)
to be extendable to a
}. ~n
C~ ~)
restriction
~E
K
n
other words we ask for a description of the K
~C~)IF
(or X~Cg~)IF ).
The problem was solved by Whitney for the space C K C ~ ) in 1934 (see [~)° In 1980 P.A.Shwartzman solved the problem for the space A % C ~ ) and in the same year A.Jonsson got (independently) a solution for the space A~+~C~) (see [6,~ ). The situation is much more complicated in higher dimensions; there is nothing but a non-effective description of functions from the space A ~ C ~ ) involving a continual family of polynomials, connected by an infinite chain of inequalities (see [8] for power majorant; general case is considered in [9] in another way). Analysis of the articles [5-7] makes possible the following CONJECTURE 3. Let
N=N(K,~,F)
be the least integer with the
following propert~ ~): if the restriction of a on any subset
H~F
with card
H~ N
function
~EC(F)
is extendable to a
K
function
~H ~ A t ( ~ )
and
HS ~
I~H I~ < co
, then ~
belongs
A~CF). K
to
Define
N(k,n)
by the formula
N (K,~) =,s~p NCK,~o,F). Then the number N(K,t~) is finite. It is obvious that N(4 ~)=~ ; the calculation of N (K,~) for k > I is a very complicated problem. P.A.Shwartzman has proved recently that NC~,~) = S' ~ - 4 ~) One can prove that N c k , ~ I ~ F ) < O0,
585
and using this result has obtained a characteristic of functions from ~ F ) , Fc~ ~ by means of interpolation polynomials (see [6]). When ~ > ~ the number N(~,~) is too large and the possibility ef such a description is dubioms. In conclusion we note the connection of the considered problems with a number of other interesting problems in analysis (spectral synthesis of ideals in algebras of differentiable function, HP space theory etc.)
REFERENCES I. M i c h a e I
E.
Continuous selections. - Ann.Math.,
1956, 63,
361-382. 2. W h i t n e y H. Analytic extensions of differentiable functions defined in closed sets. - Trans.Amer.Math.Soc., 1934, 36, 63-89. 3. D a n z e r L., G r u n b a u m B., K 1 e e V. Helly's theorem and its relatives. - Proc.Symp.pure math., VIII, 1963. 4. B p y ~ H H 2 D.A. Epoc~paHcTBa, onpe~ea~eM~e c noMom~m aoEax~H~x npE6am~eH~. - T p y ~ ~ 0 , I97I, 24, 69-I32. 5. w h i t n e y H. Differentiable functions defined in closed sets, I. -Trans.Amer.Math.Soc., 1934, 36, 369-387. 6. ~ B a p ~ M a H
H.A.
0 cae~ax ~yHE~HR ~Byx n e p e M e m m x ,
B o p a m ~ x ycaoBm0 8~r~ys~a. - B c6."Hccae~oBaHm~ no T e o p ~ ~moz~x B e ~ e c T B e H m ~ nepeMeHH~X". - HpocaaBa~, I982,
y~oBaeT~mu~ I45-
168. 7. J o n s s o n A. The trace of the Zygmund class to closed sets and interpolating polynomials. - Dept.Math.Ume~,1980, -
AK(~)
7. 8. J o n s s o n A., W a i I i n H. Local polynomial approximation and Lipschitz type condition on general closed sets. - Dept. Math.Ume~, 1980, I. 9. B p y ~ H H ~ D.A., m B ap n M a H H.A. 0n~caH~e c~e~a ~yHEIU~ ES 0606~eHHO~O rfpOOTpaHCTBa ~ r m m ~ a Ha n p o E s B O ~ H ~ i EOM-naET. - B c6."Hccae~oBaH~ no ~eopHH ~ y H E n ~ MHO~HX Be~ecTBeH~mX nepeMemmx".
Hpocaa2a~ I982, I6-24.
Yu.A.BRUDNYI
(~.A.BPY~)
CCCP, 150000, HpooJza~l.~, HpooJIaBCEH~ I D c y ~ p C T B e H H ~ yH~Bepc~eT
586 MULTIPLIERS, INTERPOLATION, AND
10.8.
A(p)
SETS
Let G be a locally compact Abelian group, with dual group F An operator T ' I p (G)--~p(~) will be called a multipliAer provided there exists a function T ~ I Qo~P) so that T(~)A--T~ , for all integrable simple functions ~ . The space of multipliers on }?(G) is denoted by Mp(G) . ~et CMp(@=IT~Mp(G):T~C(V) }.
.
In response to a question of J.Peetre, the author has recently shown that for the classical groups, C M p ( G ) is not an interpolation space between M ~ ( G ) = M ( G ) and C M ~ ( G ) = L ~ ( P ) ~ C ( P ) . More specifioall~, we obtained the following theorem (see [2] )° THEOREM I. Let G
denote one of the groups ~ ~,
Then there exists an operator T
(b) TIM(G) is a (c) T ICMp(G)
L~(F)NC(F).
bounded operator on
i_~
n o t
NCG).
a bo~de~ operator on CMp(@,p~t@.
Observe that T is i n d ep e n d e n Our method of construction makes essential use concerning ~(~) sets. Recall that a set E ~ type A(~) (~ < ~ < oo) i f whenever ~L~(T) all ~ 9 ~ E , we have ~ E I % ~ T ) . We used the sult of W.RUdd.u [1].
S >~
THEOREM 2. Let N >5
{0, I,
, and let
2,...,M} (a) F
, o r Z n.
so that
is a bounded ope,z~tor on
(a) T
~
% of p , ~ < p < g . of certain results Z is said to be of and ~(~)=0 for following elegant re-
be an integer, let N
M = 5 S-IN s-~
be a prime with
. Then there exists a set
F~_
so that contains exactl,y
(b) II ~ ~25 ~< cIl
~ II2
N
points,, an d
, fo r evex7 t r i ~ o n o m e t r i c vo!yaomial ~ ,
A
wlth
~(~)=0
(Here C
is
for
~I,E¢:F
(Suoh
are called
~
i n d e p e n d e n t
of
~-pol~momials,).
N ).
AS a consequence of Theorem 2, Rudin showed that there exist sets of type A(~s) which are not of ~rpe A(25+6) , for all 6>0 (see [I] )o An obvious conjecture arisimg from Theorem I is the following: CONJECTURE I. Let
4~ 0 . The difficulty is in proving t3at F~,N is of type ~ (p) (with all constants uniform in N ). One may seek to accomplish this by w r i t i n g an ~F~,M - p o l y n o m i a l _~ i n a J u d i c i o u s way a s a sum o f p r o d u c t s o f F - p o l y n o m i a l s , and c a r e f u l l y examining t h e r e s u l t a n t r e p r e s e n t a t i o n o f ~ . However, new e s t i m a t i o n t e c h n i q u e s f o r ~p norms would s t i l l be v e r y much a n e c e s s i t y i n o r d e r t o c a r r y o u t t h i s program. REFERENCES I. R u d i n W. Trigonometric series with gaps. - J.~ath.Mech., 1960, 9, 203-227.
588
2. Z a f r a n M. Interpolation of Multiplier Spaces, Nath., to appear. MISHA ZAFRAN
Department of ~thematics University of Washington Seattle, WA 98195 USA
Amer.J.
CHAPTER
11
ENTIRE, MEROMORPHIC AND SUBHA~ONIC I~UNCTIONS
This mid and ramified theory, the oldest one among those presented in this collection, hardly needs any preface. By the same reason ten papers constituting the Chapter cannot reflect all tendencies existim~ in the field. But even a brief acquaintance with the contents of the problems shows that the main tendency remains invariable as though more than a quarter of the century, which passed since the appearance of the book by B°Ya.Levin "Zeros of Entire Functions", has shrunk up to an instaut~We reproduce here the first paragraph of the preface to this book: "One of the most important problems in the theory of entire fumctions is the problem of connection between the growth of an entire function and the distribution of its zeros
Nmny other problems in
fields close to complex function theory lead to this problem"~ The only discrepancy between then and now, apparently, consists in more deep and indirect study of this problem A good illustration to the above observation is provided by Problem 11.6. It deals with description of zero-sets of sine-type functions and is important for the purposes of Operator Theory. Problem 11.2 is, probably, "the most classical" one in the Chapter. The questions posed there look very attractively because their formulations are so simple.
590
The theory of subharmonic functions is presented by Problems 11.7, 1 1 . 8 . Problems 11.3 and 11.4 deal with
exceptional values in the spi-
rit of R.Nevanlinna Theory. Problem 1 1 . 1 0 concerns the limit behavi~trof entire functions. An important class of entire functions of completely regular growth is the subject of Problem 11.5. "Old" Problem 11.9 by B. Ya.Levin includes three questions on functions in the Laguerre-P61ya class. Problem 11.1 is rather a problem of approximation theory The problems 11.1, 11.5, 11.6, 11.8, 11.9 are "old" and the rest are
new.
591 11.1. old
THE INVERSE PROBLEM 0F BEST APPROXIMATION 0F BOUNDED UNIFOR/KLY CONTINUOUS FUNCTIONS BY ENTIRE FUNCTIONS 0P EXPONENTIAL TYPE, AND RELATED QUESTIONS
Let E be a separable infinite-dimensional Banach space, let ~ c E , C ... be a chain of its finite dimensional subspaces such t ~ t ~l"4, Em= kt, and U ~ is dense in ~ . For ~ £ ~ we define the sequence of "deviations" from
e4 { II - iII: S
~
by
,
....
S,N,Bernstein [I] (see also [2~ has proved, that for every sequence { ~ 0 of non-negative numbers such that ~ n ~0 there exists ~ £ E , with
This is a (positive) solution of the inverse problem of best approximation in a separable space in the case of finite dimensional subspaces. Strictly speaking S.N.Bernstein has treated only the case of E =C [ ~] , E ~ being the subspace of all polynomials of degree 46-4 , but his solution may be reproduced in general case without any change. Now let ~ (~) be the Banach space of all bounded uniformly continuous functions on ~ with the sup-norm; let B ~ be its closed subspace consisting of entire functions of exponential type %~ (or, to be more precise, of their restrictions to ~ ). S.N.Bernstein has shown [3] that many results concerning the best approximation of continuous functions by polynomials have natural analogues in the theory of best approximation in B(~) by elements of We define the
deviation of
~ "
from ~
A function ~ being fixed, the function ing properties: 1. A(~, ~) ~ A ( I : ~ )
by
A(~,~)
has the followfor ? < ~ .
592
PROBLEM 1. Let a bounded function
6~-
s a t i e f ~ c o n d i t i o n s 1- 3 . Is there a f u n c t i o n
PROBLEM 2. Let 3 ~-0 B~_ 0
A(I,~)
~
~
(0 4 ~ < o o )
, ~eS(~)
be the c l o s u r e of ~ 2
iS a propgr subspace of B ~
PROBLEM 3. L et
~ F(~)
be a Boh z
such
. S~
. What is itsf~+codimension in
almostrp~@~
function. Is
necessarily a ,jump function?
PROBLEM 4. Let
A (I,~')
be a ~ump funotion. Is ~
almost-pe-
rlodic? REFERENCES I.
2. S.
B e p H m T e ~ H C.H. 0d odpaTHO~ ss~a~e T e o p ~ Ham~y~mero np~d~e~ Henpep~_BH~X ~JHmq~. - B EH. : Codp.co~., T.2, M., MS~-BO AH CCCP, 1954, 292-294. H a T a H c o ~ H.H. EOHOTpyET~BHa~ T e o p ~ ~ y s ~ , M.-~., I~TT~, 1949. B e p ~ m • e ~ ~ C.H. 0 H a r e m n p ~ d ~ m ~ e ~ Henpep~Bm~x ~ ~a Bce~ Be~ecTBem~o~ oc~ np~ n O M O ~ nexm( ~ y ~ E ~ ~am~o~ cTeneH~. - B EH.: Codp.co~., T.2, M., MS~-BO AH CCCP, I954, 371-395. CCCP, 3IO0(E, Xap~EOB Xap~EOBCE~ ~CTHTyT ~xeHepoB x o ~ s ~ I o r o
M. I.KADEC (M.~.~M)
CTpO~Te~OTBa
OC#n"I~AItY Problems 1, 2 and 4 have been s o l v e d by A.Gordon (A.H.rop~o~) who k i n k y s u p p l i e d us w i t h t h e f o l l o w i n g i n f o r m ~ t i o n .
~WW
1. (A, Gordon). Le__~t ~
A ~ , e)
~(~) , ~ 0 .
s a t i s f ~ 1N~ above. Than theze
593 PROOF. P i c k a d e n s e s e q u e n c e {~K}~>~o i n (O, e o o ) end consid e r a monotone s e q u e n c e o f p o s i t i v e numbers { ~ } ~>~o such that
K Let { tR} t e n d t o ~r ~o so f a s t t h a t the intervals T K d,,e~ K~>0 . . = ~ e ~,: I~-~K < ~ - ~ } do not overlap, St~c~srd a r g e n t s show
(~) ~ o f o r each } e [1%
~ (t-~z) and --~--- 0 .
Here ~E
(2)
stands as usual for the ohal~oteristic
function of a set
E. ~iveu )~~
~ > 0
Clearly ~ ~ ~
let
. The d e s i r e d
~
is defined as follows :
K~O The s e r i e s c o n v e r g e s a b s o l u t e l y a n d , m ~ f o r m l y on compact s u b s e t s o f (see (1)},which implies ~eB(~) . Fix 5> 0 end consider
~)
d~ ~
p(~).~(~, ~,~_~)
.
AK~ Since ~ ~ i s c l o e e d u n d e r b o u n d e d a n d p o i n t ~ s e c o n v e r g e n c e on ~ At i s c l e a r t h a t ~¢~ B~. On t h e o t h e r h a n d ( I ) i ~ l i e s the inequality
Suppose mew A(;~,o~) ~ ( f f )
of i n t e g e r s
such that
~K~
. Then t h e r e e x i s t ~:'6", ~ ( ~ , f f ) < } ~
. t~operty 2 of P
e n d (2) imply
that
We may assume w i t h o u t l o s s o f ~ e n e r a l i t y
that the sequence
594 It follows dist (~t ~, ~ g) ~ ~ refore A(.-~6") ~ ~I(.6") . @
THEO~
2. (A.Gordon).
mOOF, L e t ~ K ] above for the constant ce ~=liSK]K~
~k~
in contradiction
to
[3] - T h e -
(B~/B~_o)=+ =
and t'~K} be the sequences c o n s t r u c t e d as s e q u e n c e ~K------- 6~ . F o r e v e r y b o u n d e d s e q u e n -
O define
~)
d~ ~
~,$_~).
~K~,
k~0
Clearly ~ t ~ B ~
and it is easy to check that
-
Indeed, ~ r ~
K
~(~-4/~, 8,~) =
q ( ~ 6,~)
i n B ( ~ ) . Therefore
Usimg t h e same argumen%s a s i n t h e p r o o f o f Theorem I , we o b t a i n
~(~K,B~)~ ~ I ~I f o r ~ ~ ~ . i~ f o n o . , the f a c t o r - s p a c e ' ~ / ~ ~_ o contains a subspace and t h e r e f o r e i s n o t separable. • Note that the function ~ with ~ properly t h e n e g a t i v e a n s w e r t o P r o b l e m 4. I n d e e d , i f ~ I ~ K ~) O,
~'~ ~
•
f r o . (~) t ~ t isometrlo to ~ %
I
ohoosen ~ives ---~ ~ then
595
SOME PROBLEMS ABOUT UNBOUNDED ANALYTIC FUNCTIONS
11.2.
A famous theorem of Iversen [I, p.284a] says that if ~ is a transcendental entire function then there is a path ~ along which tends to ~e . Thinking about this theorem has led us to formulate the following four problems, which we would like to solve, but cannot yet solve. I. I_~f ~ is a transcendental entire function, must there exist a ~th
~
alon~ which e v e r y
derivative Of ~
Short of that, how about just havin~ both
tends to oe ?
and
~,ity alon~ ~?(It renews, say, by w ~ - w l i r o n
tend to infi-
theory, that i f
is a transcendental entire function, then there exists a s e q u e n c e (En) such that I(~) (Z~) - - - ~ as B - - ~ for each k= ~ , ~ , . . . , but to obtain a p a t h on which this happens seems much more difficult. ) 2. If I is an unbounded anal,Ttic function in the open disc ~ ,
~ust there exist ~ sequence (Z~)
of points of D
ev e ~
~--=
k= 0 , ~ , £ , . . . , ~
just h a v i ~
(L)
(z~)-~
~(Z~)--~
that one can always find If ~
~
? Short of that, how about
and ~t"(Z~)-,~ : (~)
such that for
(~he authors have sho~
So that both
(Z~)'~O@ and
I' (Z~)-~.~
grows fast enough, i.e., if either
or
=~,
then we can show that there exists a
(~)
(2)
with
as
~-~ for ~ = 0 , ~ , ~ , . . . , but we do not know hew necessary these growth assumptions are. Note that conditions (I) and (2) are not strictly comparable. The proof involving condition (I) quotes a theorem of Valiron, while that involving (2) uses some Nevanlinna theory.) 3.
Is there an eas 2 elementar~ proof I u s i n ~ n e i t h e r Wiman-Vali-
ron t h e o r ~ n o r advanced N e v a n l i n n a t h e o r ~ t h a t i f
~
is a transcen-
596 dental entire function. . then there exists a .sequence ~(k~(~) ---~ ~
a_ss~ - - ~
f0r
~
0,4, ~ . . .
(~)
such that
?
4, The following possibility is suggested by many examples: the differential equation
where the ~ no solutions
are polynomials in one variable, not all constants, has ~(~) that are analytic and unbounded in the unit disc . For example, the equation ~ f z _ m I can be solved explicitly by means of the standard substitution~ V ~ - ~ V ~ V ~ and it is -'4 easily seen that it has no unbounded analytic solution in any disc, lending support to the above hypothetical statement~ The QUESTION re-
mains whether that statement is true or not°
REFERENCES 1. T i t c hma Oxford S 932.
JAMES LANGLEY LEE AoRUBEL
r s h
E.C. The Theory of Functions
2nd Edition,
Dept.Math, University of Illinois Urbana, IL 61801 USA
597 11.3.
COMPARISON OF SETS OF EXCEPTIONAL VALUES IN THE SENSE O!~ R.NEVANLINNA AND IN THE SENSE OF V.P.PETRENKO Let
~
where [ @ ~
be a meromorphic function in C
and put
is the spherical distance between ~
and ~
. Demote
in the sense of V.P.Petx~nko an~ by E~C~)=IOv£~:~(~,~)>0} . the set of aeZicie~t values o~ { . It'~s ~cl~ear that The set E~(~) is at ~ost countable if ~ is of finite order [I]. There are examples of ~ 's of finite order with E~({)~ E~(~)
E'~(,,()CE#(_~).
b
41.
-
,,
PROBLEM
I. Let E~CE~cC
be arbitrary at
sets, Does there exist a meromorp..hic function I
with
most countable of f.inite order
Eech=E,?
implies
EN({)= E~({)
I~O:B:I~ 2. Let {
[6]. be an,, ,entire f ~ c i ; i o n of f.i.mii;,e,,,,order,
REFERENCES
I. 2.
3.
4.
H e T p e H K o B.H. POCT MepoMop$s~X ~ , Xap~xOB, "Bm~a m~oxa", 1978. ly r p i~ m ~ H A,~. 0 cpaBHe,~ ~e~eETOB 05(@) . -- Teop~ ~ys~, ~yHE~.a~a~. z ~x np~., X ~ K O B , 1976, • 25, 56-66. r o ~ ~ ~ 6 e p ~ A.A. K Bonpocy O CB~S~ M e ~ y ~eSexTOM o T ~ o H e ~ e M MepoMop~HO~ ~ym~mz. - T e o p ~ ~ , ~ymu~.aHs~. ~x np~., Xap~oB, I978, ~ 29, 31-35. C o ~ ~ H M.~o 0 c o o ~ o s e ~ M ~ MHOZeCT~ ~e~eETm~X 3~a~e~ s OTF~OHe~ ~ MepoMoI~HO~ ~ EOHe~O~O n o p ~ a . -C~6.MaTeM,~/pHa~, I98I, 22, ~ 2. I98-206.
598
. E p e M e H E 0 A.3. 0 ~e~eETax ~ O ~ O H e K ~ q X $ym~m~ EoHe~oro nops~a (B nepal). A.A. GOL 'DBERG
(A.A.IY~hEBEPr)
A. E. EREMENK0
(A. ~.EP~EHE0)
MepoMop~x
CCCP, 290602, JI~BOB JL~BOBC~ r o c y ~ a p c T B e ~ y~epc~TeT CCCP, 310164, Xap~oB np.JleH~Ha, 47, ~sm~o-Te~ec~ ~HCTH~yT ~S~HX ~eMnepaTyp AH YCCP
599
11.4.
VALIRON EXCEPTIONAL VALUES OF ENTIRE FUNCTIONS OP COMPLETELY REGULAR GROWTH
Let E# be the class of all entire functions of order p , ° 0 and ~(0)~--~ , where ~ is the open unit ball in 6 ~ , Thus ~(~) is convex (and compact in the compact open topology). We think that the structure of N(~) is of interest and importance. Thus we ask: What
are
the extreme points of ~(~)
Very little is known, Of course if ~ - ~
=
? , and if
c4+ 8-)Ic4-~),
(1)
then ~ is extreme if and only if #(~) ----- C~ , where O ~ T . It is proved in Eli that if S ( ~ ) ~ $ ~ and if ~ is the Cayley transform (I) of £ , then $ ~ E £ ~ ) , where E(~) is the class of all extreme points of N (~) . Let K~- (~,..., K N) be a multi-index and consider monomials K
~(~) ~
C$ K in 6~
where by
such that I~(~)I ~ ~
if ~
~
. ThuslCl~([~[)"~
we mean K,+...
let
~(~)=(~+OK~K)/(~--0K~K). It is proved in [2] that ~ ( ~ ) if and only if the components of k are relatively prime and positive. 2. We have ~ ~ ~(~) , however it is a corollary of the just mentioned theorem of [2] that ~ 6 6 5 ~ ( ~ ) , where the closure is in the compact open topology. Thus E(~)=~= c~o~ (~) . (If ~ - ~ ,
then E ( ~ ) =
c ~ E c ~) ).
It is also known that if ~ is an extreme point of ~(~) and if (I) holds(that is to say if ~=(~-~)/(~÷I) ), then ~ is irreducible. This is a special case of Theorem 1.2 of [3]. The term "irreducible" is defined in ~]. If ~ ~ , then ~ is extreme if and only if~ is irreducible. However for ~>/ ~ , the fact that ~ is irreducible does not imply that ~ is extreme. 3. The extreme points ~ in section I and the extreme points that can be obtained from them by letting ~ $ (~) act on ~(~) have the property that the Cayley transform I=(~-I)/(~+~) is
holomo~hic on ~ U 8 ~ Is this the cas,e for eve,ry ~
in E(~)
?
If the answer is yes, then it would follow (since ~
) that
824 the
F. and M.Riesz theorem holds for those Radon measures on a
whose Poisson integrals are pluriharmonic. In particular there would be no singular Radon measures =~= 0 with this property, which in turn would imply that there are no nsnoonstant inner functi6ns on 8
.
REDOES I. P o r • I i i
F.
Measures whose Poisson integrals are pluri-
harmonic I!. - l l l i n o i s J.Math. ~ 1975, 19, 584-592. 2. F o r e I i i F. Some extreme rays of the positive pluriharmonic functions. - Canad.Math.J., 1979, 31, 9-16. 3. F o r e i i i
F.
A necessary condition on the extreme points of
a class of holomorphic functions. - P a c i f i c 81-86. 4. A h e r n
P., R u d i n
W.
J. Nath.~1977, 73,
Factorizations of bounded holomor-
phic functions. - Duke Math. J., 1972, 39, 767-777. PRANK ~ORELLI
University of Wisconsin, Dept. of Math., M a ~ s o n , Wisconsin 53106, USA
COMMENTARY The second question has been answered in the n e g a t i v e mentary in S. 10.
See Com-
625
12 •3 .
PROPER MAPPINGS OF CLASSICAL DOMAINS
A holomorphic mapping
is called
p r o p e r
if
q;~
~
of a bounae d domaln ~ C ~
~
~l~"~O
~(C~C~),
~
for e v e ~ se-
A biholomorphism (automorphism) of ~
is called a t r i v i a 1 p r o p e r m a p p i n g of ~ . If .0. is the l-dimensional disc ~ the non-trivial proper holomorphic mappings ~:~-'~ do exist. They are called finite Blaschke products. The existence of nontrivial proper holomorphic mappings seems te be the characteristic property of the l-dimensional disc in the class of all irreducible symmetric domains. CONJECTURE I. For an irreducible bounded s,~etric domain in C ~ , ~
# ~
, every proper holomor~hic mappin~
~--~
is an
aut omorphiem. According to the E.Cartan's classification there are six types of irreducible bounded symmetric domains. The domain ~ p ~ of the f i r s t type i s the set of co lex
matrices
Z
,
,
such that the matrix I--Z ~ Z is positive. The following beautiful result of H.Alexander was the starting point for our conjecture.
T~o~M i.e. ~ = ~ ? , ~
I (H.Ale~nder [I]). ~et ~ and let
be ths ~ i t ball in C P ,
p>/~ . Then ' e v e ~ proper holomorphic map-
,~i~ (~:_qp,~ ~I'9.p,~
is an. autom.o.~ph.i~mof the ban.
Denote by S the distinguished boundary (Bergman's boundary) of the domain ~ . A proper holomorphic mapping @: ~ •~ is called s t r i c t i y i r o p e r if ~ ( ~ ( ~ u ) } ~)--~0 for every sequence ~ E with the property ~ ( ~ S)-~0. The next result generalizing Alexander's theorem follows from [2] and gives a convincing evidence in favour of CONJECTURE 1. THEOREM 2 (G.M.Henkin, A.E.Tumanov
le bounded symmetric domain in
,proper holomorphic mappin~
C ~ and
~ :/i--'/~
[2] ). ~
If' -0.
~ D
is an irreducib-
, then,,any strictly
i,~,,,~,~,,~,utomo,rp~!sm,
Only recently we managed to prove CONJECTURE I for some symmetric domains different from the ball, i.e. when ~ 2 ~ S .
THEOEEM 3 (G.M.Re.'~in,
R.~.~ovikov). Let, ~ c C
b e the classical domain of the 4-th type, i,e.
~
, ~~,
626
$I ={ z:z.~', ((,~'~')~- Izz'l~)V~< ~}, where
E=(~,...,E~)
and
Z l stands forthe transposed matrix
The n every prgperholomorphic mappin~ ~ - ~
is an automorphism.
Note that the domain I~,~ of the first type is equivalent to a domain of the 4-th type. Hence Theorem 3 holds for ~L~,~ e We present now the scheme of the proof of Theorem 3 which gives rise to more general conjectures on the mappings of classical domains. The classical domain of the 4-th type is known to have a realization as a tubular domain in C ~ ~ ~ , over the round convex Gone
boundary of this domain coincides with the space T h • d i s t i Then boundary g u i9 2 s contains h e together d R =t~C • ~= 0 } with each point Z ~ \ ~ the l-dimensional analytic component 0 ~ = ~ :~EC~ I~ >0}. The boundary ~ 0 ~ • ~f this component is the nil-line in the pseudoeuclidean metric ~5
=~o-~®,-...~ e n
the disti~uished bo~d~ry S
.
If ~ is a map satisfying the hypotheses of the theorem, an appropriate generalization of H.Alexander's [1~ arguments yields that outside of a set of zero measure on ~ a boundary mapping (in the sense of nontangential limits) ~ :9 ~ --~ ~ of finite multiplicity is well-defined. This mapping POsseses the following ~ro~erty: for almost every analytic component 0 Z the restriction q l ~ is a holomorphic mapping of finite multiplicity of ~ into some component ~ W " ~urthermore, almost all points of ~ are mapped (in the sense of nontangential limits) into points of ~ . It follows then from the classical Frostman's theorem that ~I~Z is a proper mapping the half-plane ~ into thehalf-plane Y v W o it follows that the boundary map ~ defined a.e. on the distinguished boundary ~ C ~ has the following properties: a) ~ maps ~ into ~ outside a set of zero measure; b) ~ restricted on almost any nil-line ~ E coincides (almost everywhere on ~ Z ) with a piecewise continuous map of finite
627 multiplicity of the nil-line S~Z into some nil-line With the help of A.D. Alexandrov' s paper
[3]
S~Wone can prove
that the mapping ~: ~ - - - ~ satisfying a), b) is a conformal mapping with respect to the pseudoeucli:dean metric on S . It follows that is an automorphism of the domain II To follow this sort of arguments, say, for the domains /Ip,~ where p ~ , one should prove a natural generalization of H.Alexander's and O.Prostman's theorems. Let us call a holomorphic mapping ~ of the ball l~p, 1 a 1 m o s t p r o p e r if ~ is of £inite multiplicity and for almost all ~ S i p , 4 we have ~(~)~ ~'~-p,4 , where ~(~) is the nontangential limit of the mapping ~ defined almost everywhere on ~Ap, I CONJECTURE 2. Let ~ be an almost Droner maDpin~ of ~ip,1 and p~ ~.
Then
~
is an automorphism~
If we remove the words " 9
is of finite multiplicity" from the
above definition, the conclusion of Conjecture 2 may fail, in virt~e of a result of A.B.Aleksandrov [4] P i ~ l l y we propose a generalization of Conjecture SCONJECTURE 3. Let ~_
be a s2mmetric domain in
from an~ product domain il Ix @, ished boundar 2. Let ~r ..
and ~zz
Gc ~-I
different
be its distingu-
be two domains in ~
intersectim~
a proper mappin~ such that for some set r. ~r
Then there exists an automorphism ~
The v e r i f i c a t i o n
and let %
~
of C o n j e c t u r e
of /~
such that
3 would l e a d to a c o n s i d e r a b l e
strengthening of a result on local characterization of automorphisms of classical domains obtained in [2] One can see from the proof of Theorem 3 that Conjecture 3 holds for classical domains of the fourth type. At the same time, it follows from results of [I] and [5~ that Conjecture 3 holds also for the balls /ip, 4 REMARK. After the paper had been submitted the authors became aware of S.Bell's paper [6~ that enables, in combination with C2] , to prove Conjecture I
628
REFERENCES 1. A i e x a n d e r H. Proper holomorphic mappings in ~ o Indiana Univ.MathoJ., 1977, 26, 137-146. 2. T y M a H o B A.E., X e H E E H LM. ~ o K a ~ H a ~ xapaKTep~saE~H aHaJ~TE~eCEEX aBTOMOp~ESMOB E~accENecEEX o6~aoTe~. - ~OE~. AH CCCP, I982, 267, ~ 4, 796-799. 3. A ~ e E c a H ~ p o B A.~. K OCHOBa~ Teopz~ OTHOCETex~HOCT~. -BecTH.I~V, I976, I9, 5-28. 4. A x e mape. 5. R u d ball. 6. B e 1 -
E i 1
c a H ~ p 0 B A.B. CymecTBoBaHEe B R y T p e H H ~ x ~ y R E n ~ B MaTeM.c6., I982, II8, I47-I68. n W. Holomorphic maps that extend to automorphism of a Proc.Amer.Soc., 1981, 81, 429-432. S.R. Proper ho~omorphic mapping between circular domains.
Comm.Math.Helv.,
G.M.HENKIN
(r.M.XEHEHH)
R.G.NOVIKOV
(P.F.HOB~KOB)
1982, 57, 532-538.
CCCP, 117418, MOCEBa, ~eHTpa~BHH~ BEOHOM~Ko--MaTeMaT~HecE~R ZHCT~TyT AH CCCP, y~.Kpac~EoBa, 32
CCCP, 117234, MOCEBa, ~eH~HCEEe rope, MIV, ~ex.-~aT.~u'~yJIBTeT
629 12.4.
ON BIHOLOMORPHY OF HOLOMORPHIC MAPPINGS OP COMPLEX BANACH SPACES
Let ~ be a function holomorphic in a domain ~ ~ 3 c ~. It is well known that if { is univalent, then ~f(~) =~= 0 in ~ . Holomorphio mappings # of domains ~ c ~ for 11,> ~ also possess a similar property: if I : ~ ~ 6 ~ is holm, orphic and one-to-one then ~ Q ~ ) = ~ = 0 at every point $ of~(~)-------(~)},K=I~" ~ is the Jacobi matrix), or equivalently the differential ~ ( ~ ) ~ ~--~-~Q~)~ is an automorphism of ~ , and then, by the implicit function theorem, ~ itself is biholomorphic. Note that the continuity and the injective character of immediately imply that ~ is a homeomorphism because ~ ~= ~_~ O ) a family consisting of all ~-preserving invertible transformations 6 0 : 8 S such that ~CCS : OJ(~¢)~~. Each O O ~ E generates a linear operator ~-W : ~ --'- ~ - (where denotes the space of all measurable functions on ~ ) by the formula ]-to ~ (~) ~ ( W ( ~ ) ) ~ ~8, ~ ~ . The elements of a set ~e~ {%. GJ ~ ~ E } are called the ~-rearrangements. Each Toa preserves the distribution of a function, hence the integrability properties of functions are also preserved. Given a subset A of ~ define the r e a r r a n g e m e n t - i n v a r i a n t h u 1 1 s of A as follows:
RHo(A) ~ N R~ (A), g~O
RH(A) = UoR~(A).
The general problem of characterization of such hulls for a given concrete set A has been posed by O.Cereteli. We refer to [5] for the contribution of O.Cereteli to the solution in some concrete cases. The following results have been obtained in [2] and [3]- Consider for the simplicity the case when ~ is [0,{] equipped with the usual Lebesgue measure. a) Let ~ = {~}~ ~ be a family of bounded functions such that for any ~ ~ L °°
Z ! c,~ ( ~; qb)t ~ ~ oo~s~ II f I1~ with a constant independent of • Here O ~ ( f ~ l : : ~ ) - ~ I { ~ t ~ , ~ - ) ~ . For a non-negative sequence { ~ ; } , ~ ] ~ such that ~ - - - 0 when ~t--~ co , define a class
where C ~ ' ~)* ~ functions in
U
denotes the non-increasingrearra~eme~t of . ~hen RHo(A)= RH(A)= IJ [2]. . For given
p, ~
p < 2
,
define a class
643
Am~ {f~ kt • I~,- (S, pll~-- o}, where S ~ ~ denotes the ~ -th partial sum of the Fourier series of with respect to (~ . Then ~ @ ( A ) = ~ H ( A ) = L 4 [3], i.e. any complete orthonormal family of bounded functions is in some sense a basis in ~ ~~ p < ~ ~ modulo rearrangements. A different effect occurs for the class A = [ ~ ~ ~4}, 4 where ~ is taken over the unit circle T and ~ denotes the conjugate function of a function ~ . In that case [4, 5]
I
T
Here
o ,
i
f(~?l~
~.
The class
~,) d,, I < ~, } arising in (1) (in [4, 5 ] this class is denoted by on non-negative functions ;with the class m ~ + J .
~ ) coincides~ . Moreover,
L ~+ L c M ~+M. In addition to (1) it has been proved in [3] that
if
I~I-Io(A) = RH (A)= M I~FJM , i.e. any function from M ~ hA~F~÷ivl can be rearranged on a set of small measure so that the obtained function has m r -convergent conjugate trigonometrical series. For any p, p ~ | , define a class m P over ( S ~ , ~ ) as follows
@
S
644
I t i s clear that L P C M P and M P coincides with L P on non-negative functions. The class MP in comparison with L P takes into account not only the degree of integrability of function but also the degree of cancellation of the positive and negative values of the function. It is known [6] that M ~ + M is linear. As f o r M ~ it has been proved in [3] that 1) rv~P~ p > 2 __is non-linear, moreover, there exists ~ M P such that ~ + 4 ¢ MP~
2) MP+L'c M P, ~< p-I. and RH(A)
,~,r the trigonomet-
rical system. is the family of Legendre Very interesting is the case when ~D polynomials { L~} on the interval [-~,+4] Tt is true [3] that .
MPc RH0(A), PROBLEM 3- F,ind ~ N o(A) and
p > _8._
(2)
•
~H(A)
f or the family of Le-
gen~e polynomials. We pose also t~o easier problems related to Problem 3. PROBLEM 3'. Is the inclusion PROBLEM 3''. Is the inqlusion
L '/'C R Ho (A) M '/~ C
~ Ho (A)
true? true?
The number ~ in (2) appears from the general theorem proved in [3]. The theorem states that if a sequence of integral operators [~-~}, ~ ~ ~ has a localization property in I ~ , and the maximal operator T '~ ~ :" s~pITn ~1 has a w e L t = e (p,p) ~ t h some p > 4 , then
645
MNrc RHo(A), where
A={
~ ~ ~ . ~i~.¢
~ )
(3)
. It is not known, whether the
power
~ in ( 3 ) i s sharp on the whole class of the operators u n d e r consideration. The maximal operator with respect to the Legendre pol~lomialo system has weak~type (e, p), e< ~ [I]. The number ~ is the value of ~--~ at p = • The problems analogous to Problems 3, 3', 3'' can also be formulated for Jacobi polynomial systems.
~/3
~
AH Fpys~cKo~ CCP
646
13.7.
NORMS AND E X T O L S OF CONVOLUTION OPERATORS ON SPACES OF ENTIRE FUNCTIONS
Given a compact subset ~ C let B(~} be the Be=stein class of all bounded functions ~ on ~he dual copy) with Fourier transform f supported on ~ 0 In fact, every function ~e B(~) can be extended to an entire function of e x p o n e n t i a l type on ~ . The linear space B ( ~ ) with the uniform norm on is a Banach space (in fact, a dual Banach space). EXAMPLE. Let K be the unit ball in ~ , i.e,,
if and only if the function ~ of an entire function on C
for some constant C. we shall consider operators ~-" form
CTI
B(K) -4" B ( K )
is a restriction satisfying
of the
=I
~ being a complex-valued regular Borel measure of bounded variation on . In other words, ~ - ~ = ~ * ~ , The function ~ ~ ~ is said to be the s y m b o i of T = ~ - ~ The representation ~ ~ ~-~& is not an isomorphism, but nevertheless the symbol q~ is uniquely determined by 7° The spectrum ~f ~coincides with the range of ~ and its norm with the norm of the functional ~ ~ C T ~ ) (0) . if K is a set of spectral synthesis then the symbol ~" .uniquely determines the corresponding operatot T - - ~" C D ) , D = ~ ~ / ~ in B(K) . Moreover, in this case
647
DEFINITION. A n o r m a 1 e x t r e m a 1 for T is any element ~ BCK) such that II~II---4 7 ( T ~ ) ( 0 ) = ll~-II. It can be easily shown that the normal extremals always exist (and form a convex set). For example, in the case ~ = ~ ~ K-- [-(~,~] O>0 , and T ~ = ~f the classical result asserts that ~ T I = ~ and all normal extremals have the following form o,e ;p ÷
l~e~p(- i, Ox). A measure
~
is called
e x t r e m a 1
for
-~
if T ~
=
and IITII" The set of extremal measures may be empty even in case of finite
K. Such problems as calculation of norms and discription of extremals go back to the classical papers of S.N.Bern~tein A survey of results obtained in the field up to the middle of 60-ies can be found in [I]. ~or additional aspects of the topic see [2], which is~unfortunately, flooded by misprints, so be careful. A compact set K in ~ is said to be a s t a r if with every ~ it contains ~ ~ for each p ~ [ o, 4] . Every star is a set of spectral synthesis and B ( K ) contains sufficiently many functions vanishing at infinity. If ~, ~ K and ~ ~ ( ~ ) [ = ~ ( ~ o ) == 4 then I~q ~ ( D ) ~ = 4 (i,e. the norm of (D) coincides with its spectral radius) if and only if the function ~ --*-~F ( ~ ÷ ~,) admits a positive definite extension to R ~ . The operator "~(D) has extremal measures. If ~(D)ll>| then every extremal measure is supported on a proper analytic subset of ~ a and the extremal measure is unique provided ~----~ o For ~ > | the uniqueness does not hold (example: K is the unit ball a n d ~ ( D ) is the Laplace operator). PROBLEMS IN THE ONE-DI~tENSIONAL CASE. ~,~ery polynomial (in one variable) is related to a wide stock of positive definite functions. Suppose that the zeros of a polynomial "~ are placed in the half-plane ~e~ ~ ~ 6 > 0 and that ~(0) = 4 , Then the restriction of ~ to [ 0~O] extends to a positive definite function on ~ . It follows that for all linear
where
BO ~-e~ B ( [ -. O , -C ~ ] - )
In t h i s case a l l normal extremals
can be easily determined and there exists an extremal measure (at least one). At the same time for pol,~.omial s qY of de~ree 2...these
648
problems still do not have a full solution. The simplest operator is provided by ~ ~ _ ~ l l . ~ , ~C~ . For X ~ the problems are solved (see papers of Boas-Shaeffer, Ahiezer and Meiman). For some complex ~ (in particular those for which the zeros of the symbol ~ satisfy the above mentioned condition) ~ admits (after a proper mormalization) a positive definite extension, so that the norm of ~ ( ~ ) coincides with its spectral radius+ Is it ~ossible to calculate the norm for al~
~C~
? How do
the " E u l e r e q u a t i o n s " l c o k in this case? Note that according to the Krein theorem the extremal measure is unique provided K = [ - O , O ] and the spectral radius is less than the norm. Of course, these problems remain open for polynomials of higher de~ree+ The Bernsteln inequality for fractional derivatives leads to the following PROBLE~. Consider on the function ~ ( ~ ) ~ ( ~ - - I ~ I ) ~
[-+,4]
o(>0
+ The problem is ~o find
If 06 ~ ~ then ~ is even and is convex on [ 0 , ~ ] , and therefore coincides on [-|~ ~] with a restriction of a positive definite function by the Polya theorem. For O~ < ~ T becomes concave on [0,|] and moreover ~ cannot be extended to a positive definite function on ~ . Indeed, if ~ is positive definite then - ~ " is a positive definite distribution. At the same time, --~" is nonnegative, locally integrable on a neighbourhood of zero and non, integrable on a left neighbourhood of the point ~ = | . A positive definite function cannot satisfy this list of properties. The best known estimate of the norm for O~ C (0,4) is 2(4+0()~~ It is evidently not exact but it is asymptotically exact for o~-+-0 and o~ +-~ | . It should be noticed that in the space of trigonometric polynomials of degree ~ PP~ the norm of the operator of fractional differentiation coincides with its spectral rsdius for o~ >I o~0 , where oCo ~ o ~ o(P~)< ~. Another example is related to the family { ~ of functions definite.
Consider the family for
teristic function
~
o~ > +
.
Since every charac-
satisfies the unequality l~(~)Im~ ~(~+I~(2~)I),
649
there are no positive definite extensions for c ~ 2 Consider now the case ~ < o ( < 2 ° The following idea has been suggested by A.V.Romanov Extend ~ ( ~ ) to (|,2) by the formula
v(1 ÷ Extend now the obtained function on ( 0 , ~ o an even periodic function of period 4 keeping the same notation ~ for this function. We have
k
~
,t
where the sum is taken over odd positive integers. It is easily verified (integration by parts) that ~ K and
0
are of the same sign. Clearly ~I > 0 Hence ~ is positive definite if
and
for
k
s.
@
(cf. [4], Ch.V, Sec.2,29). The ~unction ~(c&) decreases on (4,2)and ~(4)=oo ~(Z)< 0 . Therefore the equation ~ ( O C ) ~ 0 has a unique solution oCo ~ ( ~, ~) (Romanov's number). The function ~ is positive definite on [-I,~] if C ~ o C 0 . At the same time slightly modifying the arguments from [5], Tho4.5.2 one can easily show that for 0 < o C I < o C ~ < Z the function
4-
is positive definite on R . Hence ~4 is positive definite on [-4~4] if so i s ~ z . The "separation point" ~0 is clearly ~ o ~ Is it true that main open,
~o > o ~ @
? For C ~ > ~
the above problems re-
650 PROBLemS IN THE MULTIDIMENSIONAL CASE. For ~ ,except the case when the norm of ~ ( ] ) ) coincides with its spectral radius, very few cases of exact calculation of ~(D)~ and discription of extremals are known. The GENERAL PROB L E M h e r e is to obtain proper generalizations of Boas-Schaeffer's and Ahiezer-Meiman's theorems, i e, to obtain "Euler's equations" at least for real functionals. Our problems concern concrete particular cases; however it seems that the solution of these problems may throw a light on the problem as a whole. If ~ : ~ - ' ~ ~ is a linear form then the operator with the symbol ~ I~ is hermitian on B ( ~ ) and hence its norm coincides with the spectral radius. This again will be the case for some operators with the symbol of the form (p o ~)I K where p is a polynomial. The following simple converse statement is true . If is a polynomial and
for every symmetric convex star
K then ~ : a linear form and p is a polynomial. Does the ~ similar converse statement hold when
p o~
where ~ ia ranges over
the balls? Let
where p ~ |
and ~ C ~ ) = - ~
~
•
The operator ~ ( ~ )
is
obviously the Laplacian ~ .4 The norm of ~ coincides with the spectral radius in the following cases: n : ~ ~ : ~,p=|~ ~ , p : ° ° . The proof is based on the following well known fact: if ~ is a probability measure then { ~ " I ~ ( ~ ) ~ = ~} is a subgroup. The case e : £ turns out to be the most pathological and perhaps the most in~eresting. We have IIAIIB(Ka : ~ . In this case extremal measure is not unique and it would be interesting, to describe all extremal measures (notice that they form a compact convex set). The problem of calculation of the norm can be reduced to the one-dimensional case for operators of the form ~(~) in ~(~2) . It is possible to calculate the norms by operators with linear symbol in the space ~ (K2) explicitly, For example, the norm of Cauchy-Riema~u ope-
651 rator equals 2 and its normal extremal is unique. Namely,
However, for the operators of the second order the things are is a positive , the spectral and normal extremals
more complicated. If the symbol *'(~) -- ( A ~,~) real quadratic form then radius of Z(D) coincides wi~h that of ~ are of the form
II (D)II
where
10~1 i s the norm,
~ ~ ~
A
t~! ~
At the same time nothin~ i s known about ext.remals ...and norm of the operator
0~+ "@==
in
B (Kz)
for
~=Z.
REFERENCES I, A x a e s e p
H.H. ~ e ~ a z
no ~ e o p u
annposcmMa~a.
Moc~m,
HaT-
rm, 1965. 2. r o p a H E,A. HepsBeHc~a BepHmTe2HS C TOq~Z s p e ~ Teopaa ouepSTOpOB. -- BeOTH.XspBE.yH-TS, ~ 205. I I I ~ F J ~ S MaTeaaTz~m a MexaHME8, m~n.~5. - XaD~EOB, B ~ 8 m~oaa, HS~-BO XaI~E.~H--Ta, 1980,
77-105. 3. r o p ~ H E.A., H o!o B ~ ~ a o C.~. 3KcTpe~ma~ HeEoToI~X ~i~eloe~z~a~x one~aTopoB. - ~ o ~ no T e o p H onep~TOpOB B ~HE-~oHaa~x nI~OTImHOTBaX, MZ~OE, 4-11 ~ a s 1982. Tesac~ ~oz~., 48-49. 4. Z y g m u n d
A.,
Trigonometric Series, vol.l. Cambr Univ. Press,
1959. 5. L u k a c s London,
E .
Characteristic functions, 2 n d e d . ,
Griffin,
1970.
E.A. GOB.IN
CCCP, 117288, MOCEBa,
(E.A.rOPm~) MezaEz~o-~a TemaT ~ e C~a~ ~ a ~ T e T
652 13.8. old
ALGEBRAIC EQUATIONS WITH COEFPICIENTS IN COMMUTATIVE BANACH ALGEBRAS AND SOME RELATED PROBLEMS
The proposed questions have arisen on the seminar of V.Ya.Lin and the author on Banach Algebras and Complex Analysis at the Moscow State University. In what follows A is a commutative Banach algebra (over C ) with unity and connected maximal ideal space M A . ~or ~ ~ , denotes the Gelfand transform of • . A polynomial p ( ~ = ~ + ~ ~'~ + •+ ~ ~ ~ £ is said to be s e p a r a b I e if its discriminant ~ is in+ vertible(i.e, for every ~ in M~ the roots of ~ + ~4 (~) + +~(~) are simple); ~ is said to be c o m p 1 e t e I e r e d u c i b 1 e if it can be expanded into a product of polynomials of degree one. The algebra is called w e a k 1 y a 1 g e b r a i c a 1 1 y c 1 o s e d if all separable polynomials of degree greater than one are reducible over it. In many cases there exist simple (necessary and sufficient) criteria for all separable polynomials of a fixed degree Wv to be completely reducible. A criterion for A = C(~) , with a finite cell complex ~ , consists in triviality of all homomorphiams ~4(X) , B(~), B(~) being the At%in braid grQup with threads [I]. If (and only if) ~.< 4 this is equivalent to ~ ( ~, ~ ) = 0 (which is formally weaker). The criterion fits aS a s u f f i c i e n t one for arbitrary arcwise connected locally arcwise connected spaces ~ . It can be deduced from the implicit function theorem for commutative Banach algebras that if the polynomial with coefficients ~ is reducible over ~ ( ~ A ) then the same holds for the original polynomlal p over ~ . On the other hand (cf, [2], [3]) for arbitrary integers ~ , ~ , 4< k % ~ < co there exists a pair of uniform algebras A c B , with the same maximal ideal space, such that ~/~ = 4 , all separable polynomials of degree % ~ are reducible over A , but there exists an irreducible (over ~ ) separable polynomial of degree ~ . WE INDICATE A CONSTRUCTION OP SUCH A PAIR. Let G k be the collection of all separable polynomials ~(~)= A k +~4~ k'~ + ,.. t ~ k with complex coefficients ~'I~''" ~ E k , endowed with the complex structure induced by the natur~l embedding into ~ k ~ (~4....,~k)" Define ~ as the intersection of G k , the submsmifold {E 4 = 0 ,
653
is a finite complex. The algebra ~ is the uniform closure on of polynomials in Z~,,,,Z k and ~ consists of all functions in B with an appropriate directional derivative at an appropriate point equal to zero. With the parameters properly chosen, (A,B) is a pair we are looking for (the proof uses the fact that the set of holomorphic functions on an algebraic manifold which do not take values 0 and S is finite, as well as some elementary facts of Morse theory and Montel theory of normal families that enable to control the Galois group). Do there exist examples of the same nature with
A
weakl~al-
~ebraicall~ closed? We do not even know any example in which A is weakly algebraically closed and C ( M A) is not. A refinement of the construction described in ~4~ and [ 5B may turn out to be sufficient. If X is an arbitrary compact space such that the division by 6 is possible in H ~ ( ~ Z ) then all separable polynomials of degree 3 are reducible over C(~) . The situation is more complicated for polynomials of degree 4: there exists a metrizable compact space of dimension two such that ~ ~ ~ ) = 0 but some separable polynomial of degree 4 is irreducible over C (~) [6~. on the other hand, the condition that all elements of ~ (~,~) are divisible by ~! is necessary and sufficient for all separable polynomials of degree ~~ to be completely reducible, provided ~ is a homogeneous space of a connected compact group (and in some other cases). These type's results are of interest, e.g., for the investigation of polynomials with almost periodic coefficients. Is it possible to describe
"al!" spaces
~
(mot necessaril Y
~om~ct) for which the problem of compl~$e reducibi~t~ oTer
C(X)
of the separable pol.ynomials can be solved in terms of one-dimensional cohomolo~ies? In particular, is the condition
~
(~,2) = 0
sufficient in the case of a (com~act~ hqmogeneous space of a oommected Lie ~roup?
(Note that the answer is affirmative for the homo-
geneous spaces of c o m p 1 e x Lie groups and for the polynomials with h o 1 o m o r p h i c coefficients ~9~). Though the question of complete reducibility of separable pmlynemials in its full generality seems to be transcendental, there is
654 an encouraging classical model, i.e. the polynomials with holomorphic coefficients on Stein (in particular algebraic) manifolds. Note that the kmown sufficient conditions t9] for holomorphic polynomials are essentally weaker than in general case. The peculiarity of holomorphic function algebras is revealed in a very simple situation. Consider the union of ~ copies of the annualus ~ Z : ~-4 < I~ ~ < ~ 1 identified at the point ~ = ~ . It can be shown that a separable polynomial of prime degree ~ with coefficients holomorphic on these space, and with discriminant ~=~ is reducible if ~ >, ~0 (~, ~) , primarity of Yv being essential for Y~>/~ [I0]. If ~=~ , ~v can be arbitrary [2], and we denote by ~0(~) the corresponding least possible constant. Now if is even then ~o(~) = ~ , and so the holomorphity assumption is superfluous. However ~o ( ~ ) ~ C ( ~ ~ if ~ and ~ are odd, with C ( k ) ~ for k ~ 5 . At the same time ~o(~) -< C ~ for all • . These results, as well as the fact that ~o(?)~/P--.~ as p tends to infinity along the set of prime numbers, have been proved in [I0~. Nevertheless the exact asymptotic of ~o(p) re-mains ~ o w n ,
it is ur...known even whethe r
~@ (p)--~ oo
a..ss ~-~oo
.
If ~ is a finite cell complex with H 4 ( ~ , ~ ) = 0 then each completely reducible separable polynomial over C (~) is homotopic in the class of all such polynomials to one with constant coeffinie n t s (the reason is that ~'~$CG):0 for ~>~ ). Let X:ivIA and consider a polynomial completely reducible over ~ . Is it p ossib!e to realize the homotop.y within the class of pQlynomials over
~ ?
Such a possibility is equivalent, as a matter of fact, to 13dthe holomorphic contractibility of the universal covering space ~ for ~ . It is known [li] that ~ : C ~ ~ V ~-'k , ~-~ being a bounded domain of holomorphy in C~ homeomorphic to a cell [12]. In ~ there are contractible but non-holomorphically contractible domains [12~, though examples of bounded domains of such a sort seem to be ,~n~own (that m i ~ t be an additional reason to study the above question). Evidently ~5 = ~£ x ~ is holomorphically contractible. Is the same true for
~
with
~
?
There are some reasons to consider also transcendental equations ~W)=0 , where IRA-* ~ is a Lorch holomorphic mapping (i.e. ~ is Fr~chet differentiable and its derivative is an opera-
655
tor of multiplication by an element of A ). In [13] the cases when equations of this form reduce to albebraic ones have been treated (in this sence the standard implicit function theorem is nothing but a reduction to a linear equation). A systematic investigation of such trancendental equations is likely to be important. This might require to invent various classes of Artin braids with an infinite set of threads.
REFERENCES I. r o p ~ R E.A., ~ E H B.A. Aaredpa~ecE~e ypaBHeHY~I C Henpepm~G~ Eos~eHTaM~ E HeEoTopHe Bonpocw aare6pam~ecEo~ Teopn~ Eoc. -MaTeM.cd., I969, 78, 4, 579-610. 2. r o p ~ H E.A., ~ ~ H B.A. 0 cenapadex~m~x n o ~ o M a x Ha~ Eom~yTaTZBm~M~ daaaxoB~M~ axre6paM~. -~oEa.AH CCCP, 1974, 218,
3, 505-508. 3. r o p n H E.A. ro~oMop~HHe ~ y ~ E L ~ ~a ax2edpa~ecEoM MHOrOOd-p a s ~ ~ IIp~IBO~MOCT]~ ceHapade~H~x n~n~OMOB Ha~ HeEoTOp~M~ EOM-~ . ~ y T a T I ~ d a H a x o B ~ a~redps~. - B EH.: Tes~cH ~OF~.7-~ BceCO~BHO~ TOH.I{OH~., MI~HCE, 1977, 55. 4. r o p E H E.A., K a p a x a H ~ H M.H. HecEoa~Eo saMeqaHN~ od ~ r e 6 p a x HelIpepRBHMX ~ y H I ~ Ha ~ O K ~ H O C M S H O M EOMIIaETe. - B m~.: Tes~cH ~oF~. 7-~ Bceco~sHO~ TOII.KOH~., MHHCE, 1977, 56. 5. K a p a x a H a H ~.H. 0d a~edpax Henpep~BHMX ~ y ~ E L ~ Ha ~O-ESJIBHO C M S H O M EOMIIaETe. -- ~HEI~.aHaJI. E ero n p ~ . , 1978, 12, 2, 93-94. 6. J~ ~ H B.A. 0 IIOX~HOMaX ~eTBepTo~ CTelIeHI~ Ha~ a~redpo~ Henpep~mm~x @ym~n~. - ~ m : ~ . a ~ s . ~ . ~ ero r r p ~ . , 1974, 8, 4, 89-90. 7. 3 ~o s E H D.B. A~e6pa~ec~'~e ypa~Rem~:~ c Henpep~mm~m EOS~H-LV~eHTaM~ Ha O~H0pO~H~X npocTpaHcTBaX.- BecTHnE M~Y, oep.MaT.Mex., 1972, ~ I, 51-53. 8. 8 ~ s ~ H D.B., ~I ~ H B,~, HepasBe~B~eHH~e axredpa~ecE~e p a c ~ p e H ~ EOMMyTaTEBHRX daHaXOBMX a~redp. - f~aTeM.c6., 1973, 91, 3, 402,-420. 9. /l ~ H B.~I. AJmeOpo~m~e (~yHEs~a H roaoMop~m~e SaeMeHTH ZDMO-mon~ecE~x r10ynn EOM~e~c~oro M~o~oodpas~. -~oEx.AH CCCP, 1971, 201, I, 28-31. I0. 8 I0 3 I~ H ~0.B. HenpHBo~w~e cenapade~H~e nom~HOM~ c rO~OMOIX~-HBMH E O S ~ I ~ e H T a M I ~ Ha HeEoTOpOM I¢~lacce EOMII~eEOHRX IIpOCTpaHOTB° -MaTeM.Cd., 1977, 102, 4, 159--591.
656
II. K a ~ ~ M a H W.H. ro~oMop~Ha~ y H ~ e p c a : ~ a a ~ H ~ H B ~ npocTpaHCTBa nO~G~IOMOB des EpaTHRX EOpHe~. -- ~ . aHaJI. E ere n p ~ . , 1975, 9, I, 71. 12. H i r c h o w i t z A. A p r o p o s de principe d'0ka.- C.R.Aca~. sci. Paris, 1971, 272, ATS2-A794. IS. r o p ~ E.A., CaH~ e c Eapxoc @ep ~a~e c. 0 ~pa~c~e~eHm:~x ypaBHeRm~X B z o ~ a T ~ B H ~ X 6a~axoB~x a~edpax. -~y~.aRa~. ~ ePo ~ p ~ . , I977, II, I, 63-64. E.A.GORIN (E,A.IDPMH)
CCCP, 117284, MOcEBa ~ e ~ H c E E e ropH MOCEOBCE~rocy~apCTBeH~ YHHBepCZTeT Mexam~o--MaTeMaT~ecE~$BEy~TeT
COW~S~TARY BY THE AUTHOR
Bounded contractible but non-holomorphically main of holomorphy in C ~ have been constructed questions, including that of contractibility of space ~ , seem to rest open. A aetailed exposition of a par~ of ~ 131 can
contractible doin K14]. All other the Teichmuller be found in ~15].
REI~ERENCE S 14. 3 a ~ ~ e H 6 e p r M.r., Jl H H B.~I. 0 rOHOMOp~Ho He CT~:r~BaeM~x o ~ a ~ m : e H H ~ X o6~aCTHX rO~OMOp(~HOCS .-- %oF~.AH CCCP, 1979, 249, ~ 2, 281-285. 15. F e r n ~ n d e z C. S a n c h e z , G o r i n E.A. Variante del teorema de la funcio~n implicita en ~lgebras de Banach conmutativas. - Revista Ciencias Matem~ticas (Univ. de I~ Habana, Cuba), 1983, 3, N I, 77-89.
657 13.9. ola
HOLOMORPHIC MAPPINGS OF S O ~ SPACES CONNECTED WITH ALGEBRAIC ~VNCTIO~S
I. For any integer ~ , = ~ > ~ + ~4A~'~+ . • . + % ~ , and let consider the polynomial p(~) . Then ~ is a polynomial in ~ (~) be the discriminant of ° ,''',~ and the sets G ~ = { ~ :
p
=
uc N { z : z, = o } ,
5G~, = { ~ : ~
= o,
~(z)
are non-singular irreducible affine algebraic manifolds, isomorphic %o G ~ X ~ . The restriction ~ = ~ I~
= t.}
oGp~
being
:
@
G~ " C* == C \ [0} is a locally trivial holomorphic fibering with the fiber S G ~ . These three manifolds play an important role in the theories of algebraic functions and of algebraic equations over function algebras. Each of the manifolds is ~ ( ~4, 4) for its fundamental group ~4 , ~4 (G~) and ~4 CG~) being both isomorphic to the Artin braid group ~(~) with ~ threads and ~4 ( S G ~ ) i being isomorphic to the commutator subgroup of ~(~), denoted B (~) ([I],[~). ~ and ~ p -cohomologies of G~ are k~own [I], [ ~ , [ ~ . However, our knowledge of analytic properties of , ~ ~ G ~ essential for some problems of the theory of algebraic functions is less than satisfactory ( ~ ] - [ I ~ ) . We propose several conjectures concerning holomorphic mappings of Go and ~ G ~ Some of them have arisen (and all have been discussed) on the Seminar of E.A.Gorin and the author on Banach Algebras and Analytic Functions at the Moscow State University. 2. A group homomorphism H - - ~ H ~ is called a b e 1 i a n (reap. i n t e g e r ) if its image is an abelian subgroup of H~ {reap. a subgroup isomo~hic to Z or { 0 } }. For comple~ spaces X and Y , C(X,~)__ Ho~ C X , Y ) and H 0 ~ * C X , Y ) stand for the sets of, respectively, continuous, holomorphic and n o n - c o n s t a n t holomorphic mappings from X to ~ . A mapping ~ C ( o G~)is said to be s p 1 i t t a b 1 e if there is ~ ~ C ~ , ~) such that is homotopie to ~ ° ~ ° ~, ~ ' ~ - - - - ~ * being the standard mapping defined above; ~ is splittable if and only if the induced homomorphism ~, :~(~) ~ ~4 < ~ ) ~4 C G °~) ~ ~(~) is integer. There exists a simple explicit " description of splittable elements of H0~ , G~) [6].
658 CONJECTLhgJ~ I. Let ~ > 4 ~EH0~CG
,~)is
and ~ @ ~
. Then
splittable; (b) H 0 ~ * ( ~ ,
(a) ever~ SE n ) =
2.
It is easy to see that (b) implies (a). Let ~ (~) be the union of four increasing arithmetic progressions with the same difference ~ (~-~) and ~hose first members are According to [6], if and F ~ ( ~ ) then all ~ 0 in H ~ CG~o ,G~)_ are splittable. A complete description of all non-spllttable ~ in H0~(G; ,G~) has been also given in [6]. If ~ >~ and ~ < ~ , there are only trivial homomorphisms from ~/(~) to ~(~) [11]. Thus for such and ~ all elements of C ( ~ , G~) are splittable and all elements of g (S G~ ~ ~ G~) are contractible. The last assertion implies rather easily that H0~*(~G~ ~ ~ G ~ ) ~ ~ . It is proved in [10] that for ~ ~ ~ each # ~ Ho~G~, ~G~) is biholomorphic and has the form ~ (~,...~ ~ ) = ~ 6 2 ~ , 63~3,.. , ~ ) with 6~(~-0 = ~ A useful technical device in the 3, Let C** = ~ \ {0, ~} topic we are discussing is provided by explicit descriptions of all functions ~ E mo~ * (X, C'*) for some algebraic manifolds associated with g~ , ~ and ~ ( [6], [8],[9], [10] ). This has led to the questions and results discussed in this section. Let ~ be the class of all connected non-singular affine algebraic manifolds. For every X ~ the cardi~lity ~(X) of mo~ ( X ~ C ) i s f i n i te (E.A.Gorin).Besides, if "H~>'I¢I,G~ [~.(X),'lJ ("6(X) is the rank of the cohomology group "~ (~) for any d i S t i n c t points ~ ~ • . , , ~,14.,Ig;(~ . Using these two assertions, it is not difficult to prove that, given X ~ and ~>,~ , the set m0~ ~ ~X, ~ ) is finite. In particular, for every ~,~>/~ the set H o ~ ( ~ G ~ , ~G~) is finite. Let Top,X) be the class of all y in ~ homeomorphic to X ; it is plausible that for an,~
H4(X,Z)) thenFlo~,'~(.X,l~\{~,...,}~})
~
~
the function
~:To~ (X)--'~'+
is bounded. I even
do not know any example disproving the following stronger CONJECTURE: there exists a function such that l ~ ( X ) ~ ~)('("(.X).) A function
~: 2 + - " 2 +
~: ~ + - - - - ~ +
for all X {p..jI~ • with ~ < ~ ) ~< ~4 ~ , ~>~.
° ~ ° Hog(G~,X¢), ~ HogCX¢, G~).
CONJECTURE4, (a) Let ~4 ~ I f ~
(~)
every bounded linear operator ~ : A • ~¢ extends to a bounded linear operator from C(~) into ~ ; (E) every bounded linear operator from A into $~ is 2-absolu%ely mla..ing;
( ~ ) f o r every bounded l i n e a r opera~ors I/: ~ r A a n a u : ~ - ~ ~ the composition ~ : ~¢ ~ ~ belongs to the Hilber~-Schmidt class. (a) Por every sequence ( ~ ) in L~(~) Such that (2) there exists a sequence ( ~ )
in ~ ~
such that
for every ,.~, ~ ~ C(.T) ;
(b) for every bounded linear operator ~: A ---~~4 there exists a non-negative finite Borel measure ~
11 ÷11 ' j i sr T
on ~
such %hat
A;
667
every
(c) for every sequence ( ~ ) sequence (3~) in A with
i n ~(r~)
e a t : l . s f y i ~ ('2) and f o r
~ ""i
(d) every bounded linear operator ~" A - - * ~ ~ extends to a bounded linear operator from 0 (T) into ~ ; (e) every bounded linear operator from A into ~i is 2-abslolurely summing, (f) every bounded linear operator from ~ into has absolutely summing adJoint. (A) Every bounded linear operator ~ : A r ~i is Hilber%ian x), (B) For every sequence ( ~ ) in ~(T) s a t i s ~ ' ~ (2) and for every sequence (~k) in A with ~ ¢ ~ I#K(~)I O
such that f o r every positive integer ~,
and e v e r y ~ , ~ , , . . . ~ ~m
,
in
~=I S-eA
.here t~e s ~ ~ for ~=~,~,..., ~.
e=tends for a n seqoenoe. S = ~ % \ ,
.ithS~----+-1
Using the standard technique of absolutely summing opera~ors one can prove i
,iu
,
E) i.e. can be factored through a Hilber% space. - Ed.
668
PROPOSITION I. The f o l l o w i ~
implications hold
>co)
(÷)
>(I) c~)¢7~ CA~--> ( B ) . PROBLEH I. I..ssCa) true? PROBT,k~ 2. I ss (d) true? PROBLEM 3. Is (A) true? PROBLEM 4. I_~s (E) true?
REFERENCES I. D e i b a e n
P.
Weakly compact operators on the disc algebra.
- Journ.cf Algebra,
2. E z c x ~ ~ o B
1977, 45, N 2, 284-294.
C.B.
0d ycxoB~x ~a~op~a-HeTTzCa, nex~z~c~oro
m I~oTeH~m~a. - ~ o ~ . A H CCCP, I975, 225, 6, I252-I255. 3. P e I c z y ~ s k i A. Banach spaces of analytic functions and absolutely summing operators. CBMS, Regional Confer.Ser. in Nath. N 30, AMS, Providence, Rhode Island 1977. 4. K w a p i e n S., P e E c z y ~ s k i A. Remarks on absolutely summing translation invariant operators from the disc algebra and its dual into a Hilbert space. - ~ich.Math.J. 1978, 25, N 2, 173-181. A. PELCZYNSKI
Institute of Mathematics Polish Academy of Sciences ~niadeckich 8, 00-950 Warsaw, Poland
COMMENTARY J.BourgaLu has answered ALL QUESTIONS IN THE A F F I R ~ T I V E . A summary of his main results on the subject with brief indications of the proofs can be found in [5]. The proofs are to appear in " Acta
669
Mathamat ica". Quite recently Bourgain obtained further improvements of his results. So those who are interested in the question have to follow his forthcoming publications. We review here some "Hard Analysis" aspects of this new work. First of all Bo~trgaln has proved that given a positive ~ ( ~ )
there exist
P,IJ, tW)
W ~,~" , ,Ith ~ W .< c t ~
, and a projection
H (~W) satisfying the wsak t~e estimate
~(~).
and which is bounded simultaneously in This leads to a conceptual simplification of the methods used in [5]. Further, Bourgain has proved that any operator mapping a ref• 1 44 I I I 4 I I OO . lexlve subspace of ~ / ~ I to M admits an extension to an operator from ~ / ~ ' to --H°". This "result has an interesting applica-
~®_ there e+is+s
F ~ H®(T + )
with
I F(+,+)+ ++ +~(+)-~l+ (h,
+iT, +:+,+, ....
T REFERENCE t
5. B o u r g a i n J. 0perateurs sommants sur l'alg~bre du disque. - C.R. Acad. Sc. Paris, 1981, 293, S~r I, 677-680.
670
S.2. old
GOLUBEV SERIES AND THE ANALYTICITY ON A CONTINUUM
The collection of all open neighbourhoods of a compact set ~(~C) will be denoted by ~(K) . A function analytic on a set belonging to ~(K) will be called analytic on K . It will be called ~ - a n a 1 y t i c o n ~ (~>0) i£
~ ! ~ ~ $
for every
t,teK.
DEFINITION. A compact set K ( ~ C ) is r e g U I a r if there exists a mapping ~K: ~ + ~ K ) enjoying the following property: for every '5 > 0 and for every# S-analytic on there exist a function ~ analytic in ~K ($) and a set W ,
W ~ u(K)
such that W ~
I~KC~,),
#IW= odlW.
The set
(1)
.-4
S-- {j : j -- 4,£,...1U[o] is not regular. Indeed, putting
< 0
{- (j-'+ (j+1)-I) j-4,~,. ,.
fl
we see that and j but lytic.
~( U $)
is S-analytic on S contains no set where
QUESTION. Is every plane ¢ontinuum(i,e. s#t) regular?
for all ~values of a 1 1 ~ are anaa q0mpact connected
671 This question related to the theory of analytic continuation probably can be reformulated as a problem of the plane topology. Its appearance in the chapter devoted to spaces of analytic functions [the first edition of the collection is meant -Ed.] seems natural because of the following theorem, a by-product of a description of the dual of the space ~ ( ~ ) of a l l fvalctions analytic on K . THEOREM. Let Bore! measure on K e~K A ~\ K
, with
~
a reRular compact set and ~ such that
~(e)=
0
C~(K\e)
~
K
a positive for ever~
• Then every function
~
e
,
anal2tic in
i,s, representable b2 th e f o l l o w i ~ formula
A +,o
(~)~)0
,r,+,-,-.m
beinR a sequence of
"*+
'
L~(~)-functions and
,,++,.,+II'/"" L+(~) = 0
This theorem was proved in [ I ] . The regularity of K leads to a definition of the topology of ~ ( ~ ) explicitly involving convergence radii of germs of functions analytic on ~ . Unfortunately, the regularity assumption was omitted in the statement of the Theorem as given in [1] (though this assumption was essentially msed in the proof - see [I], the beginning of p.125). The compact K was supposed to be nothing but a continuum. A psychological ground (but not an excuse) of this omission is the problem the author was really interested in (and has solved in [I] ), namely, the question put by V.V.Golubev ([2], p.111): is the formula (2) valid for every function ~ analytic in ~ \ ~ provided K is a rectifiable simple arc and ~ is Lebesgue measure (the arclength) on K ? The regularity of a simple arc (and of every 1 o c a 1 1 y - c o n n e c t e d plane compact set) can be proved very easily, see e.g., [ ~ , p.146. The Theorem reappeared in [4] and [ ~ and was generalized to a multidimension~l situation in [ ~ . It was used in [6] as an illustration of a principle in the theory of Hilbert scales.
672
We have not much to add to our QUESTION and to the Theorem. The local-connectedness is not necessary for the regularity: the closure of the graph of the function ~ - ~ ~ , t ~ ~0~ ~] is regular. The definition of the regularity admits a natural multidimensional generalization. A non-regular continu~n in C ~ was constructed in K7]. The regularity is essential for the possibility to ~epresent functions by Golubev series (2): a function analytic in ~ \ S (see (])) and with a simple pole of residue one at every point j-~ (] = ~ , ~ o o , ) is not representable by a series (2). Non-trivial examples of functions analytic off an everywhere discontinuous plane compactum and not
representable by a Golubev series (2) were given in ~8]. REFERENCES
I.
2. S.
X a B ~ H B.H. 0 ~ a2a~or p ~ a ~opaHa. - B ~ , : " H c c ~ e ~ O B 2 ~ no CoBpeMeHH~M npo6~eMaM T e o p ~ ~ y ~ z ~ EOMn~eKcHoro nepeMeHHoro ". M., {~sMaT~s, I96I, I2I-I3I. r o x y 6 e B B.B. 0 ~ H o s H a ~ e a H s J m T ~ e c ~ e ~yHzny~. ABTOMOp~-~ e ~yHKL~H. M., ~ESMaTI~g8, 1961. T p y T H e B B.M. 06 O~HOM a~axore p ~ a ~opaHa ~ ~ MHOr~X E o ~ e E c ~ x nepeMesH~x, r O ~ O M O p ~ Ha C ~ B H O JG~He~Ho BH-nyEm~K M~omec~Bax. - B C6.~rOJIOMOp~HNe ~yHEI~H~ ~g~OISLX E O ~ e E C H R X
4. 8. 6.
nepeMe~"o KpacHo~pcE, H~ CO AH CCCP, 1972, I39-152. B a e r n s t e i n A. II. Representation of holomorphic functions B a e tions M ~ T
by boundary integrals.-Trans. Amer.Math. Soc., 1971, S 69,27-37. r n s t e i n A. If. A representation theorem for funcholomorphic off the real axis. - ibid. ]972,165, 159-165. ~K P ~ H B.C., X e H ~ Z H r.M. ~ [ ~ H e ~ e s a ~ a ~ Eown-
~IeECHOI~O a H ~ s a . 7.
-Ycnexz
Ma~eM.HayE, 1971, 26, 4, 93--152.
Z a m e R. Extendibility, boundedness and sequential convergence in spaces of holomorphic functions. - Pacif.J.Math., 1975, 57, N 2, 619-628.
8.
B ~ T y m z E H A.P. 06 o~Ho~ sa~a~e ~a~xya. - HSB.AH CCCP, cep.MaTeM., 1964, 28, ~ 4, 745--756. V. P. HAVIN
(B.n.XAB~H)
CCCP, 198904, ~eE~Hrps~ HeTepro~, F~6xHoTe~Ha~ n~omaA&, 2 ~eHHHzpa~cE~A rocy~apcTBeEH~ yH~BepcxTeT, MaTe~aT~zo-~4exa~m~ ec E ~ ~aEy~TeT
673
* * *
CO~ENTARY
BY THE AUTHOR
The answer to the above QUESTION is YES. It was given in [9] and [10] . Thus the word "regular" in the statement of the Theorem can be replaced by "connected"
(as was asserted in 11] ).
REFERENCES
9, B a p ~ o ~ o M e e B H8
8 I ~ OK!08CTHOOTB.
A.JI. AHa~aTa~ecKoe n p o ~ o ~ e H ~ e c K O H T g H ~ y ~ -- 3 8 1 ~ C K ~ H S ~ q H . C e M ~ H . ~ 0 ~ , 1981, I13, 2 7 -
40. 10. R o g e r s
J.T.,
Z a m e
W.R. Extension of analytic functions
and the topology in spaces of analytic functions. Math.J.,
1982, 31, N 6, 809-818.
- Indiana Univ.
674 8.3. old
THE VANISHING INTERIOR OF THE SPECTRU~ Let A
and B be complex unital Banaoh algebras and let , then i% is well known that
~4cB
6'ACx ) --,
an~ S ~ ( ~
B,,B(x)
~
(~
,
where ~ ( ~ ; is the spectrum of ~ relative to ~ and ~gA(X) is its boundary. Taking ~ %0 be the unital Banach algebra generated by X , in this context we say that x i s n o n-t r i v i -
a 1
i~
~
~(~
~
*
. ~ilov DJ has proved that if
~ ~.
~-
is permanently singular in ~ (i.e. ~ - X is not inver%ible in any superalgebra ~ of ~ ) if, and only if, ~ - - x is an approximate ~e~ ~sor (AZ~ o~' A i.e. i~ ~ , ~ A , ~ ' ~ , l l = ' ~ , s u c h -~hat,8,,~(A-:~--,-O
(I~, > ~,).
Let ~ ~ ~ ~ 0 ~ ~ ~ 0.
(~ a~})
be a sequence in ~
denotes a sequence of complex numbers) is a Banach algebra
under the norm N~
~ ~ ~ II ~--- E 0
~d
with ~0~---1 and ~t.t~ O ) vanishing on
REFERENCE
3. E a p r a e B -- BecTH~E ~UY
II.II. 0 Hyxsx ~ y ~ , (to appear )
nepzo~ecEHx
B cpe~HeM.
679
S. 5. old
~-BOUNDEDNESS
OF THE OAUCHY INTEGRAL ON LIPSCHiTZ GRAPHS
Let ~ be a real 0@~ -function defined on ~ ,~ in the complex plane defined by the equation~(~)~+i@(~) and
the path (~G~),
(,X~9)(~)&¢~v.p. I V(lh.(.'I+~/,pQ1b)~,'I;
(the Cauchy integral of ~
taken along the graph of ~
). I have
proved D]
II
-
I
where the finite positive function C is defined on an interval E 0, ~), ~ being an absolute positive constant,+~t~0~(~ )~_~_ ~_+e~
(prodded ~ p l~ ~ ~ ) THE PROBT~M is to know whether tion defined .... o n i.e. whether the Lipschitz graphs
in~
t h e
C
can be replaced by a func-
w h o I e
half-llne [%÷oo)
,
~-boundedness of ~T can be proved for a i I (not only for those with the slope not exceed-
). REFERENCE
I. C a i d e r 6 n A.P. Cauchy integrals on Lipschitz curves and related operators. - Proc.Nat.Acad.Sci. USA, 1977, 74, N 4, 1324-I 327. A. P. CALDER6N
The University of Chicago Department of Mathematics 5734 University avenue Chicago, Tllinois 60637 USA
680
*
*
*
C0~TARY The P R O B L ~ (coinciding with problem I of 6. I ) has been solved in [3~ : the Cauch~ integral defines a bounded linear operator in o n .... e v e r ~
Lipschitz .~raph (for a~7 value of its slope). The
proof is based on an estimate of the operator ~ (see a) in ~roblem III of 6.1) with A I ~ ~@"(~) . It is proved in [3] that ~Aall~ c(~+ ~ ) ~ ~i II~ . using results of [2] Guy David found a lucid geometrical description of the class ~ , ~ < p < ~ (we use the notation from 6.2). Associate with every simple curve ~ a maximal function M F ,
'~"~'(O,'t'~]
: ,
where I" I stands for the one dimensional Hausdorff measure. G.David proved ( [4], [ 6]) that
A f t e r t h i s r e s u l t i t seems vez7 probable t ~ t the w e i ~ t OJp d l s o u s sed i n 6.~ , i i i s connected w i t h ~ ? " A lt is proved in [5] that the singular operator with the kernel ( z ~) -uA ( ~ ) II~-~ 4 ~' is continuous in ~(~) whenever Fe ~® ( ~ ")
FI
........
and A: % - ~ m satisfies the Lipschltz condition on ~ . The proof is based on the quoted estimate of the operator A~. The work [5] contains also a proof of the continuit~ of a singular Calder6n-Zyground operator with odd kernel on ~ (~) , U being the graph of a Lipschitz function ~ : ~ > ~. Articles [2] and [6~ show that the more the spaces H + ( U ) and ~ t (0 are close to be orthogonal the more r is close to a straight line (and vice versa). And now one more interesting QUESTION. Which closed Jordan rectifiable curves ~ hav e the ~Oi!,lowin~ DroDerty: all Cauchy integrals of measures on ~ be!on ~ to the Nevanlinna class
N~
i_nn~
~ (i. e. are quotients of bounded func-
tions analytic in I ~ U )? A.B.Aleksandrov has pointed out that no non-Smirnov curve enjoys this property. Moreover if ~ is not a Smirnov curve then there exist i£ ~ (r) and a discrete measure ~ on r whose Cauchy integrals
681
do not belong to ~ r ( ~ is found by a simple closed-graph argument, the existence o f ~ uses some results from ~7]).
REPERENCES 2. C o i f m a n R.R., M e y e r Y. Une g4n~ralisation du th4or~me de Calderon sur l' int4grale de Cauchy. Fourier Analysis, Proc. Sem. E1 Escorial, Spain, June 1979, (Asociaci6n Matem~tica Espa~ola, Madrid, 1980). 3. C o i f m a n R.R., M c I n t o s h A., M e y e r Y. L'int~grale de Cauchy d~finit un op~rateur born@ sur ~ pour les courbes Lipschitziennes. - Ann.Math., 1982, 116, N 2, 361-388. 4. D a v i d G. L'integrale be Cauchy sur les courbes rectifiables. Pr4publlcation Orsay, 1982, 05, N 527. 5. C o i f m a n R,R., D a v i d G., M e y e r Y. La solution • l des conjectures de Calderon. Prepublication Orsay~ 1982, 04, N 526. 6. D a v i d G. Courbes corde-arc et espaces de Hardy g~n4ralis~s.Ann.Inst.2ourier,
1982, 32, N 3, 227-239.
7. A ~ e K o a H ~ p O B A.B. ~ asaxora TeOlOe~ M.l~cca o con!osxess~z ~ s ~ s x ~ ~poczpascrs B.I~.CM~pSOBa E P , 0 < p < ~ . - B 06. "TeoloeS o~el0aTOl0OB e TeOpSS ~ H ~ " , 1983, ~ I, HS~-BO ~rY, 9-20.
682
s 6
SETS OP ~IQUE~ESS F0~ Q C
old By Q $ is meant the space of functions on ~ that belong together with their complex conjugates to ~ C . Here, H is the space of boundary functions on ~ for bounded holomorphic functions in ~ , and C denotes C(T) . It is well known [I3 that ~ C is a closed subalgebra of ~ (of Lebesgue measure on ~ ). Thus, Q ~ is a 0 ~ -suhalgebra of ~ . The functions in ~ C are precisely those that are in ~ and have vanishing mean oscillation ~2~; see ~3~ for further properties. A measurable subset ~ of T is called a s e t o f u n i q u e n e s s for Q $ if only the zero function in ~ 6 vanishes identically on E . The PROBLEM I propose is that of characterizing t h e sets of uniqueness for ~ C
.
There are two extreme possibilities, neither of which can be eliminated on elementary grounds: I. The only sets of uniqueness are the sets of full measure; 2. A set meeting each arc of ~ in a set of positive measure is a set of uniqueness. If possibility I were the case then, in regard to sets of uniqueness, ~ C would resemble i~ , while if possibility 2 were the case it would resemble C . One can, of coUrse, inquire about sets of uniqueness for ~ and for ~oo+ $ . For ~ ~ the answer is classical: any set of positive measure is a set of uniqueness. In view of this, it is quite surprising that, for ~ C , the first of the two extreme possibilities listed above is the case. In fact, S.Axler E4~ has shown that any nonnegative function in [,0° - in particular, any characteristic function - is the modulus of a function in ~oo+ C . Concerning ~ $ , I have been able to rule out only the second of the two extreme possibilities: I can show that there are nonzero functions in ~$ that are supported by closed nowhere dense subsets of T . The construction is too involved to be described here. It suggests to me that the actual state of affairs lies somewhere between the two extreme possibilities. However, I have not yet been able to formulate a plausible conjecture. REFERENCES 1. S a r a s o n D. Algebras of functions on the unit circle. - Bull.Amer.Math. Soc. ~ 1973, 79, 286-299.
683
2. S a r a s o n
D.
l~unctions of van_ishingmean oscillation. -
Trans.Amer.MathoSec.~1975, 207, 391-405. 3. S a r a s o n D. Toeplitz operators with piecewlse quasicontinuous symbols. - Indiana Univ.Math.J.,1977, 26, 817-838, 4. A x 1 e r S. ~actorization of ~ functions. - Ann. of Math., 1977, 106, 567-572. DONALD SARASON
University of California, Dept.Math., Berkeley, California, 94720, USA
CO~NTARY T.H.Wolff~has
BY THE AUTHOR
shown that the only sets of uniqueness f o r ~ C
are the sets of full measure. He did this by establishing the remarkable result that every function in C can be multiplied into ~ C by an outer function i n ~ C . The result says, roughly speaking, that the discontinuities of an arbitrary ~ function form a very small set. Wolff makes the preceding interpretation precise and gives other interesting applications of his result in his paper. REFERENCE 5. W o 1 f f
T H
Two algebras of bounded functions. - Duke Math.J.,
1982, 49, N 2, 321-328. EDITORS' NOTE: S.V.Kisliakov has shown that for every set E c T of positive length there exists a non-zero function ~ in VMO supported on E and such that the Taylor series Z ~(~)~ converges uniformly ~n the closed disc. Let g 0
in B
but ~e ~*= 0
on
This ~ ; ~ would be the Poisson integral of a singular measure. Hence CONJECTURE I is equivalent to CONJECTURE 11 . If ~
is a
positive measure on ~
integral is pluriharmonlc, then ~ t_~o ~
whose Poisso n
cannot be singular with respect
.
Porelli [3~,[4~ has partial results that support the following conjecture (which obviously implies Iz ): CONJECTURE 2. If ~
~s a r e a l measure on ~
, with plurihar..m,,,, ~-
nlc Poisson inteKral, then ~ < < 6 ~ . CONJECTURE 2 leads to some related CONJECTURE 3. If #
HI-problems:
is holomorphic in B
CONJECTURE 3 [. There is ~ constant C ly on the dimen:iQ ~, ~ e A c B )
and ~C~ ~ 0 ,
, C< ~
(dependin~ on-
such that
(~he ~ I I
co~JEcTm~ Y;. I_Zz ~
then
all
al~ebra).
is....h.o.!omo.rphic i n B
, ~=~+
~v , aua
Clearly, 3 ; implies 3, and 3~#is a reformulation of 3 that might be easier to attack. Let N(~)
(~l~%l~Y~
o~ a l l ~ , ~ N ~ tegrable.
be the Nevanlinua class i n ~
i s bounded, as ~--*-~ ), and l e t ~ . ( ~ )
, ~orwhich ~ ~÷1~1|
consist
is u~o==~y
in-
This would imply I z, hence I. CONJECTURE I leads to the problem of finding the extreme points of the unit ball of ~I(~) . (When ~ , these are exactly the
693 outer f u n c t i o n s of norm 1.) Let ~ = { CONJECTURE 5. Ever~ ~ extreme point of ~
~Ht(~):]1~11t~ ~ } .
, ~ 6 H' ( B )
, ~th
II II,= I is
.
It is very easy to see that 5 implies 1. If ~ A ( ~ ) it is known (and easy to prove) that ~(~)---~~(B) . It is tempting to try to extend this to ~ ( E )
: Is it true for ever2 I
that the essential range of
~* on
~
, i~oo(~)
,
is equal to the closure of
An affirmative answer would of course be a much stronger result than CONJECTURE I. To prove it, one would presumably need a more quantitative version of ~ ( ~ ) ~ ( ~ ) . Per example: Does there exist O , 0 > 0
( d e p e n d i ~ onl~ on the dimension
for every ~
,~
A(E)
a holomorphio mapping c~ for almost all ~0
, with ~(0)~-0 , I~I < ~ E-~
, 00e~
CONJECTURE 6. If ~D
~ ) such that
i n n e r
?
Finally, call
if S t ~ c P ( % w ) ~
. is inner, then c~
is one-to.one and onto.
This ~mplies I, as well as the conjecture that every isometry of ~P(B) into HP(~) is actually onto, when ~=4=~ . See [5]. If "~nner" is replaced by "proper", then CONJECTURE 6 is true, as was proved by Alexander [2].
REFERENCES I. A h e r n P.R., R u d i n W. ~actorizations of bounded holomorphic f u n c t i o n s . - Duke Math.J.,1972, 39, 767-777. 2. A I e x a n d e r H. Proper holomorphic mappings in ~ . Indiana Univ.Math.J., 1977, 26, 137-146. 3. F o r e 1 1 i P. Measures whose Poisson integrals are pluriharmonic. - Ill.J.Math.. 1974, 18, 373-388. 4. P o r e 1 1 i P. Measures whose Poisson integrals are pluriharmonic I I . - Ill.J.Math.~1975, 19, 584-592. 5. R u d i n W. hP-isometrics and equimeasurability. - Indiana Univ.Na~h.J.~1976, 25, 215-228. WALTER RUDIN Department of Mathematics University of Wisconsin. 110 Marinette Trail Madison, WI 53705, USA
694
COMMENTARY The existence of non-constant inner functions in the ball of Cm was proved independently (and by different methods) by A.B.Aleksandroy [7] and E.L~w [9] (see also [18]). Both articles are refinements of preceding papers by A.B. Aleksandrov [6] and M. Hakim - N. Sibomy [8] respectively, where the problem has been solved "up to 6 " (but in different senses ). Here are principal results of [7~, [8], [9~. THEOREM I ([7~). Let
~
be a positive lowe r semicontinuous func-
tion on $ , ~ £ ~ (~) . There exists a sin~lar positive measure o_.nn $ such that 9 (~) = I ~ i~ , and the Poisson integral of~-9 S
is pluriharmoni c.
THEOREM 2 ([8]). ~or ever~ continuou s positive function and for every positive number
and ~-~ ~