Complex Analysis and Spectral Theory: Seminar, Leningrad 1979 80 (Lecture Notes in Mathematics 864)

66 is isomorphic to a subspace of ~ . PROOf. Using compactness arguments it follows that .IR.~ ~d can be divided into small disjoint subsets ~ E.}k ~ such that for any function ~ ~ (~@) the follow~inequ~allties hold

where ~ E ~ ,nd / ~ = ~ ~P'~o~, ~ • ~oo~, o~ ~ o ~ 4.~. ~.e~~ ~ ( ~ ) ~ X~ H ~d F~'.]CX ~(~,p] . ~ o m ~

~.7 ±t fonow~ that

X is isomorphic to a~bs~ceof ~ if ~ < ~ p . ~herefore ~ ¢ ~ X ~ * o o . • COROLLARY 4.8. No~imfinite-~mensional Banach space can be imbedded into H~(~ ) as a complemented subspace(0~p<~). Similar but slightly more complicated considerations permit us to deduce from the results of §3 that

EH~(~]~ ~" ~6~CI~

,

~=

A

-~-

A

-~ >o, o< p~¢

Recall that H ~ ( ~ )

aii Z ~ ~ }.

Chapter Pour. PARTIAL SUNS OF POURIER SERIES OP co~!~ous ~c~io~s ~ FouRIER c o ~ F i c I ~ s oF H t ~ o T i o ~ s (0 ~ p ~ ~ §0. Intrqduct!qn Let ~ be an infinite subset of ~ . Associate with M the strictly increasing sequence ~ = [ ~}~0 with

Denote by ~.an operator

~ ~-

~ <~k )} k ~-0

from the

m distribution space ~ (T) into the space of all one-slded sequences of numbers (i.e. sequences indexed by ~ + ) In this chapter we give a complete description of the range ~M(Hh , 0~p<1 , for sets l a o ~ in th~ m ~ - - ~ d ' s sense i.e. with ~ ~I~ , ~ , ~ M, ~ ~ } > ~ . , ~ s des-

cription

is

a simple

consequence

of a result

of

~33~ .

67 Purhter, in this chapter an analogous problem is solved for the spaces H P'°O , O
H

2U oA<

oo

}.

Let o

A-~- +o~

The_ocaseHb~(~,f. +00) cH"~c

~d

=-BM(H ~) ---- ~

is of no interest in view_ ._°f_H%HP'O0 also w e n ~ o ~ that ~ ( H % = ~ M is lac~=~v or even iZ M is a finite it is

union of lacunary sets. The case ~ = ~ is peculiar and the most interesting. As we shall see in §3 for any lacunary ~

The behaviour of partial Fourier sums of bounded (continuous) functions (or what is essentially the same the behaviour of norms of linear oombizustions of the Dirichlet kernels) is the subject of a large number of papers. See e.g. E19,23,26,39,44,

46,47~. ~1 we obtain the estimates of Pourier coefficients of otions ~ H P (or H ~'=~ ), 0
z~ §3

the sets

~--- T

ZMLH: ''~) = TM(C(T))

and R~(H~'~)=

M ([ .~0(T)) are described for lact~ary sets In § 4 we give amalogous results for the space ~ : functions ~ analytic in ~ satisfying

n)

fum-

~. of all

68 §1. Fourier coefficient9 e s t a t e s for HP,~-functions (0< P~<4) ,,,,,,,~,,,~j

,

,,,

.

,

,

,

The following theorem is proved in [4]. Here we give a simpler proof. THEOREM 1 . 1 . Let ~ P , O
PROOF. The case ~-----0 is well known; it follows, for example, from the subharmcnicity of l~lP. We reduce the general case to this particular o n e . . t'l, ~_~ ~'~'~

I t is c l e s r t h s t ~ H ~ . -

.~d ~(~;~K---~O~{O). I t remains to r e -

D~a~A 1.2. Let ~ : H ~v.' ~ 0<, p.<+ co

. Then

I~<~)I~
z,~,,r=~. 1 . 3 . ~e~ ~ P ' ~

, O
• Then

? If

~£E'H ~'00 . %hen

T PROOF. It is sufficient to use the estimates

69

c.ll~ll°e,o~(~-~.)-v~, ~,~ T

~(~,ee ~ . e ~ 1.~)

~.4. ~.et ~ H ~'~ , O< I° < } e. ~he= ~or ~ I ~("~1 ~ ~ "'/~k ~ II~I1,~,~ ~ . H ~ ,°° then J ; ( H , ) = O(ti ,IIP't) (H,,'-"~'I'O0) '

~" ~ o ~

.~IN

Further if PROOF. By Lemma 1,3.

0<~,
now ~ = ( 'l--~ ) . • ~OR~ ~ . ~. ~et M , ~ ~ ~ , , be a lacu~ary set and ~ H ~' ~ . Then :for any I~ F_ ~]~ ,

If

f~

O0

then

k~M~ k~
PROOF. By the Paley theorem (cf, E13] ) and Lemma I .3

'I

Set now

~2. ~

- I , es ,,0~

First prove a lemma. ~,~ 2. I . ~.et ~ C~°(I~)

,

. Then

li, K=_ff, where ~ (SO)~

+ oo ~(~)

~

is the Fourier trausform of ,~-~ for

anyNc~.

70

PR00P. It is sufficient to use the Poisson summation formula

~2-I.e COR0~Y

2.2. For any ~L~ ~

~l-I/p.~@~----~=-~ ~ @~)~K

~
there exists a trigonometric

and]IQ~llLp <<

such that ~ $ ( ~ ~

COROLLARY 2.3. Denote by IP the subspace of ~ consisting of all trigonometric polynomials of degree ~ 11, Let be a linear functional on LP~ . Then

PROOF. Set W~ ( ) =

Of c o u r s e ,

these

o Then

corollaries

Let W = ~' ~/ ~-

are

of interest

Co(W)~OC=~OCK}K)O: ~W~Co~ Tm~,,OKE]Ii2.4. Let

PROOf. The i n c l u s i o n I-~3]. P r o v e t h e

whereOcW--{OCKWKJk>.o"

,

inverse

R~(HI')c

)

±s right invertible.

(F~4-~)

o n e . By l a c u n a r i t y

was proved there

exists

~>0

÷_____L ~ such that ~ + ~ / ~ k > i - 8 . By the 0or.2.2 there exists a sequence ~ . ~ , . _ • of trigonometrical polynomials suc~ that

Set ~ = ~

Then I~ ~ ~ ~ ~~~d

~,Q.~K Z "*~

P~(~)=~,

•

RENLARK2.5. Let 0<. p < l . One can prove that ~/W(HP) = --~-~P( 1"t'~~'~/'P) iff M is a finite . ~ o n of l a c u n ~ r ~ sets.

by (~-~)'d He° (resp. (~.~)-d. CA,O ) the set functions ~ analytic in 9 satisfyl '~-%)@~•=- H O0

Denote

of all

ous~, H P'~ ~ ( ~-~)'~/PH ~ H~_."~ ~(4-~)''/P C~ ~ • ~om~

.

M) M C ~ , be lacu/~B.ry and U
and the operator P~M : HP---~.- ~P(~4 in

o n l y when O

be a positive sequence.

2.6. ~et M

~e ~ c u n ~

McIN ~a"'oL p < ~ . The~

RMC HP'~) -- ~mC( ~-z)u~ H ~) = ~ ~( ~ ¢ R

~

and all these operators ~M are right invertible. PROOF. We prove the first assertion the second being analogous. It is clear that ~,M(('~-'Z)'I/P~°°,) c ~I/M(~P'°°). l~rthermore ~ u ( H P , ~ o ) c ~ P ( ~ / - ~ / P ) ,, by T h . l . 4 . It "" r e , . ~ n. ~ to prove that

again g ~ 0 ~C°0(~)

=~

such that roW,~/rr~w ~. ~ ' , ~(0)------~ ; ~ p p ~ c

L~(-~)~ 1(

a~(%) =

~ c O (~' .~/e)C g , ( ( ~ ' Z )

, Let

"VP H )

E-E, ~e~OO('Flff, f'l/P)

,~k.(~11~. ,~

. Choose

. Take a function ~ - Set ~ ( ~ ) =

. Take

. I t is easy to check (using Lemma

~,M(HP'°°)--

R ~ A R K 2.7. Let 0 < p < ~ . One can prove that = ~ o o ( ~ - f / p ) ( ~ M ( H P , 0 ~ ) = Co(~"l/P)) iff M is a finite u n~on of lacunary sets. On the other hand, the equality holds for lacunary sets only.

43. Sparsely indexed pourier sums of continuous functions A

~.et M be an i n f i n i t e subset 0 # ~ r a t o r T M~ d'CS, { ~H$K~ j k~O

. Denote by ~M fro,,

of all numerical sequences, THEORE~ 3. I. Let p C boO ( T )

L~(~)

the ope-

into the ,pace

• Then

E~C~) ~ ~ H t'°° • I~ ~ C ( ~ )

and ~(~1= 0 I~/0

~'~C~) ~ c H~'~

then ,

PROOP. Again we sb~!l prove the first ~ssertion only. We can suppose ~(K)= ~ ( - ~ ) , ~eN , ~d ~(0)=0 . set

,

0

where 1 ~ ( ~ ) ~ - - ~ K
^ K+ ~ ~(k)% K~O

then A

~eBNO A

~(K)~l(

and}=

s,~),

, Let m

be an arc

72 of T

, centered at 1. One of definitions o f ~ M 0

implies

.< o~(-rj II ~ IIBM o

Z

~(I)

2

SO

and

Z

T Note that~

~,Qr~(#)~I ~ =

(~-Z)-]~(~)

. Now we cam.

0

complete the proof by the following lemma. L~IIIA 3.2. Let ~ L / 4 E O~ ~ ] . Suppose that

° ~22
I~OOl' O~ L

. . .

1~(~->I~#>,41-o(~), ~

A , -- l 0-., ~-.+# ~

~<.~,,,.ptio,, J' l'kl ~ c ~,-

(Az;~g ~

y

. so

A~

~,{-b:A.,,: It~cb#-I: I >A}

-< t,~ [ ~ A l , , ,

.,<--~- A~, [. I~,1~"~

: I~(~Jl,.A.~,'t ~

A~

.

We have C

%~A

THEOREM 3.3. Let M

~A

~A

be a lacunary set and

Them I/1, ~ '1 .

ee

73

3e ~C (~) then

If

PROOF, Use Th,1 5 and Th,3.1 . • Note that the paper ~44] contains results essentially more general than Th.3.3 Denote by ) the set of all sequences { ( ~ &>~0 such that

Denote by ,~ the s h i f t operator in the space of o3.1. one-sided numerical se%uences. Note t h a t gM((¢-~) -~ ~ea) = T M ( m ~0) , because ~M((~-7,)-'I~) = T ~ ~ f o r any ~ H ~ THEOREM 3.4. Let

M

be lacunary. Then

TM(L~°) =TM(H ~°)

--_

t~M(H~,°°)

=

, ,t>00

TMCO)--TM(CA ) --- MCHo ) =

~

©

LE~D~A 3.5. Let O~-[6~k}K~ 0 be a complex sequence, ~ lacmlary set such that ~k<~ ~*? , K ~ + , for some ~ > ~ . Three followi~g assertions are equivalent.

~6 ~+

k= ~

~-~,o0

3) there exists a sequence {~(~)~ ~,o " "~'

~ ~ %"l,t," (-' 95 ~, ?

~'---~o ~

"

PROOF. The i m p l i c a t i o n

2)-->3). L e t %%&~% ~Lo= 0 . Set ~ ~'I

cation

I

such that

v

1 ) ~ ) ~ 2) i s o b v i o u s . Prove t h e l.~y~-i-

~ (~j K=~ ~


Z ~< K < ~ ~ 0

for others

1% remains t o n o t e that OC-~-~ ~____~%~

~=Z ~

,0C(~)~~

I,t,-* -t-o0 . . . .

~

a

~

.

~(~)

;~ w h e r e

and is

the

= backward

shift

74

operator Prove

*

--

now

3)==~I):

and furthermore I g

~ '1~

{t~:~

PROOF OF Theorem 3.4. Again we shall prove the first assertion only. It is clear that T M ( H ~ ) ~ L~ . By Th.3.1 . Fur%her, PI,,~(~/~)~: G"M by Th.3.3. It remains to show that ~ C T ~ ( ~00~ . We can suppose for s i m p l i c i t y that ~ - - ~ '~,K: ~ ' / ~ " . Let @ ~ U Then ( 7 ~ E ~ ~,~ ~(~) , where ~ ( ~ ) ~ ~ and It is well known that g~(CA)-----£ (cf. [58]~). So there exists

TM(L~)~£M(H~,~).

~P.,ll~: Ib,<';'~.

Ib,~

= '3C>'"'

~q,,'=r~,//~J

I"1

.

and a, ( 0 ) - - 0

~"

''~

"

~'~'~A

~

,v~

v~{~ - -

- Let ~'~(%) .... ~-, ,',~ '~.J..~J.,-I< ...... be the FeJer kerne~ and set ~(W~-Q,.(~ ), K = - ~ V ~(~) . It is =~ #t,1,(~)('~-"~) , Then 3 ~ H ~ ~0

because

# 0~ • • coRot.t,Ary3.6. Let E c

Obviously, T ~ ~ f o r some ~ > ~

~,

o Then

TM
(2)

~n ~c~.~Tu(L~,., ) t~(~-" We can obtain the following reinforcement of 0or.3.6.

~Eo~ 3.7. ~ t M, ~ c ~ + , be lac~n~v. ~ o ~ o n o w ~ sertions are equivalent.

as-

I) The equalities (1)(or (2))holds. 2) The set M is a finite ~nlon of sets satisfying the condition of Oor. 3.6. p ~ A R K 3.8. One can prove that if both assertions of Th.3.7

75

TM

hold then all operators ~ M and are right invertible. On the other hand if the assertions of Th.3.7 don't hold then no operator 'I'M involved in the statement of Th.3.4 is right iavertible° To show this we note that the right inverbility of the operator TMI H~(TiJlC^)implies the right invertibility of the opera. Thus the space ~M -~ L ~ / ~ % T M ~or y.~_ c / ~ T , , ) can be imbedded as a complemented subspace M ~ '" M ~© into ~ (or C r. } -)" Hence the space YM ( M ) cannot conspaces L ~ ) ~ t as a uniformly complemented family the of subspaces. Consequently, the equivalent conditions of Th.3.7 hold.

torT,~LL~'(TM"''IC ~)

§4. Sparsely indexed Fourier coefficients of

We can easy deduce the following theorem from the results of ~2 and Ch. III. THEOREM 4.1. Let M c ~ + , 0
One can prove Th.4.1 not using the results of Ch.lll. The case p = ~ is not new: I)-->2) by [32] and 2)~>I) follows from [60]. o eover, there is an log _ o ° oTh.4.1 f (in this case 2) has to be replaced by I ~ , M ( ~ : ) = ~:r(~'~)) . The implication I ) = > 2) was proved in [18] (true, f o r o 6 = ~ only). The case o~ We 4/p can be considered analogously or can be reduced to o~= 4/p by means of the fractional integration. The implication 2)~>I) follows from the results of [60] as in the case p ~---~. RK~ARK 4.2. If the equivalent conditions of Th.4.1 hold then the operator ~ M is right invertible. Purthermore, in the case p~ (in contrast with the case p < ~ ) we can take the operator [ ~K~k~0~---*-k~0 ~k ~ t K as a right inverse. Chapter Five. SOME R ~ A R K S ON THE BACKWARD SHIFT OPERATOR ~0. Introduction

the

One of well known descriptions of the invariant subspaces of backward shift operator ~ H ' H ~,

depends on the notion of the so called pseudocontinuation (see

76 [7]). To recall it let ~% be a function of the Smirnov class NA. The function ~ ' ~ \ T - - - ~ , is said to be the^pseudocontinuation of ~ if I~ = ~ if the restriction ~ I ~is meromorphic with the bounded characteristic (in the sense of Nevanlinna) and if ~ ~(%~) ~ ~ ~(~) a.e. on V . Because~o~the well known uniqueness theorem each function ~ ~ ~ ~ A , cannot have more than one pseudocontinuation. Each inner function Q0 has the pseudocontinuation by the reflection principle : ~'(~) = ~-~-~ )-~~ I ~ ~ ~ ~ Let X be a subset of the class N A " Denote by X ~ N ~ t y set of all functions ~ ~ X ~ aam~tting a pseudocontinuation It is easy to see that for every inner function ~ the intersection H~N ~ N~- ~ ~%t"t ~ H ! is a ~*'invariant subspace ~ = H i) of the space H ~ . Each invariant subspace ~) of the operator ~": H~__~ ~ has such form. Let now 7. be a space of functions analytic in the disc D and let ~ NA . Let's suppose that ~ C Z • We shall say that the invariant subspaces of the operator ~ : ~ ~ are standard iff the lattice of all ~ -invariant subspaoes of ~ (except ~ itself) coincides with ( ~ D ~ is an inner funotion} and ~ ~ : SO~/Ais a constant. Note that last equivalence holds if ~ D ~oo

: •

It is well-known (see [3]) that every nonzero invariant subspace of the shift operator ~:~P--~- H ~ , o < ~ + o s (for ~=~oo the space H P is endowed with the weak-star topology) is representable as ~ H ~ where ~ is an inner function. This implies (by the duality arguments) that ~-invariant subspaces of the spaces H P ( ~ 2 < , o O ) ~ ~ / H L and~MOA are standard It has been noted in ~*.' D5] that the invariant subspaces of the operator H~ H I are also standard. In the paper [17] we shall describe a whole class of locally convex spaces ~ ~NA , with standard ~*-invariant subspaces. In particular, this class includes the spaces H ~ ( ~ * O S ) ~ ~ / H I_

l,i. ~. O I) In t ~ s chapter "subspace" means a closed linear manifold.

77

=

~ ,.~ Jl

'

}

t,1.,.-...~-+,~.~

others.

.

Therefore in this chapter as a rule we shall deal with the operator ~ * on non locally convex spaces. In ~ 3 we shall prove that the invariant subspaces of the operator ~ : H %o ~ --- H ~,ee ~ are standard. Recall that

It is worth noting that

~Linvariant subspaces are not

s t . ~ d iz Z n N~ ~ { o}

, in

cannot be represented as Z N q

aK

fact,

the m~bspace [ O}

wSthsome inner function

o In p a r t i c u l a r t h i s i s the case i f ~ - ~

~ ~

f o r some

~T. In the paper F15] we describe the i n v a r i a n t subspaces of the operator ~ * : H P - ~ - - H P O < p ~ • The description of ~ - i n variant subspaces of the space mPn H~ was of great importance for this. In ~2 (Th.2.1) for a class of separable symmetric ~) metric function spaces X we describe the invariant subspaces of the

operator ~ :

XA--4~ XA containing the m~bspace XA n X ~

.

From that we deduce a complete description of the invariant subspaces of operators ~ :X~ ~" X A for the class

{X: X = XA • X~- , X =/J}. Igoreover, §2 contains a description (in terms of pseudocon-

tin~tlons) o~ noncyclic vectors of the operators ~*: X A - ~ X A for all separable quasinormed symmetric spaces ~ such that X~ ~ 0 . A s~m~lar problem for the spaces X X LP (~p~o) was solved in [30], and for the spaces ----

{0~p<~} in bs]. In ~ 1 we define the maximal functional calculus for the operator ~ % ~ : H P ~ H P ( 0 < p < ] ) , in other words we describe the set of all functions ~ ~ 0 0 , such that the Toeplitz operator

S~

.PC~,S) ~

f ~ H~

can be continuously extended onto the whole space H P as a cont]Luuous map~") . The r e s u l t s of t h e ~lwere announced in 05]. ,, ,, ,,uu ,,,, ,, tt

,

x) See for this subject Ch.l. ~ ) Recall that by P

we denote the Riesz projection.

78 A similar problem for p = I [38], The case

p< ~

it is well

was solved in [55], see also turned cut to be more simple. W h e n ~ < p < + ~

that the operator

is continuous for

every % ~ e H ~ ,

~I. F~ctlonal calculus for the~ o2elrat£r

~*:H P

>

HP(O
I t is clear that the spectrum of the operator ~ :

HF~H P

(0 4 p < ~ ) coincides with the closed unit disc D : each point ~; i ~ ~ is a simple eigenvalue of % ~ (%~(4_~)-I= ~(~_~)-~) and on the other hand for every ~ C ~ the operator I - ~ is invertible, and

It follows that an operator R)~: a P~ HP commutes with the backward shift operator iff there exists a function__ (automatically unique and holomorphic in ~ ) defined in D such that

In this situation we shall write ~/~-----q(~). Conversely, if for a function ~ the operator ~ defined by the last formula admits a (linear and continuous) continuation as an ( H P ~ H P) -operator, we shall write ~ Before stating our results let us recall (see the list of symbols at the begining of the article) that by {j~% }~>0 we denote the standard scale of functions analytic in the disc ~ and smooth up to the boundary. THEOREM 1.1. Let p e ( O, ~ ) and let ~ ' be the backward shift operator in the space H P . Then

(~'*)HP~H P

,,,,,, ,,

m) In the present' section all operators and functionals are supposed to be linear and continuous.

79

n

~aO0~. Lift

,~(~) H PE

HP P

. Co~sider the f.=ctio-

al ~P-~--(~(~ )~)(0) ;, ~ H . I t is clear that O6((4-/~)']) = ~Q A) • Hence ~ / + p - 1 by the duality theorem of P.I>aren, B.Rpmberg and A.Shields F31~. Let now '--'~~I~" . ~ o . that ~(~*)-H P~ H P . #

:By the above mentioned dul~llty there exists a f l l n c t i o n a l c~ Define the operator

~

by the formula

~-W ,or

/

1~(H~HP

check~

it

isis. co.rant c - - c ( p , ~) for every polynomials

~

suffices

to shOW t h a t

there

ex-

such t~,tllu#llH~.
. We have

T

It remains to show that ~ = ~ ( ~ ) . • coRo~ 1 . 2 . T.et O ~ (0 ~ , U ~ / ~ V P -I

=~(A)(~-A~)-

for an-A,

. mhan there

I~-I>'~.

PROOf. The assertion can be reduced to the theorem by the isometry ~ ~'f~(~'4) of the space H P onto H f • • Now we are ~oing to consider an analogue of Th. I. I for the I ( H ~r~ HE ) . The spectrum Of ~ : operator coincides with the circle T ' ( ~ - A~o~)'1~ =

~--~ E*

%~" ~(~) on the circle T

~e~

\ T

. Let

~

be a function

. I f there exists an operator~: .~Pn H e

- ~ H P o H P_ such t~t 1 / ( ( ~ - ~ ) " ) = ~ ( ~ ) ( ~ - ~ ) "

we ~ite ~(EJ)(HPn H_P) = FIP n H P

,~,

. It should be noted that because of Th. 1,0.1 (b)othe operator ~ is uniquelly defined by the function ~ . Obviously ,~or a given operator ~ there exists a function V with = V (~#)

~o~

~.~. ,.st p ~ o , ~ ) ~

~o'

......... I(H ~n

)

80 and let

~

be a function on the circle ~

. Then

(S'~)( 14Pn FI_p ) ~ H ~'n Hf<-~--> ~ ~J~/P-~. PROOF.

Let

tu(~)(HPn H P) c HPn H.~

. Consider

the funct±onaz~CHPnHP)* , ~(,~)----- ( g ( ~ ) £ ) ( O ) . Then ~ ~ Z P - ~ because of Th.~ of D4]" ~/pLet now ~ / ~ - ~ Consider f i r s t t h e c a s e ~ . p . I f we define ~o ~-~- ~ then ~o(HPn HP)~ H.~ ) ~HPn H_P and obviouslz ~0 = ~(~=*). Let now ~ I / p - f In view of 0or. I.2 there e~st~ an operator ~ : H.P----;-- HP. s u c h that ~((4" A~v)"i) ----=~(~)(l-~z)-' for eve~ A , I A I ; , 4 . ~et #0 =

(~)l(H~n

=~I(HPn

Hf)

. Clearly, ~(g~)(HPoHP)cHPnH_P.

To finish the proof it is enough to note that

_,,L'P-'=

..

o,clic ,oc or, of the o,erator S'*:

~ec~l that N ~ THEORE~ 2.1. Let

[ S~ t: : X

~(~÷I~I)~}

be a separable symmetric metric space,

X c N , X = XA * X;~ S,,~ , ~ d ~et E b e ~ invarlant subspace of the operator : XA--~ XA, ~= XA , i f XAFIX~ E ,

then E = X A n ~ N~

for some ~ s r f~ctlo~ ~.

To prove Theorem 2.1 we need a description of invariant sub-

cpaces of the shift operator ~ : ~ ( T ) + X(~)> ~ f ~C~ ~ , where X is a separable symmetric metric space. Por the case X = LP ( 0 < p ~ +oo) such a description can be found in 3]. The general case is quite analogous to the particular one. THEOREM ([3])° Let X be a separable symmetric metric space, C N , and let E be an invariant subspace of the operator E' X. ~ X ~ Then there are followlng two Possibilitiss: I) ~ E , and then E = ~ XA , where ~ Lee

I ~ I = 4 ~.e. 2~ ~ E = E , ,~d then E =

~=~.

,,

~ X

, where ~ ~ L~' ,

The following corollary follows immediately from the above theorem. COROLLARY 2.2. Let X be a separable symmetric metric space, X C N . Set ~ ( s * X ~ ) ~s ~, X~ • Then t h e f~ly [ ~ x~ / X~- ~ , ~s ~ i-,,er function} is the

family of a l l ( =~= XA IXx)~-~invariant subspaces of the spaceX/Xx@

81 PROOF OF THEOREM 2. I. It is enough to describe the invariant

subspaces of the o~e~o~ ~(

~+ XA n ~ )

~:XA/XAnX~

i&~ ~mJ+ XA ~X~

a description let us consider the XA S_ X / ~ K ) ~ ~ ~ ~

~A/XAnX~, . To

obtain such canonical projection and the mapping ~'~, ~ The conditi-

i'~ XA/XA n XX--~X/XA,. }(÷+XanXA)

•

ons of the theorem imply ~] (XA) = XZX'~. ~ K8%~ = XAn X~ and hence the mapping ~ is To finish the proof it is enough to mention commutativity of the following diagram

x/x

;'>

X~

XA/XAn X-A ---,.- XA/XA n COROLLARY 2.3. Let E

an isomorphism. Cor.2.2 and the

@

be an invariant subspace of the ope-

rator ~ : N A ~ N A ~ E = ~ ~A~and let E = NA ~ NA. Then E = NA'O ~ N ~ for some inner function ~ . @ It should be remarked that Th.2.1 contains a complete description of inve~ia~t subspaces of the operator ~ * : X A ~ X A for all separable symmetric metric spaces X with the properties Xh N X~- T~!o}, ~ cz~V e_ndX~- XA + X~• However, i f X A A X~ =~= %v5 . Th•2.1 yields no description even for the collecton of the noncyclic vectors of the operator~*: XA-->-X~. • In conclusion of this section we describe the above mentioned collection for one class of separable symmetric quasinormed spaces X (see cor°2.6 below). Let X be a separable quasinormed space and let

~ / $ E X [ o , ~ ] . Hence X D 2 M O

. With every ~-invariant

subspace E , E ~ X~ we associate an inner function ~ E in the following way. Consider at first the intersection E n n B M O A ~- B M O ^ • Stan~=d ar~ments show that this intersection is a ~*-i~varlant subspace closed in the star-we~k topology. The well kmown description of ~-invariant subspaces of the Hardy space ~I implies that EnRMOA=.~MOAn q~Nx for some inner function ~ (defined by E uniquely up to a

82 multiplicative constant). Set ~ E ~ ~ THEOREM 2.4. Let ~ be a separable symmetric quasinormed space, ~ f ~ / ~ I~ X [02 ~3 ~*. , and let E be an invariant subspace of the operator XA-~X A , F =A ×A" Then E c XA D ~E N~ . Proof of the theorem is based on the following lemma, but in all other details it is completely analogous to the proof of Th.1 in D5]. L ~ A 2.5. Suppose the conditions of Th.2.4 are fulfilled and l e t ~ ! ~ E , k ~ H ~ , (~#). g e h~ (here I~ROOF OF ! , ~ A 2.5. repeats the proof of L e n a 1 i n ~ 5 ] . We have only to note that every sequence of functions bounded in the space B ~ 0 and convergent in meausre converges in fact in the space X. @ COROLLARY 2.6. Suppose the conditions of Th.2.4 are fulfilled and let ~ 6 X A . The function ~ is noncyclic for # t iff ~ possesses a pseudocontinuation. @

§3. !nvariant subspaces of the operator ,,,,

D

, ,,

enote by ~~,00 . the s e t of a l l f , ~ c t i o n s

• ,{I#I:,'A.1---

o(A"~)

"'°(A-,-+~)

S,

~

. Recall

/;o

thatH~,~(~

THEOREM 3.1, Invariant subspaces of the o p e r a t o r ~ - ~ H ~ ' p °'~ are standard, PROOF, Let E be a ~-invariant subspaceE~,,e ~'~. In v±ew of Th.2.4 we have

RMO A n~EN~ cE= N~:~n@AN~ It remains to show that F = H I,~ n ~ A

n ~ E N~ mation ~:T~

N~

. Let # ~

HZ~~n

Let .s cons~er a measure prese~,~@ transfor(0,~1 such that J ~ _ (~÷)~o ~

We recall that ~* denotes the decreasing continuous on the left permutation of the function I~l • Set and define the function ~A: T ~ ~ by the following equalities:

X(AI=

{

A}

83

o>

k(~)~ A(A) ~ t<~___~)k(~)~
~A c~)=

Continue the function ~A into the unit disc by the Poisson integral. being con__jugate harmonic functions, ( ~ ) ( 0 ) = 0 ~ • By Lemma 2.5 P ( ~ ~ ) ~ E for every A ~ 0 .~We shall prove that ~=l~rt'b .~(~'. ~) in the quasinorm of the space H ~0~ . Remark ~hat t F, I ~< ~ , and since ~-~ --

Considerthe outerf=ction Fa=dA*~IA, ~, ~

~It,~= ~ ~o~

=Z~(A)

; 0

forA

~ +co ,wehave

.~

F~=#~

t~

L)

It follows that ~ ~A ~ = ~( in ~ u ~ . Now it is enoug/~ to show that ~ k ~ A"l"l~+~ . ~-~, ~ ) - ~ ' . For this in view A-~+oo A - A -of Kolmogorov theorem it suffices to prove that

A~+~II(FA-FA )~IIc o =

.

~e

have

J~.A

T

J~<.A

ACA) ~.

A(A) _<

,Vs,

i

I o

A(A) &

We remark that the proof of the theorem above is very close to that of Th.6 in E16] (the last theorem deals with the Kolmogorov-Titchmarch notion of E-integral).

Appendix A GENERALIZATION OF THE BOCHNER THEOREM

ON ANALYTIC M~SURES Let ~ be a locally compact abelian group. Denote by the group of all characters of G . Let M A ( @ ) , where

84

A~F , denote the that ~ p p c A property o f ~ and

~

set of a l l m e a s u r e s ~ M ( ~ ) such We shall say that a set A ~ F has .~. Riesz (or is an ~-set) iff

M~ (G) ~ b ( g ) . J.H.Shapiro [54] has found a sufficient condition for a set to have the F. and ~.Riesz property. THEOREM (J.H.Shapiro [54]). Let G be a compact abelian group, A C P , 0
~he f~=ction~ ~--~_ ~ ( ~ ) ( ~ A U ~ ous extension onto ~YA~K~ (C~) .

(~)) Then A

h~s a continuis an R-set.

In ~ it is shown also that this condition gives a new proof of the Bochner theorem. We prove here that a generalization of the Bochner theorem follows from this condition. ~oreover, we give a generalization of the Shapiro theorem for the case of a locally compact abelian group. THEOREM I. Let A ~ + ~ ~ . suppose that for every ~ + the set [ K E ~ : ( ~ , K ) E A } is bounded (from above or from below), Then A is an R-set. PROOP, We use the result of J.H.Shapiro. Let ~ = (~{~ ~$) and let ~ be a trigonometric polynomial on ~ with ~ concentrated onAO[K~. It is clear that for each ~ $ ~ T the following inequality holds

C(k,p)~ Therefore,

,P fls l

Denote by ~ ( ~ ~ ~¢) the measure on ? with ~pv= ~L ( ~ ) ' ) . Then ~ e ~ (T~) if ~ d o~y if ~ ~ U ( T ) for all

~

~+ .

P,OOi,. it is clear that if ~ C [~ (T ~')

then~,~[~ (T)

$5 for all ~ + . Conversely, let ~ C~) for all~/E~+. With each function ~ we associate a trigonometric polynomial p~ __ such that II~,.-D I/ I < ~-~ • r~lll,' . Consider the function ~e~, ~(~)_p~ (~) . It is clear that E~(°~) . It remains to note that ~ - ~ ~ ( T ~) by Th. I. @ Note that Th,2 follows easily from a result of A.V.Buhvalov A.A.DanilevichE25~ about the ~-valued Hardy spaces H P, Analogously one can prove the following THEOREM 3. Let ~ ~ ~I be locally compact abelian groups and let ~ ~ be the dual groups. Suppose that A ~ F is a countable set satisfying the conditions of the Shapiro theorem. Let ~ M(~x~1) . Denote by ~ the measure on ~I with ~ = ~(~') . Then j(b6S~_~(CTxCTI) if and

THEO~

4

. Assume that 61~pp ~ C

~et ~ ~ M ( R )

the mea~;e'L M(I~'~), wit~ $~= ~(~.~. Then~(~'b if ~

o ~ y i f -~,.c ~ ( R ~'~

)

for

].l ~. ~ O.

.--

"I PROOF. It is ""sufficient to prove that ~ ( E )IX, ~£~1 for all 06~0. Denote b y ~ , ( O b > O)

(~i,1~-I)

if

the measure on T ~ defined by the equality ( ~., Q0 ) ~- ( ~ , ~ ( 6 ~ X I ...~ e ~ ~ ) ~ ~C~) " J - ~ ' - " ~ It is easy to see that if ~ ¢ ~ (~) then ~ 6 ~ ~ (T~) for all sufficient large values of 6~ > 0 . It remains to apply Notice that Th.4 implies a result of R.R.Coifman and B.Dahiberg [29~. Now we give without proof a generalization of the Shapiro theorem for the case of locally compact abelian group. THEOREM 5. Let ~ be locally compact abellan group. Denote by ~ the group of all characters on ~- . Let A C C 0<~<~" Suppose that for each K e ~ there exists a neighbourhood of zero in ~ such that the functional ~ ---->-~(K) ( ~ (~u{~))+~ C ~ ) ~ C ~ l ) can be continued onto ~(~U{~})+~(~) as a continuous functional. Then ~ is ~-set. I would like to express my deep gratitude to E.M.Dyn'kin, V.P.Havin, N.G.Makarov, N.K.Nikolskii and V.V.Peller for translating into English this article and for a great number of helpful suggestions and corrections. I am indebted especially to V.P.Havin for rea(Ling the Russian

86 and English versions. Also I wish to thank S.V.Kisljakov

and S.A.Vinogradov

for

helpful conversations.

References I. B a r b e y

K., K B n i g

H. Abstract analytic function

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B u s k o

E.

Fractions

adh~rentes darts L ~ < T )

continues

et

f~uction-

born4es

~ la suite de leurs s o m m e s

non

parti-

elles de Fourier. Studia Math., 1970,4 34, 319-337. L. Two remsrks on ~ and B M O . Adv.

27. C a r I e s o n

Math., 1976, 22, 269-277. ,,p 28. C o i f m a n R.R. A real variable characterizstion of H . Studla Math., 1974, 51, N 3, 269-274. 29. C o i f m a n R.R., D a h 1 b e r g B. Singular integral characterizations of nonisotropic H ~ spaces and P. and M.Riesz theorem. Proceedings of Syaposi~ in Pure Mathematics, 1979, 35, Part I, 231-234. 30. D o u g 1 a s R.G., S h a p i r o H.S., S h i • 1 d s A.L. Cyclic vectors and invariant subspaces for the backward shift operator. Ann.Inst.Fourier, 1970, 20, N I, 37-76.

88

31. D u r e n

P.L.,

R o m b e r g

Linear functionals on

H -P

S h i e 1 d s

B.W.,

A.L.

spaces with

reine u n d angew Math., 1969, 238, 32-60. e n P.L., S h i e I d s A.L. Properties of " ~ 32. D u r < 4 ) and its containing Banach space. Trans.Amer.

(o<

Math.

Soc., 1969, 141, 255-262. 33. D u r e n P.L., S h i e i d s of H P and ~ P

A.L. Coefficient multipliers

spaces. Pac.J.Math.,

34. F e f f e r m a n

C.,

1970, 32, N I, 69-78.

S t e i n

E.L

U P

spaces of se-

veral variables. Acta Math., 1972, 129, 137-193. 35. G i 1 b • r t J°E. Niki~in - Stein theory and factorization with applications.

Proceedings of Symposia in Pure Mathema-

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J.Eo Some properties

of fractional integral II. Math.Z, 1931/32, 34, N 3, 403-439. 37. X a B ~ H B.H. 06 a H a : m T ~ e c z a x ~yHEnZaX, Hpe~CTaBEMHX ~RTerpaaoM K O n m - C T ~ T ~ e c a .

BeCTH.ZeHEHrp.yH-Ta,

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S. On functions with conditions on the mean os-

cillation. Ark.Math., 1976, 14, 189-196. 39. K a h a n e J.-P., K a t z n e 1 s o n

/

o

Y. Serles de Pou-

/

rier des fonctions born@es . Prepubllcations, Paris-Sud, I978. 40. K a 1 t o n N.J.,

S h a p i r o

Universit@

J.H. An

~-space

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~(- ~)

in terms

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K. The failure of spectral analysis in

0<

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with

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J.,

de

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1976, 82, N I, 111-114.

P e • c z y ~ s k i

A. Con-

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1979, 288, N 22, AIO09-AI011.

/

45. M a u r e y

B. Theor~mes

de factorizatlon p o ~

teurs lin4aires ~ valueurs dans les espaces

les op4ra-

~, . Ast~risque,

1974, N 11. 46. 0 C E 0 ~ ~ O B

K.H. 0~eHEa n p H O ~ e H E a

no~nocae~OBaTe~HOCT~D CTeEaoBa,

cyMM ~yp~e. T p y ~

Henpep~BHO~ MaT.HH--Ta ~M.B.A.

I975, I84, 240-258.

47. 0 C E 0 a E O B

K.M. Hocae~OBaTea~HOCT~ HOpM CyMM

~fp~e

oz-

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Math.Soc., 1968, 74, 526-528, R u d i n W., The inner function problem in balls. 8an.Ha~.ce~HapoB ~ 0 ~ , I978, 81, 278-280. s h a p i r e J.H. Remarks on ~ -spaces of analytic functions. Lecture Notes, 1977, 604, 107-124o S h a p i r o J.H. Subspaces of L P c @ ) spanned by characters: 0 < p < ~ . Isr.J.Math., 1978, 29, N 2-3, 248-264. S t e g e n g a D. Bounded Toeplitz operators on ~I and applications of the duality between ~I and the functions of bounded mean oscillation. Amer.J.~ath., 1976, 98, 573-598.

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E.M.Dyn'kin

THE RATE OF POLYNOMIAL APPROXIMATION IN THE COMPLEX DOMAIN Introduction. I. Preliminaries. 2. Muckenhoupt condition and outer functions. 3. Estimates of potentials. 4. Best approximations. 5. Moduli of smoothness. 6. Faber operators. 7. Approximation of Cauchy kernel. 8. Area approximation. 9. Special cases and generalizations. 10. Some unsolved problems.

Introduction

The classical subject of the approximation theory is the connection of the rate of polynomial approximation of a given function with its local smoothness. In the complex domain this connection depends on geometrical characteristics of the boundary of a region where the approximation problem is considered. The aim of this paper is a description of the complicated interaction between these three factors. Let ~ be a function analytic in a plane region G • Usually, the rate of its polynomial approximation is measured by the sequence ~ E ~ (5) ~ ~ of the best polynomial approximations in some sense (uniform, weighted, in the mean etc). The modull of smoothness ~ (5 ~ ~) , ~ 0 , of different kinds (uniform, weighted, in the mean ... ) are used to measure its local smoothness. Geometrical properties of the boundary ~ G are expressed in terms of the conformal mapping ~ of the exterior of G onto the exterior of the unit disk. The geometry of regions with smooth boundary is essentially the same as that of the disk (0 ~ 0 4 ~l~fl < C ~ < oO ) and the standard form of Jackson-Bernstein theorem may be transfer-

91 red to these regions without modifications Alper

~ , 2~

). If the region has corners, uniform approxima-

tions and homogeneous

conditions of Lipschitz type become not

adjusted one to another. cated

(see for example S.Ya.

The whole situation

becomes compli-

and there are new effects in the case of

~-approximati-

on on the boundary or in the region. The origin of the theory, which is the subject of our paper, can be found in W.K.Dzyadyk's papers r 3 - 5 ~

, where he

obtai-

ned a constructive characterization of the class of analytic functions satisfying a Lipschitz condition of given order in a piecewise smooth region

~

° Now the theory is complete at least for

the regions with Lipschitz boundaries.

This paper contains its

systematic exposition. We have no place to describe the ~ i s t o r y of the question.Some historical comments are contained in ~ 9

and 10. Here we only

note the essential contribution into the developement of the theory by W.K.Dzyadyk with co-authors Tamrazov and N.A.Shirokov ~15,

16~

r8-14~

E3-7] , N.A.Lebedev, P.M. , V.I.Belyi and V. M. ~ k l j u k o v

, T.Kovari and Ch.Pommerenke

lius E19~ ; see also the author's papers

E17,

181

and T.H.Gane-

E20-25~

. For further

historical information we send a reader to cited works and also to surveys L26, 12, 27] . In

§I of the present paper the main facts concerning func-

tion spaces, classes of regions and conformal mappings are collected. In

§ 2 outer functions in the sense of Beurling

E28~

are

studied, whose boundary moduli satisfy well-known ~lucKenhoupt condition ~30, 3fl . It appears, that on such a way one can give real-variable proofs of main properties of conformal mappings and their level curves. From the geometric function theory we use only distortion theorems of Koebe and Lavrentiev ~32, 33~ • 3 is r a t h ~ t e c h n i c a l potentials. In

and contains some estimates of Cauchy

~ 4 we introduce the main approximation characteristics

of functions~

best weighted approximations.

The weight may be

any function on the boundary with the logarithm of bounded mean oscillation. All standard weightS satisfy this condition.

Further,

we connect best approximations with pseudoanalytic continuation of functions L34, 21, 2 ~ , which gives an appropriate intermediate language. In

§ 5 we introduce moduli of smoothness for functions of

92 _ -L P (~G) " Their main difference f r o m usual ones is the variable step, defined by conformal mapping ~ . Further, we connect moduli of smoothness with pmeudoanalytic c o n t i n u a t i o n and get an "inverse" theorem of approximation theory which gives an upper estimate of the moduli of smoothness of a function by its best approximations. For the proof of "direct" theorems, i.e. for the estimation of best approximations by moduli of smoothness, we need some constructions of polynomial approximants. At present one knows two such c o n s t r u c t i o n s . An "elementary" construction of approximants by means of ~aber operators was proposed independently by T.Kovari Da~, T.M. Ganelius [19] and the author [20~ for the uniform approximation. It is based on a deep result of Ch. Pommerenke and T.Kovari ~7~ • Another c o n s t r u c t i o n , m o r e complicated but m o r e powerful, including approximation of Cauohy kernel, was proposed by W.K.Dzyadyk E3, 4~ andros developed further by N.A.Lebedev and N.A.Shirokov E9, 11~ . In §6 we expose the theory of ~aber operators and completely describe the connection of best approximations with moduli of smoothness in the case of slowly varying weights, in particular for the uniform approximation and weight.

L p -approximation without

In ~7 we expose the polynomial approximation of the Cauchy kernel and obtain the final result for general weights. In # 8 we study the area approximation.The results of § 8 were announced by the author ofs for the first time.

[233

but are published here with pro-

In §9 we formulate the general theory of §§4-8 in particular forms for standard weights. We discuss a constructive characterization of Lipschitz classes, uniform approximation problem, approximation in the mean etc. At last, in §10 we discuss some unsolved problems of the complex approximation theory. In the present paper the recent papers by V~I.Belyi, VoV. Andrievsky and P.M.Tamrazov [16, 35-373 are briefly mentioned. They have extended an important part of the theory to regions with quasiconformal, may be nonrectifiable,bo~idaries. The author is not a specialist in this domain and sends the reader to papers [16, 35-37~ and to V.I.Belyi's monograph ~aJ In conclusion,the author expresses his deep gratitude to N . ~ Shirokov, L.I.Potepun and V.I.Belyi for stimulating discussions and to N.K.Nikolsky and V.P.Havin for their p e r m a n e n t support of

93 this work. List of sEmbols. Every symbol is accompanied by the number of section containing its definition. The notation A.B means subsection B. in section A. Throughout the paper C and C denote various constants; @denotes the end of proof.

Ap(F) 1.9

I(~) 5.2

~(F) 1.9

K(~,~) 1.1

~*

1.5

B MO([~. 9

K~p(Rw) 4. i K~p~(6,w)9.5

~^

i. 5

1.6

LP(F, W) 1.2

~ r

L~PA(p,~)I.3

~

1.6

~

8.1

C ~.~ CACB) 1.2 C~ (B) 1.2

Wz

9.1

1.4

D~(#,I) 5.~

M÷

1.9

~nc({~I~WJD5. I

2. I

EP(cT') 1.2

Mp pr

1.1

J#(%, E ) 1 . 1

E P(~W) 1"2

~Z ~

5.1

~"p(~,~,w) 4.3

~ ~(~ ) Io. 1

F~({,W)p4 . 1

Q(w) 2.4

~

1.5

E,,(.J:,W)p4.1

~(6)

I~

1.5

S~(~W)p 8. I

~p,w 6.1

Oj~¢(#,~ W)p 5.3

e~(T)p 1.3 #+; #_ 1.8

T 1.1 Tp,w 6.1

~(T,g,W)p 5.3

{+~

1.9

G

1.4

H~(~, ) 7 . ~

2.2

i.~l.,..(#,I)

9.1

~1,Te 0 6.1

IZl

1.1

V~ir

2.2

~;F

1.1

w~,w~

2.2

II#IIx

~.2

Wm

2.4

§ I. Preliminaries. 1.1 Some notations.

~s the plane of the complex variable ~ =

~=~+~

X+ ~

or

94 = { ~ , I ~'1 < ~ } i s the open u n i t d i s k . T = { ~ , I% I = ~ } i s the u n i t c i r c l e . . ~ ( ~ E )is the distance from a point % to a set E

• and

~(~,%)={~I~-~I< ~ } is the disk with the centre the radius ~ . If{ is the length of the arc ~#~__ ~ / ~ - ~ is the conjugate exponent.

~#~:~)-~/~,(~÷/~tx

+ ~, ~/~V~)

is the Cauchy-Riemann deri-

vative. I. 2. ~Mnction spaces. II~IIX is the norm of the function in the Banach space X be a rectifiable Jordan curve, W a nonnegative Let F function on F ~W)= [ ~ ~W~(F)} is a Banach space with the norm ll~llL~(r, w)__ ll÷wllL~(r) " Let

be a plane region bounded by a simple rectifiable

curve F

EPC~)

, O
of anal,tio functions wit~ Let W

For ~ p ~

, is the Smirnov class

norm H~II~p,~=

be a nonnegative function on

H~HL~F~

[28, 2~

,

F with W ' ~ E L P (F).

we set

EPCG,w)={#eE~(CT)

, #w~LP(F )~ with the

norm

II~WIILP(F) . if G = D then E P ( G ) ~ - ~ m is the standard Hardy class [28~ . If ~ is a plane compact, then ~ A ( ~ ) is the space of all continious functions on ~ analytic in the interior o f ~ with the sup~emum norm. Its Lipschitz subspace ~ ( ~ ) , 0 < b < ~ , is defined by

CA (B) = I ~, S~CACB)l~¢~)-~
and SEN P LP -modulus of , M~=~... is defined by the formula

Am÷~

95

Let

~>0

. We d e f i n e the c l a s s

L pA{p, )

on

B

by

P,

l$pA (p,~)

The functions of of order ~ in the mean.

satisfy the Lipsohitz condition

Let us define the best approximations of a function

~

,

tl P : the infimum is taken over all polynomials It is well-known that [39, 40]

~

of degree

~

.

(1.t) 1.4. Classes of regions. ~or this section see I.I.Daniljuk Let curve F Lipschitz curve F XE~

~

[41]

.

be a plane region bounded by a simple rectifiable • ~ will be called a Lipschitz region (and ~ a curve) if in a neighbourhood of any point ~ e F the can be defined by the equation ~=~(X) ~

~

I

being an interval, where

clx-x'l Such a curve

F

,

x, x[ET

(1.2)

has a tangent in almostevery p o i n t ~

~

; we

denote the slope of this tangent by % ( ~ ) . will be called a Radon region ( ~ a Radon curve) if this function ~ has bounded variation on ~ and moduli of all its ~umps are less than ~ . REMARK. If ~ is of bounded variation but some of its jumps are of modulus

~U

then

~

will be called a region of

bounded rotation. EXAMPLES. (~) All convex regions are Radon regions. (~) A region with a piecewise smooth boundary without edges is Radon region, but with an edge on the boundary it im ef bounded rotation only. ($%~) All Radon regions are Lipschitz regions, but not vice vers~. 1.5. Conformal mappings. For a Lipschitz region ~:C\~-~\ ~ ) > 0

~

we shall denote by

@

,

~ , the conformal homeomorphism such that ~ ( ~ ) = o O , , and by ~ its inverse mapping. F is a quasicon-

96 formal curve ~ because the length of any arc of ~ is commensurable with its diameter. Then the mappings ~ and ~ admit a continuation to quasiconformal automorphisms of the whole plane. By Lavrentiev theorem ~ and ~ hawe the so-called ~ - p r o perty [33] : the image of any circle lies in a ring with a # bounded ratio of radii. The ~ -quasiconformal reflection [42] with respect to ~ will be denoted Z ~ Z * . On the other hand, boundary correspondences of ~ and T by ~ and are absolutely continious and ~# and ~ are outer functions in the sense of Beurling [28, 29] , in C \ G and C \ D • Now we remind some consequences of the ~ -property and Koebe distorsion theorem.

~) ~or ~ e C \ G

¼1 ~,~)1 ~,P(<'P(~,? ) / f c~, F) ~ 4 I'~r(~) I, (c{) sot ~ set

^

~ ~

(

C \ F~

~.)^

(1.3)

set 4-- ~(m(~)llm(~)l ) and for 1~e C~ ^

. The point ~ i s not the nearest to

~

on

I~-~I ~f(~,g)

, but (~) A point ~ arc I ~ ~ (and

C\ T

~ will be called reciprocal to the will be called reciprocal to ~ ) if

J( ~,I )
( 04 and C~ are constant in each discussion). Thus, if ~ is reciprocal to ~ then @(Z) is reciprocal to ~ ( T ) and vice versa. Further, we h~ve

I ~oQ~I

~ I <,P(I)l/II1

-

III'f

-rI I~'1 •

(1.4)

1.6. Level curves. Let Z ~ \ ~ The level curve ~ By the

Let

, %70 . Set Z ~ = ~ I ( ~ + % ) ~ ( Z ) ~ . is defined by ~ = I Z T ~ } _ - - { Z ~ I ~ ( Z ) I = ~ % } "

~ -property,for ~ C ~

F

be a function on

we have

~

. Then

97 eo

C~G where tion

I~(~)1~

e

&

~ . T h i s f o r m u l a i s y i e l d e d by a s i m p l e combina-

o f t h e change o f v a r i a b l e

% ~

~(~)

, o f the a p p l i c a t i o n

of polar coordinates and of the inverse change of variable. By the quasiconformal reflection we get for any function F on oo

I .7. Carleson embedding theorem. For ~ C ~ I(~) , $5~ ~

I

We need t h e f o l l o w i n g

variant

, set

o f C a r l e s o n embedding theorem ( t h e

proof is standard F43~ ). THEORE~'I. Let J~ be a measure in G following two assertions are equivalent.

cllsil#(r)

and

~<~OO

, ;r LP(r-)

. The

•-

Such a measure will be called a Carleson measure. The length on some curve ~ is a Carleson measure with respect to auay region, if the length of the portion of ~ in any disk of radius is less than 6 ~ . 1.8. Cauchy type integrals. Let ~ ~i ( ~ ) , r be a Lipschitz curve. Two functions

F are analytic in ~ and C \ ~ respectively. Further, if s ~ P ( g ) , ~ ~ p~
98

long to

EP

and

11~4E~(e+ll~-II~a~,,)~~11~11~(~)

•

•

Recently A.P.Oalderen [44] has proved such an estimate for some Lipschitz regions, but for general Lipschitz regions this problem is open. 1.9. ~uckenhoupt condition. Let ~ be a Lipschitz curve, W be a function on F , W ~ 0 . We shall say that W satisfies ~uckenhoupt condition (Ap) , 4
Z

I

The whole theory of ~0, 31] can be transferred to the Lipschi~ curves without changes. In particular, if w ~ A p C F ) , then

w~EI CAP~F(F) and

are

I w(~) P F,,I

for some

~el

o~>~

F

, f(~

PI< ~

~ I )~$1II

IZI P

and further

c~.9)

Z

then

i~aRadonregion

zf

,

i~_~lp I~1 ~ c(~) ~ wP.

For le L~(F)

~f w~ApCF)

, and for an

II M~IIG(F,,~ ).
e

aria W e A p ( F )

then

D] (1.1o)

II J:: IILP(F,w ) ..< c I1~11Lp(r~ ~) where

Finally, we define a class

ACF)--{ w, w~e A~(F) One can prove like in [30, 31] ~4:~ W~ B M O ( F ) , where

BMO(F)={~, ~z*P~ z

A (F)

of regular weights

for some~>O,~
- II---/z

iff

99

In particular, if W I

w~/w~

AC[)

,

W~(~)

,

and

then W { ~ $ ~ ( D

•

2. Muckenhoupt condition and outer functions. This section

is an account of some of results of the

author's paper [4~] 2. I. Muckenhoupt mappings.

=et ~ and % be ~ipschit~ ourves and ~ppose ~ : ~ -~r~ an~ ~ _ - < ' : r ~ - ~ r ~ are a~so!utel~ oontiniou~ ~omeo morphisms; ~ will be called a Muckenhoupt mapping or, more precisely, will be said to belong to the class ~ p , ~
if I~'I~IP~ Ap(~)

then W C ~ I P ~ (

,

wher~

)

P~ooF. =et p : P4 P~ , I be an arc I g: ~ , J = =~(II¢ V~. ~y ~{~l~er ine~ualit~ w±th the exponent (p-~)/(~-~) '

I

I

Y

I

I

T

COROLLARY. If W e A ( F ~ ) then LE]~9~A 2. i f ~J)~_ M~ , then ~ :

{>{

PROOF. By

[31]

for some

~

0

W~ ~eA(~ ( ~ ' ~ M(~

)

"• f o r some

and for any arc I C rl

,I

T

I

III I

Y

Now we can prove the main d i s t o r t i o n ~ e o r e m

for ~ucke~oupt

map-

100

pings. THEOREM i. Let ~ and % be Lipschitz curves, --*'~ and ~---~-~-~ be ~luckenhoupt mappings. If ~ ~

,Ill P< I~(I}l

~

: ~--~ ,

<%(_~_) III "~/~

(2. ~)

.

PROOF. The right inequality is implied by the left one for . We shall prove the left inequality. First,I~(~)I----I~(1)l #

]J,z I~'I and by

+ have

(

IIl

(1.9) for ~ e l

)P Y\l

IIl P

,3(~,~'I)~III,

we

c II~'l cl~(1) J.

Y'I

I

2.2. Estimates for conformal mappings. The results of ~o 2. I give simple real-variable proofs of properties of conformal mappings. Let again ~ : ~\ ~--~ be the conformal mapping, normalized as usual, of the exterior of Lipschitz region ~ onto the exterior of the unit disk, and ~ ~-~ be the inverse mapping: LEN~KA 3. ~ E ~ % on T • @~ L i p ~ PR00F (cf. [41] ). Let schitz region and there exists an ~ ~ 0 such that

@(t)= ~I,'%,~'C~'~"~)

I @¢t)-0c~')l-
for I t - t ' l < ~

quently, there exists a function %0 ~

C~[-~,~]

Consesuch that .

Ie-@01<~/z . But ~r is an o~ter function and so V:f is the conjugate function for @ = @ 0 + ( @ - @ 0 )

~pl

Further,the conjugate function for @o is bounded and ~t~p#@-S01< < ~ / ~ . Then, I ~zl4/bcA$(~) by Helson-Szego theorem ~1] . •

C0ROLL~YI. ~ M ~ for some p>1 . @ coRoL~,~Z 2. w e B M O C T ) iff W o ~ B M 0 ( F )

@

RENARK. One can prove, that if ~ is a convex region, then ~ ~ for all p m ~ . In the next section we shall see that if i~,r. 4/P A p ( ~ ) , then I ~ P I I / P ~ A p ( ~ % ) for all ~ > 0 uniformly. Using this fact, theorem I, " ~-property and estimates (1.5), we obtain a number of estimates for distances to level curves ~% , % ~ 0 . LE~F~ 4. Let ~ be a Lipschitz region. There are numbers Ik and V , 0<%r~ V~ ~ ~ such that

101

f(~, F) where

~=

'

~- Ir/V .

2.3 Estimates for outer functions. LEMMA 5. Let ~" be a Lipschitz region, ~ 0 a n d { ~ Et(C.~-). is an ar~, reciprocal to ~ (cf. If Z ~ ~ and I c F n ° 1.5), then

c~ !

IIl~ F~I

But

~

~_~

and so

F

t~-z*I×t~-~1 on F ' I coRonAry, l~ l ~ C M* ~ Let now W e ~ ( F ) ons ~/~ and W Q

in G

- • (cf. n°1.7). •

. The function W defines two outer functiwith the boundary modulus ~/ respectively

an~ C \ G L~.~A 6. Let

~

be a Lipschitz region, is an arc recip-

focal to

55

, then

(2.7)

102 PROOF. By (2.6) and (1.9) for

W~/°

we have

I

I

The inverse estimate follows from (2.6) and (1.9) for --W

JW~(Z)/~(Z)J~C/f(Z~[-).0

COROLLARY 2. If COROLLARY 3- If .@ formly for % > 0 2.4 Regularization. Let ~ be a Lipschitz region and

° Set f o r Z ~ ~ . This func~>0 , W~(Z)---- j We(Z~)J regularizaticn of M/ • tion ~ on r will be called The following assertions follow from lemma 6 and previous esti-

mates. (i) Let I C

W~I~CO~"~

I

r

W~A(F)

III< f ( I, F~ )

be an arc. If uniformly on I • If

I~

Ill ~j2(~

then

~)

then

I

w ~ Ap(F) • In particular,for where uniformly for % -~-0 . (ii) There exists a critical exponent ~'(w)

c~(~/~f~(w).<w~/w~. 0

(4,

such t~t on F

0<0"< ~ .

0
)

(2.8) ,

(2.9)

3. Estimates of potentials. 3. I. Formulation of results. Let ~ be a Lipschitz region. Let ~ , positive numbers and "t < ~ . For ~LLI(F~)

be set

103

r~

1@<1~

"~E]- .

I{-~1

%

We need the exponents ~ ~- OF(,CT) Q(w) for weA(F) (of. n ° 2.4). LE~S~A 7. If ~ is a Radon region, WEAp(U)

I' (cf.

n°2.2) and

~'(W),

then

I~td:MA 8. If ~ ~ p ~< ~ Q (w) then

,

w~ A(F)

and ~'1- 6 (#-OF)

II* l/, cr ) then

RENARK. The Radon condition in lemma 7 is needed only to assure the ~-est~mates of the Cauchy integral. 3.2. Proof of l e ~

7.

Consider an outer function W in the interior ~ level curve ~ with the boundary modulus I Wol / E analytic function F w in G ~ we have

of the • For

104 By (2.6) and Carleson embedding theorem it is enough to Check that the measure ~ on 9

#(E}= f

wPI~I-P

EcF~

~

E is a Carleson measure in t ~ , ~ ~ 0 . If and ~ ( K ( t ~ ) t ) ) ~ 0 property there is a finite s#ch that ~ ( ~ t ) N F~ 0

~ I L I ~0t ~d on ( / ~ k ,

Gm . Consider a disk K ( ~ $ ) , t
. ~y (2.~) on/~,I~IP~I(TKLI-'# lw~l P , I w~l P× till-'# (I~), N N

and~(K(~,~))~
I:~[I.l-'~wP}-'~< cE II.l
I~

QI~)~

3.3 Proof of lemma 8. By (2.5) and (2.9) for ~ e F %

f(% F)~< ~7(~,F~)~-~ I ~-~ t~ and

so

where

•

•

I~ , ~

, ~(~)~< cl w~(~)ll ~-~l~7(~If ~

W~(~)G(~)~
M=~.6-Q)(4-¢)+~, ~ o Consider the operator

U

' L P ([%)

~

LP ( F )

defined

by

It is enough to verify that its norm is bounded for all p , ~ p ~ co . By Riesz-Thorin theorem [46] we must prove this assertion only for p ~ ~ , co . if p = ~ then

~(~ F~)M

llull On the other hand, if

p----- ~I

then

I~_~IM,I I~1 But by (2.4) # ( ~ , r ~

#(~,F )M(~-~ I~_~IM(I_~.~-1~1 ~
and

•

105

3.4. Proof of lemma 9. Let E4(~) = { ~ F ~ , I~-~l ~ ( ~ ) ~ , ~(~) = =~ k E~(~) • Divide ~ into two parts ~ and ~ , namely the integrals over E4(~) and ~ ( ~ ) respectively. At first, estimate ~I • We have for ~ ~i(~)

and so

E4(%) Now we can repeat the reasoning from the previous proof for the

operator

q

:

L~(F~,)_,,.LP(F), E~(~)

and get

Now estimate and ~ ~--- ~

H~

• By (2.5), (2.s), (2.9) for , qyEF , we have

~

.p{~> r<~)~ c#(,r, r,s- )~-<~ 1 ~- z 1 ¢,

and, finally,

0 . h e r e M - ~ m , - ~_ q, ments again. @

E~c~) . We can repeat our previous argu-

I06 ~4. Best apRroximations. 4.1. Weighted best approximations.

~p

classes.

In this section we turn to the main subject of the article, the theory of polynomial approximation. Let ~ be a Lipschitz region. Analytic functions in ~ have many different approximation characteristics. For example, one can consider for a function ~ EP(~) , ~ p ~ oo , its best approximations by polynomials of degree ~ in the E ~ -norm. If p ~ o o it is the classical uniform approximation. On the other hand, the problem of the constructive characterization of Lipschitz classes leads to the best approximations with weights ~(Z)~/~)~ which are not analogous to the uniform ones. As we shall see, there is a rather simple general construction containing all these cases and their h P -analogues. Let w~A(F ) . Remind that it means ~ W ~ 0 ( V ) . Let , ~p~ co . For ~ = ~ . . , ~ define weighted best approximations of ~ by the formula

#eEP(G)

E~(-I,W)p

(4.1)

~t# ][(-#-#, )W~I~, bp(y ) p~

where infimum is taken over all polynomials ~ of degree ~ . The quantity (4. I) contains the regularization W4/~ of the weight W . If ~ is a Radon~region and w c A p ( V ) (this condition is stronger than W E ~ ( F ) ), we define nonregularized best approximations o

E~(~,W)~

~

II(~-.Pt, p) W ,LP(r )

(4.2)

Evidently, E °~+~ ~ E ~° for all Pb . This is not the case for E ~ , but for ~ ~ k ~ ~P~ we have c 4 E ~ < E K ~ o $ E ~ . Let now ~ < ~ p ~ < o o , ~0 and r(w~ ,w~A(F)~ be the critical exponent from n°2.4. For 6 ) ~ ( W ) set

Kp(6,w)=[TeEP(G), if

G

E~(f,W)p = 0(~-~)} .

is a Radon region,

~< p < ~

, w~

A p (F)

and f

The f o l l o w i n g lemma removes p o s s i b l e c o n t r a d i c t i m n s . L~A IO. I f ~r i s a Radon r e g i o n , ~ < p ~ oo then for any polynomial P ~ of degree

WeAp(F)

and m

107

tl& JIL r> IIP ML r IIP u n i f o r m l y w i t h r e s p e c t to ~ - - - ~ ~ . . . . PROOF. Both functions ~ are analytic in 6

, Then 11 ~ . . P Cbl " , I , = and T1 1 i p(F~/vl, )

--IIP~wll¢< n , II¢~IIEP----IIP~ w,/~llwcr;

ll%llLetr,/ )

~

IIP we II LPCr /,

c i p l e lIP, w~ll¢cn,,> On the o t h e r hand f o r

~c" ~

. By the m a x ~ u n p r i n -

{IIP~wlILP(r)

~1

=

> llP~w~/~llm(rl}

•

and i n v e r s e e s t i m a t e s f o l l o w f r o m ( 2 , 6 ) and lemmas 7-9. @ COROLLARY. Under c o n d i t i o n s of lemm. tO f o r any ~ e E P ( ~ )

E~(÷,WJp=O(~ -~) ~} E~(#,W)p:O(~-~). • 4.2. Examples. (i) Let W ~ ~ . Both (4.1) and (4.2) define best approximations in the ~ ? norm (uniform for ~-~-oo ). The classes K ~ ( ~ \ J ) were discussed in works ET, 18-22, 47-49 ] . P

(ii)

w : I~rl ~

~et The function ~f

. By n ° 2 . 2 being outer~we have

W~A(F)

]~ ~(z, ~t~)' l; LP(F) Kp(G,l~'lb a=e classical objects

and~(W)<~.

(4.3)

The classes of the complex appro~tion theory. If p = ~o the~ K ~ ( G , I ? I ) is t~e ~ips c h i t z c l a s s of o r d e r ~ (see [ 3 - 1 6 , 26, 27 ] ) . f o r l < ~ p < co see E24, 25, 50, 51] • (iii) Let ~ be a Lipschitz region with the boundary A and level curves { A % } and @ : C k ~ " C "~ be the confo=al m ~ p p ~ . Set W = I~'I ~ l C q - ~ . It ~s eas~ to ~rove that W ~ A (r) and for ~ F

w,c~,×

~?[@c~, A~] -~, P~

~

f(¢(%Av~) ~ LP(F)

For ~ = D these E ~ ( ~ W)p are usual uniform approximations, for ~ = ~ (4.4) coincides with (4.3). In general case

108 (4.4) gives a continious transition from uniform approximations to ( 4 . 3 ) [52~ • (~V). I f W ( ~ ) : I ~ - ~ o l d" , %o~ F , then w ~ A ( F ) for

any ob

and W~ A ~ ( F )

t h e wei

t

ne

w( l-

~eas~e ~

for - ~ I p < o4 < ~ / p [

J

belongs to

. In general, with

A (D

a

finite p l a -

. Finany, if Wl,W~ ~(F~

,

then W~ W~ • ~ ( r ) , i.e. the definition (4.1) contalus different cases of mixed approximation [53, 52, 49] • 4.3. Pseudoanalytic continuation. 6 Our main problem is the description of classes K p in terms of the local smoothness of functions. We begin with the translation of the definition of K ~ into the language of the pseudoanalyric continuation. Pseudoanalytic continuation theory I34,21,251 describes the smoothness of a function ~ (belonging, say, to E ~(~) ) by means of its continuation onto the whole plane such that ~ ~ decreases with a prescribed rate near r . Por example , ~e iff it admits such a continuat i o n a n d l ~ ( ~ ) ] ~ C ? ( ~ I-)~-t , ~ C ~ ~ . There exist different constructions of continuation connected with different characteristics of ~ . So, pseudoanalytic continuation gives a general language for the comparison of the different definitions of function spaces. LEMMA II. Let ~ be a Lipschitz region and ~ E I (&) . For any ~ > ~ three following assertions are equivalent. (i) There is a function ~ C W ~ ( C ) such that ~IC-~-~ . (ii) There is a function ~ ~ W ~ ( ~ ) s u c h that F%IF--~I P . (iii) The function ~ admits a representation

E34~

;)e("7.,)=Jf

g~ (G)

~(~)(~-%)-Id~

, /~eL~(C\G)

.

(4.5)

REMARK. In section (ii) of l e ~ II ~ /F is defined in the sense of embedding theorems [40, 54, 55] and ~ I r in the sense of boundary values K28, 29~ . In both cases they may be regarded as angular limits almost everywhere. PROOF OF LE~AA. The implication ( ~ ) - - > (t~) is evident. If (iii) holds then the right side of (4.5) by Calderon-Zygmund theorem [42, 55] defines a function from W ~ ( C ) and so (i) holds. Finally, suppose (ii)holds. For ~ G

109

C~e So, taking angular l i m i t s , we get (4.5) almost everywhere on F with A = @ ~ F ~

. But both sides of ( 4 . 5 ) belong to E~(G)

and so coincide in the whole G . • If the assertions of lemma II hold, we say that the functi). on ~ admits pseudoanalytical continuation (from ~ Such a continuation - i.e. functions ~ or F~ of lemmaV II will be denoted by the same symbol ~ • It doesn't lead to contradiction but we remind that continuation is not unique and all e s t h e t e s mean c e r t a i n pseudoanalytic continuation with the same estimates. By (1.6) we can transform (4.5) to co ~(~)-~-] where

°

-%~

I

AI ( ~ ) ( ~ - Z ~ f ~

(4.6)

%1

Finally, let w ~ A (F) . Introduce the following characteristic of the pseudoan~lytic continuation:

w)=

1

In particular for

'

0

(4.7)

p ~ co

F.,: RE~. The representation (4.6) and the norm (4.7) involve the con_formal mapping ~ : C ~ ~ C\ ~ . We could formulate similar definitions for other systems of curves, and first of all for the level curves of the usual distance. But we shall see that the best approximations (4.1) and (4.2) lead to the estimates of pseudoanalytic continuation exactly in terms of (4.7). 4.4 Continuation by approximation. Let ~ E ' C G ) and l e t [ ~ } ~ be a sequence of polynomials of degree ~ ~, . converging to ~ in E l ( G ) . For example~it may be the sequence of best approximation polynomials of ~ in the sense of (4.1) and (4.2). For Z ~ C " ~ and ~ - 6 ~ I ~(Z)I -~<~ ~ + 4 set

110 THEOREM 2. Let ~ be a Lipschitz region and ~F , [.LOll,, } and ~ be as above. If ~ e ~ C ~ k ~ ) , ~>~ , then # admits a pseudoanalytic continuation such that

18TI
C~ [ O, + ~ )

for 0 ~< ~ 1 , ~(~)= 0 continuation for ~ ~ \ I~ by the formula

for

~

~

. Define the desired

(4.9) i

It is clear that # ~ C°°(.l~kl~) and I~# 1~ O A . Introduce defined by (4.9) for I ~ ( ~ ) I _ ~ -N and a function for I~(%)j_~<~-N or in ~ . Then coinciding with ~Z~ ~ FN ~- 0 in the interior of the level curFN _e C ~ ( C ) ve I,~- N and I ~ F N I = I ~ l ~ c ~ in its exterior. By the Green's formula for

~-N

We get a representation (4.5) for ~ taking N ~ o 3 in the last equality. @ ~,t~. zz then L P C F, w ) c L°FCI- ) for sufficiently small $ ~ • Further, if then

w ApCF) I-

IW4/l~l>O. Ft-~(W)-on

and

llglILP(1)~cm~(w)ll$w4/~IlLP(r)

Thus, the polynomials of best approximation in the sense (4.2) converge to # in E ~ ( G ) . The polynomials of best approximation in the sense (4.1) converge to ~ in EIC~) if

~,~'{W~En,(.-J',wlp = 0(.4). THEOREM 3. Let

G

(i) I f W e ~ C F ~ ~0 , then ~

be a Lipschitz region, ~ e Ft(,(~) • and EtC(.J:,W)p--~ 0(I~,-?'r'w}''5) admits a pseudoanalytic continuation such

that

(it) z~' l < p < ~ ~ 0 such that

, then

, w e A p CF ) ~

G

andE~C~,W)p=O(E~),

admits a pseudoanalytic continuation

111

(£ %w)~ c.E [U,~] (~, W)p

(4.II)

CCT,W)

COROLLARY. Any function ~ E K~ doanalytic continuation such that

¢p C~, % w ) = 0 ( ~ b

admits a pseu-

(4.12)

•

PROOF OF THEORE~ (i) Let us consider a function

Evidently,

C E E ~C~\G ")

II¢ItEP~CE~(~)~ I

2 ~ t

~*~

x Iw~ ~ ) 1

clI¢IILP(F )

~ d in

and

• On the ~,~ ~ ~1 ~

terms o f

l~Cco) ~- 0 other hand on

check the condition A ~ L ~ ~ C % m and by (1.6) and (4.1)

C~G

F~

I w~¢~)I

(4.8)

we

C~\~)

,

×

have IIzAw~IIG(~)

. It is well-known [29] and the theorem 2 gives a continuatlonwith To

o Further,

we

thatll~llG{E)~ll@llEP

remark

that](F,F~)~

~=o

for sufficiently small ~ ~ ~ . (ii) This case admits just a

similar proof, but

~

must

be

defined as ~ = [ P~.+,-]P%~ ] We ~-~'+' • We shall see that the condition (4.12) is exact: if (4.1~) holds then ~ K~b ( ~ W ) In the next paragraph we shall connect (4.I2) with local characteristics of ~ such as moduli of smoothness~ This roundabout way will be simpler than the "direct" connection of approximation and smoothness.

§ 5. Modull

of

smoothness,.,

The theorems of approximation theory connect the rate ofthe polynomial approximation with the local smoothness of functions. 0 The best approximations E~ , E~ from ~ 4 measure the rate of appro~mation. Here we introduce the moduli of smoothness for measuring the local smoothness. There is a number of different definitions of these moduli. The definition by Ju.A.Brudnyi

112 56, 57]

is the most convenient for our purpose.

5.1. Local polynomial approximation. Let [ be a Lipschitz ourv% ~ ( ~ ) and ~ C ~ be an ar~. Consider the best approximations o~ ~ by polynomials of degree ~ , ~ ~ 0~... in L ~(~) :

p~

(I)

In the local approximation theory FF5 is fixed and ~ is variable in contrast with the global approximation theory of §4. For a fixed I"11, we can replace (4.1) by the following interpolation construction. Let ~o be such a polynomial of degree t~ that i o

o

F o r a Lipschitz arc

~

with the ends

6_~ID.

~ , {

set

• Then for any polynomial ']I of degree~

J F be an arc ; divide I into ( ~'I" '1) arCs IO,,... of equal length. Let 0v0;... • (b~ be their origins• Pinally, for ~ ~(I) denote by ~ ~ the polynomial of degree ~ with values ] ~ ( ~ ) ~ w (Z) ~ Z at the

Let I ~

points for

~K

'

IK ~-----~ " ' " ~

any polynomial

T

of degree

1II

$(%l)
c

~epends on ~

• It is clear, that ~ T ~ FFb and for any fl > 0

and ~

) I÷1

T we have (5.2)

). So

s IILcm and we can replace the approximation definition (5.1) by an interpolatory one. As in §4 we consider a weight analogue of (5.1). Let @ be a Lipschitz region, ~ < ~ co and w E E,P(F) • Set for ~ EP(G)

113

~.~(#,I w)p

=

{~÷ ll(~-:e~)wIILP(i

There will be two cases of interest for us: if if ~ 4 ~ W~ ~{~ W . In both cases I I

) .

w~ApLF)

9~(#,I )-
and

(5.3)

with the cmnt. Set for

(5.4) THEORE~ 4. Let ~ be a Lipschitz region, S E E J ( ~ ) and let ~ be defined by (5.4). If ~ E L ~ ( ~ \ G ) , ~>I , then ~E E@ (~) and ~ admits such pseudoanalytic continuation that I'-~-~ I ~
and further

[ ~e~,r~, f~~%/(~,r)/irl_<#~+~}, ~=o,t . . . .

~ =

if ~ e ~ ( 1 )

then 9m(},l) -
Consider the sequence of partitions

F

into arcs F =

~ I~

14'= 0~4~,.. where I ~ = F and every following partition,,~, arises from halving all arcs of the previous one. Set m ' - - ~ OO -LK ~v on I k . Almost everywhere on r 0

~=T%E(T~+~-T ~)

~1.4 K

K

114

(ii) Consider the whitney partition of unity ~=~-]%K in \~ [55] • Let ~ K be the centre of the suppKort of ~ K and let ~ K be an arc on F with centre t~K and length J/JO f ( ~ K ~ F )

. In

For any polynomial

F =

~\G

K

],

+

For T=

set

7F=

IM CI(~) for k w-lth ~ k ( ~ ) ~ O implies l ~ F l ~ C~ . In particular, ~ ~ ~C\~) . Further, I F ( ~ ) I ~ o M ~ ( ~ ) , but M ~ E ~ (P) and we can pass to the limit when % ~ 0 in the formula ~,

.

. ~

2~z)

,

~e

C\~

, the i n c l u s i o n

N So we get the representation (4.5) for #

because

FIF=÷#F.•

5.3. Definition of moduli of smoothness. By ~ 4 t h e global polynomial approximation gives the estimates of ~ # in LP (~%) , % ~ 0 . Butwe see by theorem 4 that ~ ~ can be estimated by local approximation of on arc I(~) reciprocal to Z . When ~ runs over ~ ,~(~) runs over F in such a way that II(~)l is always equivalent to the local distance to ~% . So, a local characteristic, adequate to global polynomial approximation, must involve some integral estimates of ~ K ~ ( # ~ I ) on such a family of arcs. Let G be a Lipschitz region, W ~ ( ~ ) and # ~ E P ( ~ ) , ~ p~oo ° Por ~ 0 consider all partitions of ~ , ~= ~IK into arcs { I K } such that ~ / ~ J ~ ( ~ k ) l ~ ~. Set

115 N

where supremum is taken over all such partitions. ~ (~j ~ W ) p will be called the confox~l modulus of smoothness of the f m c t i I,P(F) of order with weight w . Tf W Ap(D, < p < CO , we can consider an alternative definition N

k=~

(5.6)

without regularization of the weight. As in § 4 we need not distinguish between these definitions. REMARK (i) Let ~d" be a piecewise polynomial function Evidently, (spline) which coincides with ~ k # on I K

Thus, we can define the conformal modulus of smoothness in terms of the spline approximation. (ii) The definitions (5.5), (5.6) are of conformal nature, because the supremum is taken over the partitions of F into arcs with equal length of their conformal images. We can formulate a completely analoguous definition with the supremum taken over all partitions ~ = ~ T K with ~/~ <~ I I K I ~ d- . Then we get "metrical" moduli of smoothness. Following to Ju.A.Brudnyi E56, 57~ one can show that these metrical moduli are equivalent to usual moduli of smoothness defined in other terms (cf. n°1.3). But they are never equivalent to conformal moduli if the boundary of ~ has corner points. In the approximation theory we need conformal moduli of smoothness. (iii) If ~----0 then OJ~(#j ~ W ) p is equivalent to usual (weighted) modulus of continuity of # = ~ in ~ ( T ) • But for ~ > / ~ such a connection between 6 0 ~ ( ~ ~ W J p and characteristics of # o ~ does not exist. 5.4. Estimates of continuation. THEOREM 5. Let ~ be a Lipschitz region, WC ~(F) ~ p <~ cO and ~ EP(C.~) (i) If O0~Kb(~ ~-j "0(~F(@)~&)~ ~,>0 , then admits a pseudoanalytic continuation such that

W)pc

50 , w)p.

,

116

(ii) If 4 < p 4

= 0 ( ~ s)

,

co

6~0

,

(÷,¢,

° W)p and C0~(~ a a ~ i t s a pseudoanalytical con-

WeApCF)

, then ~

tinuation such that

PROOF. The p r o o f s o f ( i ) and ( i i ) are c o m p l e t e l y a ~ l o g u o u s . ~or example, by theorem 4 we have on ~

By ( 5 . 3 ) we get

w) o I

w )p

F~

W)p=

We need the condition O0~(S, C 0(~w)+&)only to check the inclusion ~ ~ L~(C\~) in theorem 4. Thus we can repeat the corresponding argument from the proof of theorem 3. • THEOREM 6. Let G be a Lipschitz region, W E ~ (~) , ~ ~ CO and suppose ~ 4 ~ @ ) admits a pseudoanalytic continuation. (i) For ~ sufficiently large co

(ii) zf G then

for

i~aRadonregio~, 4
sufficiently

large

andWeAp(F}

)I~

cO

Here M = ( m + ' ~ ) V " (for IY~---lY(Cr) see n ° 2 . 2 ) . PROOF. (i) For ~ C ~ define a polynomial of degree

where

%o

I~;-~ol~-~IWl ~<-- o is the centre of

~

. On

By definition in (5.5)

I II x o

f(~ F~ }

I~_~1~*~ for

~ ~ T

and so

117

where

% Now the estimate (5.7) follows by lenm~a 9 for $ < ma 8 and (2.2) for ~ . (ii) Set ~ [ ~ I~(~)l~ ~+ ~ and

and by lem-

o

Further argument is just analoguous to the previous one with the use of lemma 7 instead of lemma 9. • COROLLARY. For any 5 ~ 0

In p a r t i c u l a r , i f

~e

Kp(~, w)

then

oOm({,% W)p = O(g~).

@

Now we have got an "inverse" theorem of approximation: the estimate ~ ( ~ ) ~ 0 (~'~) implies 60~u(~ ~ The "direct" theorems of estimates of ~ ( ~ ) are the subject of two next sections

§6. Faber ,operators~,,, In 44 we have got the lower estimates of the best approximations by means of the pseudoanalytic continuation. For the upper estimates we must have certain construction of polynomial approximants. Two such constructions are known - the elementary construction by means of Faber operators and a more complicated but more powerful construction of the polynomial approximation of the Cauchy kernel. We expose the first of them in this section and the second in the next one. 6. I. Definition of Faber operators. Let ~ be a Lipsohitz region, W E A p ( F ) , ~
REMARK. Evidently, (~o ~) ~f~/e W g ~ ~

and L~WA we

so

Tp,w eE@(G)

12. If

ADCI) [

~

L~(~)

is a Radon region,

then

4 < p < O0

f o r some and

118

P~oo~. ~vid.ntZ~,ll{~o~}~'W%~'WllLPCr)

--- It;11,~

So, (6.1) follows from Muckenhoupt condition. • For p ~_ ~ , CO we cannot prove such lem~a, because Cauchy integral is not a bounded operator in LPc F) . However, if W ~ '~ then we can formulate the following result.

Denote ?4 =

T4, 4

~4

and ~oo LEMMA I3. If

and

TOO ~-

Tea, ~

, i.e.

are classical ~aber operators without weights. ~ is a region of bounded rotation, then

¢i) UTea J:ll

'II :IIcA¢

,

CAC- )

Proof o f this lemm~ is due to Ch.Pommerenke [58,17] and one can read it in J.E. Andersson's work [49] • • R~MARK. In contrast with lemma I2, the condition of bounded rotation in the last lemme is not connected with the boundedness of the Cauchy integral operator. Let us construct now an operator ~ p , w as an inverse of Tp~w . For ~EP(~, W) set

T (ii) ~p,w is an inverse operator of Ip, W " PEOOP. (i) is evident, because Cauchy integral is a bounded operator in (ii) If ~ :

~(T) Tp~w

~

, ~ < P < oO then

.

$= Cfo~)sor,/p w;~+~, ~eE*C~,G-), k c ~ ) = o (~o ~)CW~o~)~/"VP_ ~,C~,oU/)CW~o~,)~r~IP

119

But the second term belongs to E P ( ~

oo

so E"m ~

.

=

#

0)

and vanishes in

•

REMARK. Clearly, the assertion (ii) of this lemma holds for p~_~ , O0 and W ~ ~ • Thus, operators TD. ~v and ~ realize an isomorphism of spaces H and ~ J ( ~ W) . But, evidently, for its image ~ ~ is again a polynomial any polynomial w C r ~ of the same degree, onsequently, if ~ is a Radon region, ~ p < co and then

w~Ap(F)

E~(Tp,~L W)p ~ %C~)p , IeH ~ By lemma 13 if

p =

"~ CO

and

~J ~--- ~

(6.2) then

O

The inequality (6.2) resolves the problem of the upper estimates for the best polynomial approximations. Now we shall discuss the action of Faber operators on the pseudoanalytic continuation. REMARKS. (i) By lemma IO (~4) an estimate of type (6.2)

holds ~or E n ( T m w ;L w)p (ii) It is known that the summation of the Taylor series gives good approximents (of the best order) for functions in P . The operetor ~Ip~W transforms this process into the summation of the Faber series (with respect to generalized Faber polynomials Tp~ w ( ~ ~ ) ). Thus, by (6.2) such a sL~mation gives polynomial approximants in ~ P( ~ w ) which are of the best order. 6.2. Faber operators and smoothness. If 5 ~ ~ admits a pseudoanalytic continuation then by the Green's formula in

%w Then

II ~p~W ~

[

•

admits a pseudoanalytic continuation and

1 --aTf,,w ~1 ~ is°rl ~+'VP I w~ i-~ 1 (a~)o~ I Consequently,

6"pC~,w

;~; ~,, y~) ~ C,6~p(-~, ~b, '~)

All arguments are applicable to the inverse operator SO

"

.,.,,w

~,

W

and

120 REMARK. It is clear, that (6.3) does not depend on the condition ~ < p < o ~ ; it holds for ~ = ~ CO and W-~- ~ • L E ~ A I5- Let ~ p ~ ~ , ~ ~ 0 , ~G ~ ~ . Three following assertions are equivalent

(i) J~L~PA

C p~ ~) .

(ii) ~eI(p C ~)~ ~) • (iii) ~

admits a pseudoanalytic continuation such that

~(÷,

~,, '/) :

0(~).

PROOF. By n ° 1.3 (i) is equivalent to (ii). By theorem 3 §4 (ii) implies (iii). Finally, let (iii) hold. Then for ~

151=4-$ The l a s t i n e q u a l i t y

=lass

IJ~PACP,~)

is a well-kno~n characterization

[55] •

o f the

•

REMARK. Of course, the proof of this simple lemma may be done without reference to theorem 3. Now we can state the first complete theorem about . THEOREM 7. Let ~ be a Radon region, ~ ~ p ~ oo , for 1 < p < ~o and W ~ ~ for p = ~ oo Let 6 ~ 0 and ÷~ E~(~ w) The following assertions are equivalent. 5 o

w Ap(F)

0

(ii) ~ 0 ~ ( S ) ~) W)p : (iii) £ p , w ~ e

O(g~)

for

m~

5/%r-g

.

LipA ( p, ~}

PROOF. By theorems 3, 5, 6 and (6.2), (6.3) all these assertions are equivalent to the existence of a pseudoanalytic continuation of ~ with the estimate (4.12). • REMARKS. (i) Condition W ~ means a slow variation of W on F . If W = J~ZI~ then ~ A p ( F ) only for sufficiently small b . (ii) For ~ < p < co the Radon condition to G is connected with the botmdedness of the Cauchy integral in ~ P ( F ) and is not necessary.

Ap(F)

121 ~7. Approximation of Cauch 2 kernel. 7. I. Construction of approximation. It was shown in @ 6 that Faber series summation gives almost best polynomial approximations. Here we shall use this summation in order to give more precise pointwise estimate for the polynomial approximation of Cauchy kernel. Let ~ be a Lipschitz region and suppose ~ 0 is an integer. THEOREM 8. For amy ~ ~ ~ \ ~ there exists a_polynomial ~ ~( ~ ") of degree k~ such that for % E

i ;_zl PROOF. Let ~ ~ ~(~) . Assume at first that Iq~174+~/~, i.e. ~ E ~ , ~+4/~ . Let ~(~)__~(q~_~)-4 where r~l1 be the Cauchy kernel. Clearly, ~4 ~ is the Faber operator from @6, i.e. ~ I ~ [(~°~)~f}t " Consider the standard Jackson kernel E39]

where

"K~

is defined by the condition

It is known that

oK

YK~v ~

C K" ~

-I~

and for

I $ I ~ ~/

I

The function

is evidently a polynomial of degree ~ ~ ; it is a partial sum of Taylor series for (~-I~) -I obtained by its summation by the Jackson's method of order k . Applying the operator ~1'i we obtain a polynomial of the same degree

122 The polynomial ~ is a result of summation of the Faber series for the Cauchy kernel by the Jackson's method. Now the equality

I~ JK~ ---~

~

implies q5

-~

-~-

1 ~(tJ-zll~-~l

~ot E~--- { ~, I ~ct)-~l~i~-~l}

,

,or 1~ E~

1~ct>-~ll~-~l

E~--_F-~:~]"E~

]~-~l ~

But according to theorem I ~2 and n ° 2.2

I~(b-~t-~ct~-~/.IC*r~ltl) ~, o
(7.3)

and thus for Zk ~ Q

I Jk~,c~ E~

I~ck>-~l

4¢ ~ o I~-~/~,1 I~(t>-~ll ~-~1 I~-~1~

I~-~1 /i~ Hence, for ~k~"

Q

]'ZK~Cb I~(t)-~t

d.t.< ~__m_I 1 ~ct~-~ll~- ~1 I~-~1 i~l.<,~C~<-,i 7/Q

E~

so for ~ k ~ Q a polynomial ~

-~

~d

1

The assumption ~

~ ~ F~, of degree

. ' 'i,~ Zk~

I r~-~/~l [~

, % ~ 4/~

-

4/~

I~,-q/.I there exists

such that (7.4)

implies I ~-~Irf(~, ~4/~ )

123 Let now

~

~\ G

lynomial approximant

but ~

earlier we have i ~ ( ~ ) I

I~(~)I-< I + ~

Construct the po-

for the Cauchy kernel ~C~(~

~ / ~ )-I

~#/~

. As

and by (7.4)

I r-,- ~;~/~, I i~_el

~

.

At last denote

1-1~(~;,~ ) - -

{ ~- [~-(~-~mc,~,~] <~ ~] '~ }

~;_

<~-~'

It is easy to check that H ~ I , ( ~ " ) is a polynomial of degree at most • satisfying (7. I ) and (7.2). @ REMARKS. (i) The polynomial (7.5) is no more the result of some kind of summation of the ~aber series of the Cauchy kernel. May be such a summation and the elementary approach of ~6 cannot provide so strong approximation. (ii) It is clear, that for ~ 5 ~ we can obtain from (7.2) the estimate I K ~ - ~ Q ~ ' ) I ~ C N ~-N for any N • The approximation in the interior of a region is better than at the boundary. We shall use this circ~,mstance in the next section 7.2. Direct theorems of approximation theory. Let G be a Lipschitz region, % V c A ( [ ) and 4~ p~ 0 o0

o

PROOF. Apply theorem 8. Set

_p(.)

=

-

_

q~ _

Clearly,

F_~,C,J', w) f>

o lIAr+ k+ As÷ A~_ILL,{r' ~,~ )

where

~o

124

~o

1~-~4/~1 "~

-

But for

by <2.4)I ~-Z;~/~,I~j~(~,~/~'~'l~'~l

~ ~-q~4

~

Therefore

o and by lemma 9

E

~3 and by Minkowsk± inequality

o

×/(~,F)/~ . ;or ~ and Minkowski inequality oo

~uffio±ent~y large by lemma 8

II A" IIm.r,,,,,.,,~ ~C](I'~%)"N~(~c,"t,W) ~/~

~

.

@ I--

~K.

if

~3

0

_

An a ~ l o g u o u s e s t i m a t e can be provea ~or IZ~(£ W)p

WeAp(F) O0

9

125 by using lemmA 7 instead of lemma 9. We shall not discuss this subject because the estimates for E °~ ( ~ W) p follow from lemma IO or ~6. Now after theorems 3, 5, 6 and 9 we can formulate the following final result. THEOREM IO. Let ~ be a Lipschitz region, W ~ (~) , 4 ~ p ~ O0 and ~ ( W ) . For three f o l lowing assertions are equivalent.

feEPc@)

(i)

eKp (G,w)

(ii) ( ~ ) l ~ ( ~ , ~ W ) p = (iii) ~

-s)

, i.e. 0 (~)

•

for sufficiently l a r g e ~ .

admits a pseudoanalytic continuation such that w)

=

•

~8. Area approximation. 8.1. Formulation of results. Till now we considered approximations in various boundary L~-metrics. There are some questions of great interest concerning area ~[-approximation. However, as can be seen from the results of ~ 4 - 7 , best approximations depend first of all on the behaviour of the conformal mapping ~ : ~ \ ~ >~kD • This is defined on the boundary of the region, but how can this mapping of the exterior of ~ be included into estimetes of the interior approximation? Now we shall see the way of overcoming this obstacle. Let ~ be a Radon region, For a function {~ E~(&) tion characteristics:

&

~

ma"

G

where infimum is taken over all polynomials ~ of degree ~ • The main feature of (8.1) consists in the presence of the measure ~ ( % ) ~ I~/(~*)l~×~ in the area LP-norm. using this definition we are going to prove an embedding theorem, which is an a p p r o x ~ t i o n analogue of classical embedding theorems of different dimensions E40,54,55] . THEOREM II. Let ~ be a Radon region, I < p ~ o o ,

wAp(F)

and ~

E4(~)

. Then for

~-4,Z,...

126

O

0

(ii) E ~ ( ~ w ) p ~

C ~ K= l,i,

COROLLARY. ~--Kp

k-11PZFJ~(#'>W)p•

(.6, W)

iff ~ ( E , VJ)p =

O(PL"3"I/P). •

REMARKS. (i) Roughly speaking, theorem II asserts that

E~tS,W)pW)41/P g~(S,w)p o

exponent, by 4/p (ii) LP(G)

, i . e . the boundary smoothness as it ought to be in embedding theorems, is diminished . Approximation without weight, i.e. in the norm of , corresponds to ~/~-j and comes under the

I~/I-~/P IQpfl-4/PEAp([-)

action of theorem II for (iii) Theorem II gives rather complete solution of the area approximation problem becsase the structure of the class Kp was completely described in theorems 7 and IO. 8.2. Proof of assertion (i) L E ~ A I6. l~or any

~

EICG)

W)p .<

11 llLP¢r,

(i) follows immediately from lemma 16 by the application of this lemma to the function 9 = ~ - P ~ where P ~ is the best polynomial approximant of ~ in the sense of E ~ ( ~ W)p • PROO~ OP LEMMA I6. ~or ~

~

denote

pc.)_ ~-~I! 1(~)~,,(~, ) 4~ where the polynomial H ~ ( ~ , ' ) is defined view of (7.1) and (7.2) we have on F

in theorem 8

Two first temrs don't exceed M 9(%) be estimated by (1.10). So we have

and the third term can

§7. In

127

Note now that by (1.7) co

~,~.~ JIl~~l~i,~i~0I~ i~),~l ~

~8.~)

Consider first the region % < 4 / b in ( 8 . 3 ) . But ( ~ - i D ) V / ¢ ~ ~4(~) and we have by Carleson embedding theorem and (8.2)

P

r,~

r ~--4/~

Therefore the part of (8.3) corresponding to the region does not exceed ¢-,~4~ [l~II P

It remains to estimate the contribution of the region 4 / ~ < % ~ co . But for ~ ~J" , ~d/b it is evident that

F (for ir see n ° 2.2). Divide now F into arcs,

N

F-- U I k, I

~J~)l~<%. Let ~'K = C(~K )'I")~ F~ . Clear that for ~SE ~K

Thus f o r a l l

'

such that "~//~

be c o r r e s p o n d e n t

•

~I~

arcs

on

, I~(~>-_P(~)I-

cITE:I

c

~p

C

q

<

j~

r

At last the contribution of the region does not exceed o0

~

#/

to ( 8 . 3 )

,..

8.3. Proof of the assertion (ii) We shall state the following polynomial inequality.

128 LEMMA I7. If ~

is a polynomial of degree

I1211~--~°~ (,J'I~lPw,:lPd-~)

where

~

, then

4/I~ .

Now for completing the proof of the theorem it is sufficient to apply lemma I7 to every term of the seriesI-p~=~[~2~÷~-~K~) being the best polynomial approximants for ~ sense of (8.1). @ PROOF OF LEMMA I7. Consider the functional

in the

4

r It is sufficient to show that

IA(~)I~O[Plle

Jill hP~r,w-t) ~/p

First of all if ~ E (~\~) and 0 • Therefore_ tion, defined in G : F(~)~P(~)~@(~)~

A(~)--~--

A(9)~

A%)

(8.4)

'

~(OO) ~--- 0 then Consider the func( ~

By the Green's formula

c~ However

-

1¢

--

The quasiconformai refieotion belonging to the class have [8~*I X~ and by the Holder inequality

I A(t,~I
By Carleson

CI

I ~',~L-~))

,

we

".

G

o

d,~ ,. [ I~, w~' l P C4*A~P~

embedding theorem and in view of

!jI ~,+w~,, iP~ c ! i ~,. w-' P'

t~,+W~4~EtCCT)

12g

If

weAp(F)

then

w

qF)

and (8.4) holds. O

9. ~p~cial cases and ~eneralizations. 9.1. Approximation in

~(~)

.

Let ~ be a Lipsohitz region. For a function ~ C A (G) and an arc I ~ define the oscillation OO~(~) I ) of on I (of order ~ ) as the best approximation of # in ~(I) by polynomials of degree ~ or, equivalently, as the distance in C(1) from ~ to its interpolation polynomial of degree ~ with interpolation points @ O ~ . . . ~ M ~ I , i ~-

~I~

Ill / ~(f~+~)

. In particular, ~ e ( ~ , l ) - - -

=~/$

$~mI#(~)-~(~)l is simply the ocsillation of ~ on ~ . D e ~ n e for W ~ ( F ) its mean value ~VI on I by the equality WI~-IWe(~I) I where the point ~I is reciprocal to I in the sense of n ° 1.5. Now we can formulate theorem IO for ~--- co as follows. ! THEOREM IO. Let be a Lipschitz region, W ~ , ~ ( F ) , . Following two asser~ ~(W) and # ~ C A (~-----T) tions are equivalent. there exists a polynomial ~ of degree (i) For any such that

(ii) For any arc I c

w

F

c#,l) w I

and every sufficiently large

I

(9.2)

REMARKS. (i) In P.M.Tamrazov's book D2] a number of different constructions of moduli of smoothness for continious functions on curves are discussed. Here we use the simplest variant of O J ~ ( ~ I ) . The "right" definition of moduli of smoothness for complicated sets is far from evidence, but now the situation is clear after the work of P M~Tamrazov ~2,59], Ju.A. Brudnyi E56] and the author E20,34,60~ • (ii) The estimate (9.1) concerns the boundary of ~ . We remind that there is a stronger estimate at interior points of . For example, we can show by means of constructions of 97 that for ~ C A(~) and ~ E @

130

I

I

(9.3)

for any M > 0 • This observation is due to N.A.Shirokov D O , I l l (see also E26,27] ). We have used it in ~8. It is well-known that in ~ we can replace ~ - M in (9.3) by ~ ' , 0 < $ < ~ . N.A.Shirokov has proved (unpublished) that if ~ has co~ner points then one cannot replace ~-M in (9.3) by ~ - ~ , were ~ , 0 < ~ < ~ , depends on ~ . It is unknown whether it is possible to replace ~ - M in (9.3) b Y 6 "~ ~ for piecewise smooth regions. It is unknown whether it is possible to improve (9.3) in a general Lipschitz region. (iii) The proof of theorem I0 shows that (9.2) holds for ~4~-~ where ~ , ~ > 0 depend on W and ~ . In particular, we can set ~ = 0 and remove moduli of smoothness of higher order for sufficiently small ~ only. 9.2. Lipschitz classes. Set W = i~[l ~ . Then (9.1) becomes I~(Z)-P~(Z)I~

Cf(Z'C/~

)~

~ I

(9.4)

and (9.2) becomes

w

cS

l)

It is easy to prove that (9.5) is equivalent to the usual Lipschitz condition of order G for 0 < ~ ( ~ and defines respective Holder-Zygmund class for G ~ I . Thus, theorem IO' transforms into the classical W.K.Dzyadyk's theorem [3-5] about constructive characterization of the Lipschitz classes. This was a standard case for a long time and the main progress in the theory was understood as the extension of the class of regions for which this characterization holds. After the work of W.K.Dzya-

dyk lyi

[3-7]

,

ALebedev and

AShirokov

[9-11,13]

, V.I.Be-

E15,16] and others the corresponding "direct" theorems were proved for very general regions. V.I.Belyi ~ 6 ] has proved such a theorem for any quasiconformal region (may be with nonrectifiable boundary). On the other hand N.A.Lebedev and P.M.TamrazovES,12,59 J have proved an "inverse" theorem, i.e. implication (9.4)-->(9.5) for any continuum. However, this generality was superfluous. N.A. Shirokov E14] has constructed an example of a region of bounded rotation (with a zero interior corner)

t31

such that (9.4) is stricly stromger than (9.5). We shall descuss it in the next paragraph. 9.3. Uniform approximation. For W------~ (9.1) and (9.2) become

l -p I

~(},I)

,

(9.6)

~ c I~(I)l ~

(9.7)

The problem of description of the function class (9.6) has attracted the attention of specialists for a long time (see [7,18-21,26,27] ). For ~ F ~ = O (9.7) means that ~ o ~ C ~ ( ~ ) . W.K.Dzyadyk in 1962 has conjectured that the condition ~ ~ ~ ~C~(~) is equivalent to (9.6). We see by (9.7) tha~ this condition is sufficient for (9.6) because C O ~ ( ~ I ) ~ ~o(~I) , but it is necessary only for small G . For example, if ~ ~ J , one cannot set ~--- 0 in (9.7). W.K.Dzyadyk and G.A.Alibekov [7] have proved sufficiency of ~o ~ C ~ ( ~ ) for (9.6) in the case of piecewise smooth regions with some restrictions on 6 . Further this subject was discussed in [18,19,61] , where exceptional values of ~ depending on ~ were indicated. At the same time some attempts to find an alternative language for the description of the class have been made fi53,20] . Finally, the author [21] has introduced the condition (9.7) which allowed to solve the problem completely. If W = l ~ f l 6 1 ~[I-5 where ~ : C\G-*~\~ , being a Lipschitz region with level curves [ then, as we have seen in n ° 4.4, (9.1) and (9.2) become [52]

A%}

I

, oo~(S,l).< c j ¢ ~ I ) I ~

If

~

=

@

this is a ~ipsohit~ class, if ~

is a uniform approximation problem. If ~ = ~ =~ ~ a problem of non-uniform approximation in the disk.

=

D

this we have

9.4. Approximation in E~(G) . Area approximation. The ease of EP(~) corresponds to W ~- ~ in theorems 7 and IO. Here the difference between conformal moduli of smoothness and usual ones is essential (except for the modulus of follows author's order O). Our description of KGm ( @ ~ ~ ) [

132 work [22] where Paber operators and conformal moduli of smoothness without weights were introduced. If D%~--- 0 t h e n ~ ( S ) ~ # ) P is equivalent to the usual modulus of continuty inLP(~) of o ~ and so one can prove the sufficiency of the condition ~o ~ ~ ~ 5) for ~ E K ~ ( ~ ~) . This result A has been independently proved by means of Paber operators by J.I. Mamedhanov and I.I.Ibragimov ( [47] , see also [62] ) and J.E. Andersson [48,49] • But the necessary and sufficient condition for large ~ cannot be found in such a way. In the problem of area approximation the main difficulty is to include the conformal mapping ~ of the exterior of ~ into the estimates of interior approximation. As we have seen, the exterior mapping controls polynomial approximation. By this cause the previous results of S.Ya.Alper [63] and V.M.Kokilashvili [64] , who have operated with interior conformal mapping ~ : ~-* --~ ~ , are complete only for regions with smooth boundaries where ~ and ~ have identical boundary behaviour. -I/~ Approximation in ~ ( G ) corresponds to W : l ~ f t in theorem II and we obtain a condition I~fI-I/PE A ~ ( F ) • There are two cases when this condition holds. (i) G is a convex region and p ~ - ~ ,~:~(G)~ 0 • (ii) ~ is a piecewise smooth region with exterior angles ~i ~ ~ ' ' ' ~ ~ N ~ 0 < ~ <~ in its vertices,

and

rj

9-5. Besov classes. Our classes ~ ( ~ ) ral classes of Besov type

are particular cases of more gene-

Kp~ so,

K;(~ , w ) = K; ~

( G, w)

.

I t is easy to show l i k e in ~ 4-7 that # e ~ $ ( ~ , cO

d

o

and

~

W )

admits a pseudoanalytic o0

I 0

continuation such that

< + co

iff

133 If ~=/= p then condition (9.9) is connected with the system of level curves [ ~%} and can be expressed in terms of conformal smoothness only. But if ~ ~ p then one can rewrite (9.9) in a more invariant form --

-3 p

Cr Condition (9.10) does not involve any choosen system of curves. On can rewrite it in the form of repeated integral with level curves of the usu~l distance and express (9.IO) in telxas of usual metrical moduli of smoothness. For example, in such a way we can get a constructive characterization of class AI ( G ) of functions of E P Q @ ) whose boundary values belong to 3ov class [54] E ~ ( F ~ . see author's work F25~ ; this characterization is the following: A E ~ ( ¢ ) = K . _ (~. l ~ q ~) . Sobolev

classes Fp(.GT)=[ eEPcc )

EF(G) }

also

admit some constructive characterization, but of another form (analoguous to the theorem of dyadic decomposition in Fourier analysis) [24,25] . 9.6. Approximation on a segment. The segment [ - ~ ~ ~ ] is not included into the domain of applicability of this article, but it is just formally. Usually this case is considered by pure real-variable methods [39,65J • But it is easy to check that all constructions and assertions of our article (except ~ 8 of course) may be repeated for a segment. In such a way one can get the results on Lipschitz classes, uniform approximation, Besov classes and so on. 9.7. Mixed appr0~ximation. Let W ~ ( F ) and w Ap(F) , Replace in all definitions of E~(~ L, W)p , ~m(~-~W)p etc the re~n/larization W ~ by the combination ( W ~ ) % W Z . This o construction contains E K and E ~ as particular cases (when W ~ J or %4~ ~ ~ ). One csm show that all results,.Aof do not § § 4 - 8 hold for this "m/xed" case. But the classe ~ depend on W,I and W~, but only on the whole weight W-~-W, 1 W9¢ • Lemma IO is really an example of such independence. § IO. Some unsolved problems. 10.1. N.A.Shirokov's counterexample. Let ~ C ~

be a bounded continutun with the connected

134 complement, ~ : C \ ~ - - * - ~ \ D , ~ ( 0 o ) = o O , be a conformal mapping and ~@={~\B , I~(~)l=~+ %~ be its level c u r v e s . ~ h e c l a s s e s CACB) and C ~ ( B ) , 0<~<~ , were defined in n ° 1.2. Define now an approximation class ~ C ~ ) , O< ~ < ~ , consisting of all functions f ~ Z O A ( g ) such that for any It,-----~ ~... there exists a polynomial ~ of degree ~b for which

We have seen in §9 that 1 7 ~ ( E ) = C A ( B ) if B = ~ i s the closure of a region ~ with the quasiconfermal boundary or if

g=E-~,~]

• ~.~.',ebede. ~

~.~.~zov

[~,~]

that N~(B) c CA(E) _ f ° r a n y continuum E N.A.Shirokov has observed the following fact. THEORE~ I2. Let E = [0~U ~0,~]

~oo~. Set

:go =

that for ~

F~=

. But in 1977

t~1-~/~ },

~ n { ~,

,

have s h o ~

q

N ~

.

It

is

easy

to see

~o

and hence S ( ~' ~ ) ~ 3 ¢ %, F~)

•

Let ~ C 17 ~ ( ~ ) . By theorem 2 ~4 ( B is not the closure of a region, but it does not influence the proof) ~ admits a pseudeanalytic continuation such that in ~ "~

Applying the estimate (2.6)to the f u n c t i o n C P ~ . 4 - - ~ ) ~ -~÷4 in ~ \ B we obtain 4

135

C~

As we have noted above, the class ( B ) corresponds to the estimate (10.3) in the whole of ~ \ ~ . But the estimate (10.2) is strioly stronger than (10.3) and so ~ b ( B ) = 7 ~ C ~ ( B ) • The formal proof is as follows. Consider the Cauchy type integral ~ ,

Q(~) = I ~¢~)(~-~)" ~

, ~eC~ B

R It is easy to check from (10.2) by the Green's formula that

$

vow any function { e C A ( B ) which does not satisfy (10.4) gives counterexample. For instance we can take

~:('~=I 1~t~' ~e[o,~], I ~ l~e~p~-~,, ~e[o,~] ~ and get i , ~ , k - O ~ * 7 , ~ O__ , O< ~< i , for this function. @ Generalizing the idea, N.A.Shirokov [14J has constructed a region ~ such that (i) the boundary of G has a tangent at all its points except one vertex, and a slope of the tangent has bounded variation ; (ii) there exist two one-sided tangents to ~ at its vertex, making an angle 9U (zero interior angle at vertex);

This counterexample disproves the conjecture, discussed for a long time, that (10.1) is an universal constructive characteristization of the class ~A ( ~ ) . The proof of this conjecture for general regionswas the main subject of the development of the theory before 1977. The theorem I2, i.e. the failure of (10.1) for unclosed arcs, is rather elementary fact. The late understanding of this fact and its consequences was caused by

136

neglecting the approximations on arcs. It is not known, how the approximation problems must be posed for general sets. It is not known either whether there exists a constructive characterization of C A ( ~ ) on arcs by means of some modification of (10.1). Note that in all proofs of ~ 4 - 9 one can limit the consideration by polynomial approximants of degrees ~ - ~ ,~ > ~ , k: ~ .... only. N.A.Shirokov [66] has shown that this is no longer true for B - ~ - ~ L 0 ~ U ~L0~ ~ . All this means, probably, that any transformation of (10.1) into some estimate of type

(lO.5) is useless. The question of constructive characterization of 6~(~) for general B is open. 10.2. Uniform approximation. Let again ~ be a plane continuum with the connected complement Denote by K ~ ( B ) , ~0 , the class of all functions ~ C A (~) such that

[ on ~ for some polynomial ~9 one can describe K S ( ~ )

region) or

P~ of degree ~ = 4 , % ~ . . . . By for ~ = ~ ( G is a Lipschitz

in ter s of looal oscillations: c{

I

(lO )

The question of description of K s ( B ) for general ~ is open. This case has some psychological advantage over n°I0. I because the language for the description of K ~ has been found quite recently even for simple regions and therefore this direction appears to be natural. Here we shall discuss only one difficulty of the problem. Let ~ ~ , region ~ having a piecewise smooth boundary with one corner point (a vertex) and suppose exterior angle at this vertex is ~ , 0 ~ / ~ to describe such behaviour.

137

For c~ ~

~

(zero interior angle) there are no difficulties,

but for ~ ~ 0 (zero exterior angle) the condition (10.6) is not satisfactory for any ~ . EXAMPLE. Let

F

=

~

be the interior of the cardioide

~

,

{

is not a Radon region because it has zero exterior angle at the origin but ~ is a piecewise smooth curve and so all proofs of ~4 and § 6 hold. Clearly, ~ ~(~) iff ~ admits a pseudoanalytic continuation such that

clq 'lCl ol-4)

(~o.7)

or iff

{=~oo~, ~eOA(D)'

(lo.8)

where ToO is Faber operator from n ° 6.1. But we cannot express conditions (10.7) and (10.8) in local terms. These conditions describe a class of functions whose smoothness changes very sharply along F : from Lips to coo

.

Really, the limit smoothness of ~ ~i(~) at a vertex corresponds to some Gevrey class. It is unknown what local language is applicable to the description of so rapidly varying smoothness. REMARK. Local and global approximation theories have the general concept of best approximation E~ (~) ~o on I by polynomials of degree ~ , but in the local theory is fixed and ~ changes, and in the global theory ~ is fixed and ~ increases. There is no theory for estimates of

~

(~T)_ uniformly with respect to both ~ and . Such a method could yiels a solution of our problem, but now local and global approximation techniques differ too much. 10.3. Direct theorems without Fourier analysis. In ~6 and ~ 7 we have constructed polynomial approximants by means of Faber operators and some artificial tricks. In the end, these approximants were connected with summation processes of Fourier series of a Faber-transformed function (or the Cauchy kernel). We propose now two problems. (i) Is it possible to construct polynomial approximants in a region without any isomorphism of Faber-type with a function

138

space in a disk? (ii) Is it possible approximation

to prove Jackson's theorem on uniform

on a circle (or on a disk) without use of the Fou-

rier analysis? The approximation

on disconnected

sets (for example,

on the

union of two regions) gives an example of problem for which the usual approach is impossible and the solution of our problem (i) could be of interest.

But now we don't see any way to solve

these problems.

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70--90. 46. D u n f o r d N.,Sc h w a r t z J.T. Linear operators. N.Y.-London, Interscience, 1958. 47. ~ 6 p a r ~ M o B ~.M., M a M e ~XaH 0 B ~.~{. KO~cTpyETEBBa~ xapaETepHcT~Ea HeEoTopoFo F~acca ~ . ~oz~. AH CCCP, I975, 228, ~ I, 85-87. 48. A n d e r s s o n J.E. On the degree of polynomial approximation in E ~ ( ~ ) . J. Approximation Theory, 1977, 19, N I, 61-68. 49. A n d e r s s o n J.E. On the degree of polynomial and rational approximation of holomorphic functions. Ph.D.Thesis, Univ. of Goteborg, 1975. 50. A H ~ p a m E o M.M., K o ~ e c H ~[ E SI.H. HpHgx~XeHze B

cpeAReM a ~ T E e c m E x

{~WH~

B 06J~CT2X C yrxaMz. B c6.

"MeTpEecF2e

Bonpoc~ TeopEH ~ p J H ~ E OTOOpaEeH~", B~n. 2, KZeB, 1970, 3-20. 51. A H ~ p a • E o M.~., I{ o JI e c H H E ~.~. KOHCTpyETHBHSJ{ xapaETep~oT~Ea HeEOTOpOrO Ho~y~acca ~ F~acca E p B

06~aCT~X C yr~aM~. B C5. "MeTpH~ecz~e BonpocM T e o p ~ OTO6pa~eH~q~", Bkrn. 6, K~eB, I975, 8-I5. 52. ~ M H B E Z H E.M. K o6~e~ sa~a~e n p z 6 x m ~ e ~ M H 0 r o ~ e H a M z B ~ 0 p ~ 0 B ~ X 06~acTEx, 3 a n . H a y ~ H . c e ~ . ~ 0 M H AH CCC]?, 1976,

T. 65, 189--191. 53. ~ z p o E o B

H.A. 0 BSBemeHHNX ~p~6~I~eHN~IX Ha SaMESyTNX MHo~eCTBaX C y r J ~ n . AOF~. AH CCCP, I974, 214, ~ 2, 295-297. 54. B e c O B 0.B., M 2 ~ E H B.H., H H E ox~ c E ~ C.~. EHTerpax~H~e npeAcTam~eHm~ ~ y a ~ r ~ z TeOpeM~ B ~ o ~ e H ~ . M., "Hayma", 1975. 55. S t e i n E.M. Singular integrals and differentiability properties of functions. Princeton Univ.Press, 1970.

56. B p y ~ H ~ ~ D.A. HpOCTpaHcTBa, o n p e ~ e ~ e ~ e c noMo~m ~O~H~X np~6~n~eEm~. T p y ~ MocE. ~aTeM. 06-Ba, 1971, T. 24, 69132.

142 57. B r u d n y i

Ju.A. Piecewise polynomial approximation,

embedding theorems and rational approximation.

Lecture notes

math,, 1976, 556, 73-98. 58. P o m m e r e n k e Ch. Konforme Abbildung und Fekete-Punkre. Math. Zeit.,

1965, 89, 422-438.

59. T a M p a s o B H.M. KOHe~Ho--pasHOCTHMe r ~ O C T H ~ nOOI~HO~H~e npH6J~xeH~z. HpenpHHT M~-75-I0, ~-TMaTeM.AH YCCP, K~eB, 1975. 60. ~ H H ~ K H H E.M. r ~ O C T T , HHTerpa~OB THN2 K O ~ . 3all. Haym~.ceM~H.~0~ AH CCCP, 1979, T.92, 115-133. 6I. A H T 0 H D K

II.E. K paBHoMepHoMy Hp~6~xeR1~0 ~ y m ~ u ~ , H e -

npepLmm~x Ha SaMEH~T~[X MH(I~eCTBSX C yrzaMH. ~OF/[, All YCCP,

1971, A, 2, 487-489. 62. H 6 p a r ~ M O B M.M., M a M e ~ x a H o B ~ . M . Hp~Mue H o6paT~e TeopeM~ npH6am~H~a B EOMIDIeECHO~ o6aacT~. B c6. "Teopz~ n p M ~ e P m a ~jH~", M., "HayEa", I977, I90-I94. 63. A ~ ~ n e p C.H. 0 n p H d m ~ e H ~ ~ T H q e c K ~ x ~JH~ B c p e ~ e M no o6ZaCTE. ~OEa. AH CCCP, 1961, I36, ~ 2. 64. K o E ~ a a m B H a H B.M. 0d a n n p o K c m ~ ~T~eCFa~X B c p e ~ e M no o6~aCTH. T p y ~ T O ~ c c E o r o ~aT.~H--Ta AH Ppys.CCP, 1970, 38, 65-72. 65. 11 0 T a ~ o B

~.K. 0 C T p y E T y ~

xapaETep;~oT~Eax E2aCOOB

¢ ~ c ~aHH~M n o p s ~ O M ~ e ~ o MaTeM.~-Ta AH CCC~ ~.B.A.CTeF~oBa, 66. ~ ~ p o ~ o B H.A. Annl0ozcc~aT~B~e Hyy~a. 3an.Hay~H.ce~.20MM AH CCCP,

np~6J~meHm~. T p y ~ 1975, 134, 260-277. CBO~CTBa O ~ O r O KOHT~1979, 92, 241-252.

l!

V.P.Havin, Bo Joricke

ON A CLASS OF UNIQUENESS THEOREMS FOR CONVOLUTIONS

I. The Hilbert transform. 2. Riesz potentials. 3o Fractional integration. 4. Newton potentials. 5. Sets of uniqueness. 6. Semirational symbols. 7. Semirational symbols and Carleson sets. 8. Another approach. 9. Logarithmic potentials of compactly supported measmres. 10.Semirational symbols: free interpolation. 11. ~ ~

~ ) -property and Zweikonstantensatzo

12oMore about M.Riesz potentials.

144 This article is a n

exposition of a seminar talk and gives

a survey of results (mostly without proofs) published partly in Eli or prepared for publication (seeE2~ ). Its theme is a phenomenon of q u a s i a n a liticity exhibited by many operators commuting w i t h translations.

This phenomenon can be roughly described in

following terms: if a function and its image (under the operator) vanish both on a sufficiently large set then this function vanishes identically. Or in other words the knowledge of a function and of its image on a "big" set is sufficient to know the function as a whole. Here the words "a sufficiently large set "or "a big set" can be replaced very

often by the word'~pen".

What they

really mean is the problem we intend to deal with. I. The Hilbert transform. Let us begin with a well known example (which to be frank is as a rule - unfortunately - our single tool). The symbol fen

~(S)

will denote the Hilbert trans-

of a function

+o3

=

v.p.

10-00

Suppose E

~

and

~(~)

vanish both on an i n t e r v a 1

. Then the Cauchy transform

-- ~

I

~(~)~t

~(~) : ~(~)(~)~---

(~E C \ ~)

vanishes identically by the slap-

-c0 lest uniqueness theorem for analytic functions. But

= ~- - ~ 0 -'~'-(~(~)(9~+~)-~(~)(9C-~)

whence

When

~=0 ~

~(~)~

for almost every real

90

•

is a set

}IE=%(÷)I E = O

of positive

L e b e s g u e m e a s u r e and

the same o o n c l u s i o n h o l d s ( i . e . { s O )

as is seen from the same reasoning. We only have now to use a more subtle uniqueness theorem, namely that there is no nonzero H ~ - f u n c t i o n vanishing on ~ . 2. Riesz potentials. tentials

Another example is yielded by Riesz po-

tx_tl

_.

. We suppose

145

06~(~)~ ~ is a (signed) measure sufficiently small at infinity (to ensure the convergence of the integral). It is not hard to see that if ~ is an interval,l~l(E)=0 and ~ f ( ~ ) = 0 for all ~ E , then ~ = 0 - In [lJthis'<'--assertion is given three different proofs. Here we choose the shortest. PROPOSITION I. Suppose ~.> 0

l.~l([-~,q)=

0

, u,~l (.~.,~)= o o~ ~-~-~-~ into

PR00~. Substituting we obtain

0=1o~1~'~

"

--~ I~1

t-~,

~ 1-~

('l-v)

"

h~

(to'1 ~'v)

I~'-~I~-~

~Y being a measure concentrated on Then &-I

T~en .~=o

the integral d e f i n i n g ~

¢~)= Isl~-~dv"(~) R

0=~[~"/~C~')

~l~J('~)

÷~

3q t ~ -~=,5,4-~% ~.

-6~ 6-I

}=o

-~

for all sufficiently large ~ o All moments of ~ being zero (recall that ( ~ ) = ~ 0 , ~=0~.o. ) we have ~ = 0 whence ---~0 and ~ = 0 @ Unlike the preceding example (concerning Z ) it is not so clear how to replace here the interval [ - - ~ ~] by a set without interior. We are able to do it under some thickness conditions imposed on the set (see section 12 below). We don't know whether the mere nonvanishing of its Lebesgue measure is sufficient for the uniqueness (but we suspect it is n o t). Let us discuss briefly the values ~ (~) . Our proof works without changes for ~ > ~ I ,~=~,~... (the uniqueness fails for positive integer ~ s )° When ~ becomes negative ~f has n o sense but i t , s still defined o n a set E "free

from ~

" i.e. provided I l~-~l~-1~l~l(b<+~

(~6E)

. ~e

-CO

set of all measures ~

satisfying the above condition and such

~/~41~.iIf~ E

that I ~ I ( ~ ) = 0 will be denoted by contains an interval, ~ 0 ,~ ~ and ~ = 01 then ~---0 (the same proof as in the proposition). The situation for E~s without interior remains unclear - with one exception. If ~ is a negative odd integer we can associate

146 with ~J a function analytic in the upper half-plane, namely ~ - - ~ ( Z - ~ ) ~ - ~ ( ~ ) . Suppose ~ is finite, ~ E j ~ f and ~ f l E ~ 0 . Then this analytic function has zero angular limits on ~ and (E3], p.304) vanishes identically by the uniqueness theorem of Privalov provided ~ 6 ~ ~ 0 ~ whence ~------0 (see[l], p.146 for the details). So we have the following PROPOSITION 2. Suppose c ~ = ~ - ~ (~=~)...)~ ~ ~ 2 5 ~ ~,0 , ~ C ~ h a finite measure, ~,~J E = O

~en . Th

3. ~ractional integration. Now we turn to the fractional int egrals :

Fc~)_ C~-~f"~

C .e ~ ).

Here we meet the analogous uniqueness phenomenon but in a slightly modified f~rm. The vanishing of ~ and ~ on an interval, ( 0 ~ ) say, does not imply ~-~-0 but it is not hard to see that it does imply ~ I ( - o o ~ 0 ) = 0 . indeed,~%0(q~) = ~ ( q ~ ) (~(0,8)) where % 0 ~ = ~ I ( - O O ~ 0 ) , and we may use proposition I. Therefore the fractional integral I @ "remembers the past of ~ ": knowing ~I(0~8) and (0~) we authomatically know ~I(-00~0) . The usual integral I~ behaves differently. We shall give another proof of the above property of I 7 (0<~<~) . This proof was communicated to us by B.S.Rubin Suppose ~ is continuous, small enough at infinity, 'F

F~) I C~-~) ~-~" A simple computation shows (sac [ 4 ] ,

(~0) p.378) that

o

where~C~)=~(~)* ~ o ~ The function

J7

~-t

0~¢

-00 being zero on C0,8)

(~- o). we have ~I(0, E ) = O

147 (because of the Abel inversion formula: --'~=~-% (~)~% ~I(~) ~ O) and therefore ~ ( ~ ) I ( 0 ~ ) = 0

(t<0>

). But , where

. so our uniqueness problem

for 1 7 is reduced to the analogous problem for the Hilbert transform ~ discussed in section I. 4. Newton ~otentials. There are multidimensional situations where the same uniqueness phenomenon holds. A very interesting and intriguing one is connected with the Newton potential. Suppose ~ ~ and let ~ be a (signed) Borel measure in ~ concentrated outside an open set G ~ ~ ° If the potential ~ :

I vanishes on ~ , then j ~ = 0 . This fact can be proved by a reasoning analogous to the proof of proposition I. The same approach is ap~,!icable to the multidimensional Riesz potentials (i.e. to ~ ' ~ with (~-~) replaced by ( ~ % ) , ~ : ~ , ~ $ = ~ . . . ) . Another approach is the reduction to the Cauchy problem for the Laplace equation (this approach seems to be appropriate only to ~ = ~ ). Namely, ~ can be naturally defined as a function harmonic in the upper half-space~+ = ~ ( ~ f ~ . . . , ~ S O ~ ÷ 1 ) : ~ + 1 ~ 0 ~ ~ being its boundary. This function and its normal derivative ~ S~.~ sionj~ of

~t

+

~

vanish on from [~ ~+I ~

C~

lity ~ a I ~h+t

_

0

. Therefore the odd and the even exteninto are harmonic in , wh~-nce ~/~I ~R*I_ = 0 - The equacan be proved analogously. These equalities T

imply But things are so easy only when ~ is open. It is not known for example whether I ~ I ( G ) = 0 ~ ~I G 0 imply ~ = 0 in case P S d ~ ~ 0 , ~le~ ~ being the ~-dimensional Lebesgue measure. (Recall that the integral defining ~ is absolutely convergent almost everywhere on ~ with respect to ~ e 6 ~ , so that ~ c a n be restricted to an arbitrary ~ with ~ C ~ 0 )o The last remark is not too important because the problem does not become simpler even if we consider measures __J~=~" ~e6 ~ onl~with ~ continuous and compactly s~orted. We are able to prove uniqueness only for @ thick enough near a point (see [5] and [I], p.141)• 5. Sets of uniqueness. The following definition sums up all preceding examples. DEFINITION Let K be a distribution in ~ X a class

148

). Suppose the convolution ~ * ~ of distributions (in ~ ~:~X . The set ~ C ~ ~ will be calhas a sense for every

led a ( K, X ) -

set if

S X,

lE=o

There are two obscure points in this definition. The first is the expression " K* ~ has a sense" and the second is the restriction of ~ and ~ to E . But in every particular situation these points become clear from the context. Sometimes the distribution ~ (called also "the kernel" or even identified with the operator ~ ~ ~ ) admits no interesting ( ~ , X ) - s e t s at all. So for example every differential operator ( ~ a linear combination of ~ and its derivatives) p r e s e r v e s open zero-sets: ( ~ ) I m = 0 whenever ~I E = 0 , E an open set. ( ~ X)- sets are met for essentially non-local operators ~ , i.e. such t h a t ( ~ ) I B is influenced by the whole of ~ and not by 5 I E only as in the case with differential operators ( ~ being an arbitrary ball). Usually it is not hard to see whether an operator is nonlocal in this sense. But the description of a 1 I (~)-sets (including those with empty interior) is much more interesting and difficult. Such description has a purely quantitative aspect discussed below in section 11. 6. Semi rational s,ymbols. The existence of A (K~X)-sets depends on properties of the ~ourier transform ~ (the symbol) of K . Suppose for example that K is a tempered distribution in ~ , ~ E ~oo , so that the convolution ~ * ~ ^ is defined (as an [Z -function) whenever ~ ~ ( ~ = ~ ~ ). If ~ is rational then no interval (5£ ~ ) is a (~)-set. Indeed suppose ~ = p ~ - 1 , p and ~ being polynomials. Take an arbitrary interval ~=/: ~ and a compactly supported nono~ zero function ~ C vanishing on ~ ; let ~ be the differential operators with the respective symbols ~ q . Then

A

But if K is composed by two d i f f e r e n t rational pieces (i.e. its restrictions onto rays ( - c ~ ~b) , ( ~ , t O o ) are two d i f f e r e n t rational functions) the situation changes: such an operator has many (~,~%) -sets. DEFINITION. Let ~ be a Lebesque mesurable function on ~ ,

149 ~6e~ , ~6 I. kl(O,+OO ) =

. Suppose %1(6,+OO),

%

being a rational function;

~. ~,~[ ~C-oo, e-) : "~(~) = ~(~)} =

O.

Then we shall call ~ a s e m i r a t i o n a 1 function; will be called its rational part. The most important example is ~ ( ~ ) ~ - ~ (the symbol of ~ ). Another example is ~ ( ~ ) ~ ~ 1-4 (this semirational function can be interpreted as the symbol of the logacO rithmic potential: ~ - * - l ~ ( ~ ) ~ I 0 6 - t I ~ Y -~O

).

A

Put W ~ - - [ ~ " ~ ~ % } (~=~,~;...~ denotes the independent variable). The set ~ can be described as the class of all ~,~-functions, (~-~times continuously differentiable with ~(~'~) absolutely continuous and ~ ( ~ ) ~ L ~ . Every semirational function k determines an operator ~ , defined on the set ~K d,~,:~.~~ L~ ~,,~v(,P~ L~ } and mapping this set into ~ by the formula ~(~9 ~_ ~ (we shall write

).

K c ~ ) = K,~p. THEOREM I. Let tional part p~-@ mon zeros, ~_.C~,.~= Then every ~ set:

k

be a semirational function with the ra, p , ~ being polynomials without com-

t~'I, , Io~C.~Lk(.~)I~.C t~l YY/-'(~'~[R). with M d ~ E

~ 0

OK, ml PROOF.

Put ~ - ~ - ~ ' ~

CQPEWI41,1..I~)K)

is a ( K , W ~

g] ~ K )

-

E=O .

Then

~enoe _~C W~ • ( ~ e ~ d~ ~ ) . Therefore ~ = ~'~'~, Q bei~ the differential ope-rator wit~ the symbol

• The function Q(.~) can be computed classioaly, all needed derivatives of ~ being well-defined, 0 (i.e. ~I E = 0 ). Then Q C ~ ) ( ~ ) = 0 for almost every t E E - The same reasoning shows that P ( ~ ) ( ~ ) = 0 for almost every t E E ( p being the differential operator with the symbol ~ ). The function ~ d e ~ QC~)_p(~) belongs to ~gv and ~ C ~ ) - - 0 ( ~ 6 ) (the number C~ is taken from the definition of semirational functions). Therefore the function to the Hardy class H _ ~ , and

suppose(K,~)IE=

t-~-~(t) e~c-~%)

belongs

150 the vanishing of h on E implies ~ = 0 • Thus(~-p)~= ( = ~ ) ~--- 0 . But ~ ( ~ ) # ( ~ ) - p ( ~ ) ~ - - - 0 almost nowhere on the ray (-eO~ ~) (see the definition of semiratiohal functions), and ~ [(- o0~ ~)~-- 0 . This means that the , and function ~,~ ~ ~ ( ~ ) ~ C . ~ $ ) belongs to H + now ~1 E = 0 implies ~= 0 • +~

It is easy to see that in this case our theorem implies the following assertion: +00 +00

-cO

~65 ~

~0

-0o

, then

E = #

1E = 0

>

o

We shall return to the logarithmic potential later 9 below). oO

(see section

2. Put (I{)(~)=(V p }I ~{@) ~_~ -OQ

It is easy to see that I maps ~ into itself~ the symbol of I being semiration~l: ~ ( ~ ) = 0 (l~]<~)~ I ( ~ ) = =CO~G~.~ C { ~ l ~ ) . In this case ~ = 0 , and every set of positive Lebesque measure is a (I~ ~,~) -set. 3. Suppose the symbol ~ has the form G ~ ~ ,~and being polynomials, and ~ E ~oo , so that ~ K = ~ . If

0

deep

Ec , E > , 0 , Theorem 1 does not permit to deduce the vanishing of ~ C ~ from conditions ~I ~ = O~ (K*~)I E : 0 only: we need some regularity condition to impose on ~ ( ~ e W2"~ , ~ = 46~ p ) . We suspect these regul a r i t y conditiqns cannot be dispensed with. Take for instance the symbol-_~'. • ~ _ ~~. ~ ~ ~ . The corresponding operator reprepresen~sa perturbation of the Hilbert transform (as does the operator in the preceding example). Theorem 1 guarantees every set E with M4¢sE ~ 0 is a (~z I) -set. Is it a C ~ ~) -set? We don't know but the following theorem shows the answer is y e s provided E has a supplementary property. 7- Semirational sEmbols and Carleson sets. Let 4~ be a compact set of real numbers, ~Ce) the set of all complementary intervals of 6 o Put

151 ~+

~E~Ce} {in the 1~st sum only ~½ set

e

will

be c a l l e d

THEOREM 2 . L e t responding

operator,

~ith

1~f<+ ~

~e

a C a r 1 e s o n

~

be a semirational Then e v e r y C a r l e s o n

s e t

p r e s e n t } . The if~(e)<+eo.

function, set

E

K

the cot-

with ~/~E~0

is a

(~K) -set. The proof of this theorem is much more complicated than the proof of Theorem Io We shall only sketch it (for the complete proof see [I], p°p.156-167)k

,~ L

Suppose ~

~

, ~IE=C~)IE=

0,

E) < + c O , ~ e ~ F ~ 0 . We have to prove ~ = 0 • Assume O-----0 (see the definition of semirationality), so that

(here ~ ~¢-~ ~

e

, p,~

are polynomials,

rational part o~ ~ " ~ 1 { 0 , * ~ ) = We cannot assert now that

jo1~}1~(4+1~IV N~

pc-'Ko).o~)

~C ~

~+00

p~-~ i s the

, but

f o r a positive

N

o This

allows us to consider the Laplace transform 0

-00 It is sufficient to prove O = 0 , because then y = 0 and the proof will be finished exactly as in Theorem Io To prove the vanishing of ~ we construct a s e q u e n c e ~ } of ~_-functions satisfying

(b) there is a positive

(-~<~¢~<0, (c) ~

~.eo

of

E

C

such

t ~ t I ¢- ( ~ )Cl ~ < C - l~l

j = ~,%.,.) ;

I Cj I~---0,

E

being a Carleson subset

with ~ c ~ E > O.

Conditions (a), (b), (c) imply ¢ : 0 by a theorem of S.V. H r u~ev ( [6], p.p.143-146). (By~ the way, S.V.Hru~ev has proved that the existence of a part =E of E satisfying the Carle~on condition and with D%&& E >Ois necessary for (@)~(~)~ (6)

152 to ~mply ( P =

0 ).

To describe further steps of the proof let us suppose ~ ~ __~ ~ (otherwise some details must be changed), so that our symbol ~ - ~ 5 ~ can be still considered. Replace the letter ~ (i.e. the symbol of differentiation) in ~ ( ~ ) and ~ ( ~ ) (see (1)) by ~ ( ~ ) = { ~ - I ( 4 - 6 ~ ) , ~ being a small =

positive number (~(~) is the symbol of the difference quo4, 2% tient with the step ~ ). We obtain the function ~=~(~)~-p(~k)r~ which is near (in a sense) to ~ , which is the Fourier transform of a function vanishing on ~ N ( ~ - ~ ) but needs not vanish on ( 0 ~ + 0 0 ) . So we pass to the function ¢%s ~'~(-eO~ O) (~(-0~0) is the characteristic function of the interval (-00~ 0 ) ) which is capable to serve as a Q~ when ~ is small enough. 8. Another approach to the problem discussed in the preceding section is also based upon the cited H r u ~ e v theorem. It is easy to deduce from this theorem the following COROLLARY. Let E c ~ be a Carleson set with P~e~ ~ > 0 and let ~ be a function belonging to H~+ and suppose that the derivative of ~] E (computed along ~ ) is zero almost everywhere on E , Then ~ is constant. P r o o f see in [I], p.167. Let us consider now the operator ~ :

I

-O0 Here $ is a positive number. The symbol of ~ is semiratiocoast hal (namely ~ @ ~ + -v ~-$~ ) and in accordance with Theorem 2 every carlesonian Ec~ with ~ 6 ~ E ~ 0 is a (~,~)set. We are going to give a direct proof of this fact. Suppose ~ ~ , ~I E = K(~)[ E = 0 . Introduce the Cauchy transform of ~ :

function

(35) ~a

coincides a.e. on E

~ =

C(M~6~ ~ ( ~ ) l

with the f

nction $

E

. Now the

Ce. $

whose derivative vanishes almost everywhere on

E

J . The corollary

153

above implies the constancy of ~ which is possible only when The proof is finished,

. Thus F(~)=

6~-~-

We conjecture the C arleson be dropped°

0

ct~t6~p~&~)(~2~)

(because

FeH

).

condition of Theorem 2 cannot

9. LoAarithmic p0..t.entials of compac..tly supp.orted measures. Till now we have worked with convolutions defined on ~ (or on some parts of ~ ). This was not a mere matter of convenience; the widening of X is not always very easy when we are interested in the (K~X) -property of a set. We shall illustrate this by the example of the logarithmic potential, Let

and ~

~

be a complex Borel measure on

~

satisfyi~4~

its potential:

This integral is absolutely convergent for almost all real ~ o The set of all measures satisfying (2) will be denoted by J ~ , THEOREM 3, Every set of reals with positive Lebesgue measttfe is a (~l~seto This means that v

Ec , I£1(E)=o,

E=o,

The proof of this theorem goes along the same lines as the proof of Theorem 1 (see also the example I in section 6), but is much more delicate, The function ~ 41J is in general nowhere differentiable and therefore we cannot reduce our uniqueness problem to the uniqueness of the Hilbert transform by means of the usual differentiation as was the case with the logarithmic potentials of ~ - f u n c t i o n s . But fortunately we can use the so called approximate differentiation. Recall its definition. Suppose a function ~ is defined on a set E ~ ~ xQ~E o A number ~ is called the approximate derivative of ~ at X 0 if there is a set ~ E Z L E such that Xo is a density point of E and

154 X~ ~X~Xo

×-

Xo

In this case we shall write L = ( O ~ p ~)(zo) Now we need the Hilbert transform of a finite measure ~ , namely the function ~ ( ~ ) defined a.e. in ~ (with respact to the Lebesque measure) by the equality

Our Theorem 3 is a corollary of the following formula valid for every

(~o~p'~'~ ~'J~(~(~,)

aoe.

in

~

(3)

This is a slight generalization of a theorem of Titchmarsh E7] concerning the approximate differentiation of conjugates of absolutely continuous functions. Its proof is rather long. We omit it (it will be published in ~2~) and restrict ourselves by the deduction of Theorem 3 from (3). If ~ I E= 0 then ( ~ ) ( ~ ) = 0 for almost every ~ E o Thus ~ ( ~ ) ( $)-- 0 for almost every ~ E . The Cauchy potential ~ :

belongs to ~ ~ for every value of and lower half-planes. Now

in both upper

~--~+ 0 for almost every real ~ ( 144,denotes the Le~esgue measure). The equality I ~ I ( E ) = 0 implies ~ (~)= O a.e. The uniqueness properties of H~-functions imply ~(~)=0 ( ~ ~) and we have only to note that the Poisson trans-

155 form of

JL

computed at a point

~ (~

> O)

is equal

s,ymbol@: free interpolation. Let X~ be some spaces o$ distributions in ~ , Xc ~ . Suppose a linear operator K maps X into ~ . We shall call a set Ec~ interpolating (with respect to the triple ( ~ X , ~ ) if for every pair ~ of elements of ~ there is # ~ X satisfying 10. Semlrational

IE=,

IE,

E

(4)

(the meaning of the restriction onto E will be made clear in every concrete situation). The solvability of equations (4) (with "the unknown" #) for all pairs ( ~ , ~ ) ~ ~ X ~ is analogous to the well known phenomenon of the free interpolation of analytis functions (see e.g.[8]). The main object of this article is the homogeneous system (4) (i.e. with ~ = ~= 0 ). It is natural to consider the corresponding non-homogeneous system as well. Interpolating sets are (in a sense) opposite to (~X)-set~. so for example, no set E c ~ with H ~ ¢ 6 ~ > 0 is interpolating with respect t o ( ~ 7 ~ £ ) . Indeed let E A c E . ~5~i~0 ~ .

then no

#~b ~

can satisfy (4) with is

because

, a

(X,

9=0

and K =

-set.

In this section we shall be interested in interpolating sets for some perturbations of the Hilbert transform, namely for cor~olutions with semirational symbols ~ of the form A

(5)

I~ being a rational function bounded on ~ and bounded from zero on ~ . We have met such symbols already. Eecall that the question whether all sets with positive Lebesgue measure are (K~ ~ psets remains open. Nevertheless our result~ concerning interpolating sets will be more satisfactory. Roughly speaking the possibility (or the impossibility) of the free interpolation is very weakly influenced by the perturbing factor ~ and the interpolatory properties of operators with symbols (5) coincide essentially with those of 3 . We intend to restrict our functions and their images under

156

K onto sets of zero Lebesgue measure. So we have to reduce our class of functions to be in a position to ascribe a value to and ~(#) at every real point (not merely at a 1 m o s t every point). Let ~ denote the real Banach space of all real functions continuous in ~ and vanishing at infinity. DEFINITION. We shall say that a function ~C ° belongs to the class ~ if the integral

A-~T oO

It~l >A 0

exists for every ~ ) e ~ and ~ C ~ ) ~ C~ Suppose a rational function ~ satisfyies

It gives rise to the operator A ~ mapping the space tempered distributions into itself by the formula A

~/

of

2%

5"5. It is not hard to see that

A~J

C0

is a one-to-one mapping of

0° the space (of all functions continuous in E and vanishing at infinity) onto itself. Suppose ~ ( ~ ) = ~ ( - ~ ) (~E ~) and consider the operator

KR: ~ ~ C ~

0

defined by the equality ~ ( } ) = A ~ ( ~ ( ~ ) ) hout loss of generality we may assume that

(~c@)

. wit-

157

integer and complex non-real numbers. Then

(6) F'I

o

where ~ is the identity mapping of [)~ , } a convolution with a real summable and bounded function. Therefore ~R(~)-----~(~)+~(~(~)) (~ ~) . Noting that (~fJ~-=

A~°I

we have also

:B

(7)

I • :},'-,

I being the identity mapping of the space ~ % of all real (on ~ ) and ~ a convolution of the finite Borel measures variable measure with a summable function. DEFINITION. Suppose A is an invertible operator of C~ onto itself satisfying (7) (with A instead of A ~ ) . Then the operator K : g ( ~ ) @ e # , (~ ~) is called an a 1 m o s t H i 1 b e r t t r a n s f o r m. THEOREN 4. Let ~ be an almost Hilbert transform, E a compact subset of ~ . The following assertions are equivalent: I. E is interpolating with respect to ^ "[ ~K~) C[) ;

2.

me6E=

O.

This theorem represents a generalization of the Rudin-Oarleson interpolation theorem. The implication ~ can be proved by the usual argument (involving the F. and M.Riesz theorem on measures orthogonal to the disc-algebra). The inverse implication requires more efforts. In the classical situation (of the pure Hilbert transform, i.e. when ~0 ) one does not even mention it because of its triviality d u e h o w e v e r t o the uniqueness property of all sets with positive Lebesgue measure (see the beginning of the section). But if we cannot use this property when } = 5 h 0 (and we suspect it does not hold at all). Using a standard duality argument we conclude that the assertion 1 is equivalent to the existence of a pasitive number satisfying

for every pair of real measures

~ ~

supported by ~

Here

158 ~4 denotes the class of all real functions ~ summable on with the summable Hilbert transform ~( ~) , ~ denotes the Lebesgue measure, ~ is defined by (7) (with A ~ = A ).

To p~.vove # = > Z ( ~E Then

M(R.F) =

suppose

denotes the characteristic function of the set ~(~4--~)+

~ ( P

~+ ~ ( ~ ) ~ ) ~ E

0

~

)o

I~I

and (7) implies

E

0~

E

E

for every ~ vanishing off m . we shall be done if we construct a family ~ & } (~0) of functions of the class ~ vanishing off E and satisfying

~-~+0

E

~+0

¢E

To do this we remark that without loss of generality we may assume the origin to be a density point of F and E to be symmetric with respect to the origin (if not we shall consider E ~ (-m) instead of E )- Now put E 8 = E N [-8,~] The~estimate (a) is true because 0 a density point of E . The proof of (b) is somewhat more complicated and we omit it (it involves some standard estimates of singular integrals). The inclusion ~&~H ~ is almost obvious: ~ ( ~ £ ) ( X ) = @ ( X "S) for great X because of the symmetry of E and ~ ( ~ 6 ) ~ Now we are going to state an ~-version of the second part of Theorem 4. Suppose ~ is a function of the class L ~ ( ~ ) ~ ( ~ ) ~ 0~ o Define the operator ~ by the equality A

(

SeL ).

(8)

As an example we may take the operator (5) where the rational function ~ is bounded on ~ (but not necessarily bounded from zero). This enables us to compare the result stated below with theorems I and 2.

159 THEOREN~ 5. No set E C ~ with ~ e 5 E ~" 0 can be interpolating with respect to the triple ( ~ ~,~, L ~ ) ( K is the operator (8))° We shall only sketch the proof. As above we may assume to be symmetric with respect to the origin which is a density point of E . Denote by ~ ( E ) the set of all ~--ftLuctione vanishing off E and put Every pair R ~e~ ( ~ ) ~ L ~ ( E ) x ~.~(E) gives rise to a family of numbers [ ( ~ ) } (~ O) :

~E(F)=(F~E~(F)~E)(FE~)

.....

where k ~ ( ~ ) ~-

~En [-~,~] (~) S ~ (~aB, a>O).

The proof consists of two parts. In the first we show that whenever

~ RE (~)

~"E depends on Ir only. In the second we note that for every function ):(0~{) ~(0,+cO)) tending to zero at the origin there is a ~ e ~ ( E ) satisfying

~--,-o ~ ( ~ ) ~ f-1 where I I= ~U,~) -~EQ ~ ) -and of (4). To construct of positive numbers and put

(9)

, so that this pair does not belong to ca~uot be interpolated on E in the sense such a ~ we take a decreasing sequence [ ~...K} so that )(&l~)< ~(-3 ( K= ~ , . . )

k=1 Then

--+oo

~En [_~K,~K] ( ~ ) ~ x

160 and (9) follows. 11. ( K ~ ~ ) - p r o p e r t y

is a

(K,~)-set,

and Zweikqnstant~nsatz. S u p p o s e ~ C ~

~e-L~

,

~IE=

0

andllK(h~LXE )

i s S m a 1 I. Then ~ as a whole nrast be small (the c o rap 1 e t e vanishing of IIK({)II~(E) implies ~ = O, so it is natural to look for a kind of the stability connected with the
I1911 o

is not necessarzly" small when ~I E = 0 and IIK({)~I$(~) is small. It is possible to guarantee the smallness of ~ ~ ~ in a weak sense. DEFINITION. Let ~ be a Lebesgue measurable subset o f ~ , K a linear operator mapping Li into ~ , ~ a linear functional defined on L ~ . The function

k,E : (o,~-oo)--.- (o,+ ~) defined by the equality

will be called the ~ - r i g i d i t y of K w i t h r e s p e c t t o E . Recall that L%(6) stands always for [~.$:~I(~\6)=Of" We shall show that

IIKcll

whenever E is a (~,~)-set. So when ~ being pressed down to zero and ~ vanishes on E the quantity I~(~)l does not resist, does not exhibit any rigidity and goes to zero too. We think the notion of the rigidity is very important for the investigation of uniqueness properties discussed here. If we can find good estimates of ~ be it only for a total ,E set of functionals ~ we have automatically the (~ ~)property of the set E o Moreover the following remark is obA vious. A eo Suppose [ K ~ }

is a sequence of

~ -functions, K E ~

.

161 satisfying

^

A

aoe. i n ~ 2) E

; is a

(g~,~)-set

^

(I=~,Z,...)

;

3) { ~ , ~ I ~ . E ( ~ ) = 0 ~ ~ n i f o r m l y for every ~ e < ~ ) * ( o r at least for every ~ e ~, being a total set of functionals).

Then

E

is a

i n j.

(~,~set.

So if we only could find good estimates of rigidities of operators with semirational symbols (i.e. estimates taking into account-in am explicit f o r m - geometric characteristics of the set F and parameters defining the symbol) we should be able to extend the uniqueness theorems of sections 6 and 7 onto much wider classes of s~unbols. Indeed the class of semirational symbols (or even of symbols (5)) is very rich and it is very often possible to construct sequences of such symbols [ R~} approximating a given i in the sense I); the property 2) can be assured by Theorem I (or 2 ). The real difficulty arises when we try to obtain 3). Our proof of (10) (provided E is a (K~set) will be based upon considerations too abstract to serve a source of quantitative results needed in 3) above. Consider two Bausch spaces X and ~ and a continuous linear operator A mapping X into ~ . Put

Let ~ be a l i n e a r functional in X . Put LEN}~. Suppose X is reflexive. The operator to-one iff

~

A

is one-

C~(~) ----0

%--,-0 for every ~ X*° The proof is easy (we have to use the weak compactness of the closed unit ball of X ). COROLLARY (the Zweikonstantensatz for ( K ~ ) - s e t s ) . Let E be a Lebesgue measurable set of ~ ~ E ~ f ( ~ ) ~ ~ L ~O.

If

E

is a

(K,Lb-set and

~e(L~)*

then

162

I

=>

-00

where

~

~--~ 0

(~Y~)~ C}

~0

K~E

The proof follows from the definition of the rigidity and

from Lemma ( X = ~

(CE),

Et=L~(E),A(~)=~E. (!(*~) ( ~ h ~ (CE)).

We conclude this section by two examples. It will be convenient here to replace the line ~ by the unit circle ~ (the definition of the [~,X)-property can be rephrased in a ~eneral group-theoretic context but we didn't do it because for the simplest groups our knowledge is too scanty). The open unit disc will be denoted by D , the normalized Lebesgue measure on by ~% ; the same letter will denote the linear functional in

EXA3,~PLE 1 o Put

~(~)(~;)=J~--

v.p. I

~(~)~

(~L~(~), ~V).

The following theorem yields a rough estimate of the of

~

with respect to a set

~oR~

E c'~' .

6. Suppose E ~ T ~ m CE) ~ 0

o Then

This equality follows from the estimates

~CE) re(E)

~
~ " (~(CE))

m(CE)

~ - s ~(E)

m-rigidity

163 The right estimate is a simple consequence of the Jensen inequality, the left requires some work with outer functions and with the Ahlfors function of the set E • EXA/~PLE 2- Let ~ denote the operator transforming every ~ E ~ ~ (~) into ~(~) :

~e:F

( ~;eT). T

So here we are going to deal with the ~-rigidity of the logarithmic potential. The results of section 6 (modified from the -situation to the ~-situation) together with the abstract considerations exposed at the beginning of the section (see (10)) show that

E ~ T ) ~ (E) ~ 0 ~I:IIII~} III ~ ~~

&-*o

B ~ +,,E ( B ) ----- O.

It is however unclear which properties of E really influence the rate of decrease of ~,~E (6) when 6 tends to zero. Let us begin with some philosophy conCerning the possible character of ~ provided every set with (K,L~)-set (I'II,), K 0 i K,E s a a linear operator mapping ~ into itself and commuting with rotations, ~6(~)* ). It is very natural to conjecture in this situation that (given K and ~ ) the smaller is ~ ( E ) the faster ~E is tending to zero. Speaking more precisely one could expect the following inequality:

Er-,T (here lu%--~L~

"~(E)>

P,,~,E(~)~UK,~ (~, re(E))

(~.o, E=T, ~(E)~,o)

where ~K,~ is a positive function defined in(O¢oo)x (O,+oO) and such that

'~"1t1,, ~rl~p[ I,~/K~,~(E,1j,):I~I;F'}= 0

g-*O

((;F)'O)

(11) ; b , , 4

This conjecture is confirmed by the preceding e x a m p l e ~ k = ~ = Namely we could take in that case

'~

~).

164

But in general this conjecture is not true and ~ % 0 can tend K,~ to zero non-uniformly with respect to the class of all sets E with ~ ( ~ ) fixed and positive. This possibility occurs when K=~ , q = M% . we have no explicit estimate of ~ E (~) but we can show this estimate cannot have "the Hadamard form" (i.e. depend on ~ C ~ ) o n 1 y as is described by (11) and (12) and as is the case with ~ , E )" Put ~4 = ~ ~ {~ ~ : ~ 7/0} and let ~-~ ~ be the pre-image of E4 under the mapping ~--~ (~=~,~r..) TKEORE~ 7- The sequence [~,E~\--~/l~=4~ is bounded away from zero (though M4.C~)~---~ ~ PROOF. It is easy to check that

~---4,Z,..

)"

(13)

where ~ is an arbitrary Indeed if ~ = ~j~@ ~ ~

~-function and~(~)----A(~)(~T). we have

0

0

Substituting q7 ~ O~+ ~q~ into the last integral we transform it to the integral f~u~(~qy)~<~I~i~ ~ e l ~ V > and (13) follows, o Take now a function ~ E [,~(~)

Put ~----~ function

is

M%[F~ )---~ D%(~) ~I~(F~+I)I=~

(~) zero

C~ off

satisfying the following

F~(~ II

(~r, e=t~,...) • The

Ile

,= II llm

Using (13) we obtain % ~ 6 1 ~ ( F ~)I so that .

4OO

165 because

I~(:~.l~
Therefore the

IIt(F,~)lt~(E,~ffJ~ (,=,~,~,...)anathe

#1-ri~idity implies ~ ( ~ ) =

< ~o. definition

~F~(F~ )~ ' ~ , E ~ k - ~ - J

of

(~=4'~'"')~

It is easy to show that the phenomenon described in the theorem (i.e. {~4~ ~ E (8~) ~ 0 for a sequence [ 6 ~ } tending to zero) occurs for every operator ~ : l,~ ( ~ ) > ~(p(~) defined by the equalities A

A

A

A

provided Ko = 0 ~ d i . l ~ . ~ ~,} = 0 ( OC._ denotes t~e ~ - t h Fourier ooe~ficient of OC ). ~r~e~ K = ~,~ we have ^

o 4

-t

^

K#=-~I}I (j~o), Ko=o. IVe have no good upper give an upper estimate of which characteristics of Consider the subset functions 0t absolutely

(~r(~{t~) d 6~: , ~

estimate of ~ , E but we shall its modification. This estimate suggests E may be involved into ~ _ W4 of [,~ (6¢) consisting of all continuous on T and with 0t~ ~,$ ~6~)

~(et~))

o Put

K Introduce now the "small dang ~ : %~t- (~)

~1 E= Now ~ut

llt(~lllL~,(E )

E(~

c"

~r~ ~=

E

g---,-O

not exceeI

. an~

~OE(¢~ ---- ~ ( E ", E() It is known that ¢ ~

(= ~)

((>o).

(:0 E (0c) ~--- O.

being an absolute constant. The proof of this theorem is a quantitative variant of the argument used in Theorem I. Roughly speaking we differentiate ~(~) and obtain the IIilbert transform ~(X)whose estimate was considered in the Example I. To be a little more precise we not just differentiate ~($6) but take its difference quotient with a step ~ depending on ~ . In this very moment appears

166 see that the rate of decrease in the right-hand side is m • . w but by e(0 E also. The funcinfluenced not only by ~Nj(E ) tion 0 J ~ characterizes "the smoothness" of E . It is interesting~o note that the sets ~ of the theorem 7 are "not t! A... "uniformly smooth : ( ~ [~'1~ ~ __ 4 f,~--~ O ~ . Probab-

ly

this is the cause of the "non uniform rigidity" of ~ (in ) with respect to E ~ . we think W E must appear in the estimate of ~ ~, E too.

And now we add that after TheoremS7 and 8 the proofs of Theorems I and 2 seem a bit less artificial: the apparition of (DE (or of other possible characteristics of ~ connected with its s h i f t s) justifies the use of differentiation to reduce the uniqueness problem to the uniqueness of analytic functions. 12. MORE ABOUT M.RIESZ POTENTIALS. Put ~ ( ~ )

-~- I~l ~ - ~

(0<~(~> OGC ~ ) , so that ~ @ j ~ = ~ , the Riesz potential of the (signed) measure ~ (recall that the integral I I I~-~1 ~'~ ~ ( t ) converges absolutely a.e. a n d ~ f E ~ w~enever ~ rem).

is finite as is easily seen from the Fubini theo-

Suppose E is a Lebesgue measurable subset of ~ with a "strong" density point (the origin, say), i . e . ~ - ~ l ( - ~ ) ~ E l tends to zero very rapidly when ~ goes to zero (we shall write here lel instead of ~ ( 6 ) , the Lebesgue measure of e ). Then following the reasonings utilized in F5I it is not hard to see that

Le(.A A)

for

a

and an

A ;~0t

or that

E

is a CKc~L4 N L%C)- set

(~>~) .

(I4)

But we preferh top resent here another version of the uniqueness theorem for ~ . This version shows that (I4) holds for some sets E whose a 1 1 density points are arbitrarily"weak". We shall prove that E satisfies (I4) whenever there exist very small intervals "almost filled" with the points of E . But these intervals need not contain a fixed point. The weak point of the theorems we are going to prove is that they seem essentially

167

one-dimensional, whereas the methods of [5.] work in well. Suppose E c ~ is Lebesgue measurable and put

IP'EIIPI ,:

,

being the set of all intervals THEOREM 9. If

p

~

as

o) with I P I=

£

@

£--~ +0 then (14) holds. To illustrate this theorem take a strictly decreasing sequence ,~0O~ ~ of positive numbers tending to zero and place a set E ~ into each segment I ~ = E ~ K . ~ 0 0 ~ ] so that IE~I is very close to _IIKI but all density points of E W are arbitrarily "weak". Choosing [ So that II ~ I tends to zero at the origin very rapidly we can make the density of U I saalso arbitrarily "weak". So we obtain a set ~ E----~elU E K t i s f y i n g (15) but whose all density points are as "weak" as we please. Theorem will be deduced from the following THEOREM IO. Let ~ be a finite (signed) Borel measure in ~ . Suppose there are a positive number ~ and a sequence !hP~t!a of intervals contained all in a bounded interval and s t

~K}

(a) ~ $ ~

I Pl I =

measurable subset

; (b)every ~

contains a Lebesgue

satisfying

E}

and

~. --t.- GO

~hen

E ~

0

I

#, = 0 w

We begin with the deduction of Theorem 9 from Theorem IO. Put ~ ~ (recall that ~ denotes the Lebesgue measure) where ~ I N ~%0~ ~I -- 0 and prove that satisfies conditions of Theorem IO. Take a sequence ~ of positive numbers tending to zero and such that

E ='~ I E

.~

~ ~(Or~)=-OO(k~

~

). To every

~

corresponds

I68 a ~jC T ( ~ )

where

satisfying

~ > 0

is chosen so small that ~

~¢~(~(~)+8#')=-~.

We may assume ~ C ~ ) #-*~ "~ ~ 4 so tha t iEm~P~ / ~ t t e ~ b L d e ~ d (~=~,..) , and theeboundedness of E ness of I] ~ . The condition (I6) is trivially satisfied. Turn now to (I7). If ~ is large enough we have

C=C(#,p,~), ?=

pP

Taking logarithms and noting that , ~ m

[

= 4 (see

(I8))

we

show that (17) is satisfied, and ~ ~---0 by Theorem I0. PROOF OF THEOREM I0. We may assume both sequences of endpoints of ~: 's tend to a point, say ~ . Introduce functions k~ on

o

tep~

the set ~ f ' ~ s ° satisfying shall) assuage $ ~ ~(~)= cause

~pL ~-'T~ 0

(17) i - ~ ? e ~ # ~ l ~ l ( ~

infinite we may (and . ~In this case ~ ~[ [ ~ = 0 , be-

I= 0

I~b

J-

for all

large values of ~ we may assume (taking subsequ~nces if neces' " ~, . + ¢,I~d. sary) that ,~¢114 k,~( ) exists and is 4 or e -. We Let ~ 9 0 e ; ~ I ~ ~ 0 : ~ = l ~ l d 'e, O~e~,z shall write ~ J ~ r [ ~ l ~ t f i ~ ( ~ > 0 ) . Put

~,

C~.t)~_~,

( ; ~ : : , o , $=t~,,...).

These functions are analytic in the upper half-plane

~,~ k'~.{~÷~.~)( ~-*÷0 ~ ~ --~-,-~''C:~ ))= l,t~ ~'("~,)whenever

~

, and

~ is an interior ~oint of I-~, b. , J~, ~ e 2 " ~ ~ "~\p:~ ¢/~ (~6 being the characteristic function of the set ~ ). Consider now a Jordan region ~ ~ C + with the smooth

169 ( C~ where

, say) boundary ~ such that ~ n ~ = ( - ~ ~ 3 , ~ such that Up~ = ( - ~ ) It is not hard to see that ~ belongs to the Smirnov class EI(~) (see E3] , p.203), ands~ I~4(~)]I~ I ~ , ~> being a constant depending on ~b , ~ ~ an~d ~ only. If 06EPI" we have K ~ ( ~ , ) ~ - - ' ~ ( ~ , } ~ - ~ " P ~ , ) _ I'~,I~,~IP~ ( ~ ) . But "a

-PI ~

-

<~ ~ I~I(D-)~ C:C(~,~)>~

and (I6) ~ p l i e s

~ 11~(~)1a~ ~<(C~l~)t~l (p~) (t=t~,...). c ~et ~0% be the harmonic me~sure on ~ ( ~ i t h respe t to I~ ) computed at a point ~ ~ ~ . It follows from the smoothness of ~ that ( ~ is comparable with the Lebesgue measure (the azc-length) on ~ :~'I EI~0j~(E)~.IE I ~ E being an arbitrary Lebesgue measurable subset of the curve ~ ' ~Z and ~ positive numbers not depending on E . Now

where

~,~ ~ o_~ ,f

~(E{)>-~ I~_ I E~ .~,~

-~

. This estimate, the inequality

and (I7) imply

t(~(z)= 0

~ut .~'~, ~-(~)=f

--.--~

(~). ~

(~

~ ~'0)

, where

It is not hard to deduce from the last identity that it holds with o~--0 too (see EI~ , p.144) and therefore the measure , ~ being ~ is ~-absolutely continuous, ~ = ~ ~ afunction from the Hardy class ~ " But l,r~ T-~

~ II~l~

,=n~

~1~ I ~ ~<

170

-I P~l~e~ t~ 1 * IP~ 1#e~t~tcP~) • Clearly I P.~IBe~I~I(P~).
[

~I~I--

We are ~one, because

-~ o~ ~ h i c h

I ~(#)I:

4

, and (=7~

is impossible if ~ 0 whenever t f f & ~

•

, and if

~(~) =

0 , then I ~ I [ ~ } = 0 • The most essential feature of the Riesz kernel ~ used in this proof is that K~ is composed of two analytic pieces: K~I(O' +00) , Ko~ l( "00' 0 ) ooincia~ng both with boundary values of functions analytic in ~ , (namely of ~ ~ ' ~ and o f ( - 6 " ~ ° t ) , ~5~-% ). The article ~2] contains uniqueness theorems applicable to many kernels of this kind.

References !. E p ~ E E e B., X a B ~ H B.H. H p ~ m m n Heonpe~exeHHOCTZ onepaTopoB, nepecTaHOBOqHNX co C~BZrOM.!. - 3an.Hays. ceMEH.~@~, I979, T.92, c. I34-I70. 2. E p ~ E E e B., X a B ~ H B.H. H p ~ m m n Heo~e~eXeHH0CTH onepaTOpOB, nepeCTaHOBOqHNX CO C~BEPOM.H. - will be published in the same "3aHEcI~". 3 H p E B a x o B M.l{. ~p~HHtKHHe CBO~CTBa aHSJL~T~eCEHX ~yHEL~fi. M., 1950, 336 c. 4. P y d E H B.C. 0 HpOOTI~HCTBaX ApOdH~X E H T e I ~ O B Ha ~p~-MOJG~He~HOM KOHType. - I~3BecCE~ AH ApM~HCEOfi CCP, I972, T.7, 5, 373-386. 5 M a B ~ a B.~., X a B H H B.H. 0 pemeH~HX 3 a A a ~ l{o~n~ Ax~ ypaBHeHZH ~an~aca (eAEHCTBeHHOCTB, HOpMaJIBHOCTB, a n n p o E c ~ -TI%MOCEOBCEOPO MaT.O6--Ba, 1974, T.30, 61--114. 6 X p y m ~ B C.B. H p o ~ e M a O~IHOBpeMeHHO~ aHHROECE~21S~E E CTHpaHHe OC06eRHocTefi HHTePpaJlOB THII~ K O ~ . -- Tp.MaT.HH-Ta AH CCCP, 1978, T. I30, 124-195. 7. T i t c h m a r s h E.C. On conjugate functions. - Proc. London Math.Soc., 1928, voi.29, p.49-80. 8. B ~I H O r p a ~ o B C.A., X a B H II B.H. CBO60~HZLq ~[HTepe0 noam~ B ~ Z B HeEoTop~x ~pyr~x F~accax ~ H E L ~ H I. - San. Hay~.ceMEH.~0~,i974,47,I5--54; ~ - idem ,I976,56,I2-28. o

VV..

S.V°Hruscev,

S.A°Vinogradov

FREE INTERPOLATION IN THE SPACE OF U N I F O ~ Y

CONVERGENT TAYLOR SERIES.

O° Introduction. 1. An axiomatic approach to the Banach-Rudin-Carleson

theorems.

2. An asymptotic formula for the distribution function of the boundary values of Cauchy integrals. 3. Interpolation in the space

~A

@

4. Two applications of the asymptotic formula.

t72 Introduction The main subject of the present paper is the space consisting of the functions in the disc-algebra

CA

whose

Taylor series converge uniformly on the unit circle T . It is clear from the definition that U~ is a proper subset of CA . Indeed, for every subset E of ~ with the zero Lebesgus measure there is a function ~ in CA whose Fourier series diverges on E (see ~] , p.58). It is possible, of course, to mention other properties showing the distinction between the spaces

CA

and

~A"

For example,

U A

is not an algebra

[2]. Our main aim in this paper is the study of proper±ies common for C A and ~ A both. Having originated in the classical papers of Banach [3] , Paley [4] , Rudin [3 , [6] and Carl eson 7] , the theory of the disc-algebra was continued in the direction of our interest in the series of papers [8-14] . The things were not so good for the space ~ . But now they are looking up thanks to certain recent progress mainly connected with the papers [15-2~ and diminishing the disproportion of two theories. The authors of the present paper hope to publish in a future a survey of harmonic analysis in the spaces ~ A and CA . We collected here some new results and some new approaches to the subject which appeared during our work on the above mentioned survey. For the more detailed exposition some notation is needed. Let ~ denote the usual normalized Lebesgue measure on T The Fourier coefficients of the integrable function ~ on the circle are defined by the formula

Then the symbol D~* ~ ~(K) ~K denotes the p a r tial sum of the Fourier series of ~ . By Weierstrass theorem CA is a closed subalgebra of C ~T) continuous functions on the unit circle formula

, the algebra of all T . So the following

173

will not mislead the reader. The space ~ A in 0A satisfying

Herell~ll.:~.~ It

is

a

I*(~ I

Banach space

II Let U ~ t ional

consists of functions

stands for the usual norm in C ( T ) .

with

respect

to the

norm

II

denote the conjugate space for the Cauchy transform of

~ . For every funcis defined by

~0 it is clear that ~ ~ is a holomorphic function in the open unit disc D ~ - { % E ~ : l~I < ~ } . Let ~ P , ~ 0 , denote the usual Hardy space in ~ . The next theorem is of prime importance for what follows. THEOREN (S.A.Vino~radov D5], [16]). If @ 6 U ~ then ~ C P~
0<~
174

We give a simple proof of this theorem. It is based on Vinogradov's theorem mentioned above and on the following result whose proof is given in ~ 2. THEORE~i 2.1. Let ~ be a finite complex Borel measure on and let ~ denote its singular part in the Lebesgue decomposition of ~ with respect to the measure ~ o Then

Here

~jI~('~)

stands for the Cauchy integral S

~J~'(~) -~_%

"

Our proof of the Oberlin's theorem may be extende~d in two directions. The first approach leads to the theorem about a simultaneous interpolation and to an axiomatic theory which differs from the Bishop's one [25] • The second generalization of the proof strengthens the Oberlin's theorem. Referring the interested reader to the section I we shall formulate now the simultaneous interpolation theorem for the space U A "desT°do this remind a definition. A subset ~ of the set ~ ~ - ~ 0 ~ . . . } of all non-negative integers is called a ~(~) -set if there is a positive constant C ~- CyL such that for every polynomial ~ e ~ ~ (v~) ~ ~ with the frequencies in ~ the following inequality holds:

']l"

"B'

For a closed subset E of ~ and for be a linear operator defined by

~/k

, A=

~

, let

THEOXEM. The equality

holds if and only if

~E~

0

and

~

is a

~(~)

-set.

In section 2 we deal with some extensions of the theorem 2.1. These results are applied in § 4 to some problems of the interpolation theory in U A ° Here is one of these applications. For every square-stummable function ~ , ~ L%(T) , let A =

175 denote the orthogonal projection of ~ into the Hardy class H i . The next theorem deals with the space

Here K is a closed subset of T a n d the symbol ~%~p (~) stands for the support of the function ~., THEORF~, Let ~ be a closed subset of T , let ~ ~ > 0 and let E be a closed subset of the set of density points for K , ~ = 0 • Let _/i be a subset of ~ with Hadamard's gaps. Let, finally, the operator ~ be defined by

a÷

s v(K).

C V ( K)~-- C(E) x///q'(.~) . There is a definition of the space V(K) equivalent to the previous one. If # E V ( ~ ) then the Cauchy transform ~ ("t3-~) -'l #(~) ~ % ( ~ ) is holomorphic in ~ k ~'. So the space of Cauchy transforms ~ ($-Z)-~(~)~(~), ~ V(k), K is explicitly the space of holomorphic functions ~ in ~ \ ~ such that ~I~ ~U A ~nd the restriction ~I ~ \ belongs to the Hardy class outside of the unit disc. It is not trivial that V(K~ =#=[@~ if R t ~ > 0 and if the set ~ is nowhere dense on ~ . The above theorem asserts that the space V(K) is large if ~ > 0. In section 3 the properties of the space ~ A are discussed in more details. We prove an analog of the ~. and M. Riesz's theorem for ~ . Together with one theorem of J A.Pe~czynski this implies the existence for every E~ #%F:- 0 of a linear interpolating operator T : C ( E ) ~ ~]A" Being identical on the circle, the classes of interpolating sets for CA and ~]A ar~ different in ~ . We give an example of a closed set F such that ~ N ~ = [ ~ } O~ I ~ : C ( ~ ) but U A I ~ =~ C ( ~ ) . some sufficient conditions for U A I E = C(E) are also given in §3The final part (~4) of the paper is devoted to applications of theorems 2.1 and 2.4. One of these applications was already mentioned above. The second application depends highly on the following identity

Then

176

also established in§4. Here ~ ( U ° ° ~ stands for the set of all multipliers of the s p a c e U / A ~ H ~ : ~II =0(~)~ and 11%(~/) denotes the multiplier space for ~ . This identity allows to extract a useful information about the space ~ (UA~U~ . I n particular, we prove that the transformation ~ ~ I-~ maps the space ~ ( ~ / ) onto $4CA_) for every subset ~A~ of ~ with Hadamard's gaps. We prove also that Blachke product whose zero set satisfies the Frostman condition (see ~3 for the definition), belongs to ~ CU / ) ~ * on the other hand, if ~ is an~ --inner function in l~tU~then the radial limits ~ M ~ ~ ( ~ ) = ---~ ~(t) exist ~everywhere on ~ and l~(t)l=~%~ 4-0 E~ . This implies that ~ is a Blaschke product. In conclusion we announce a theorem generalizing a recent result due to de Leeuw, Katznelson and Kahane ~ . Combining the method of ~ 8 ] with the method of S.Kisljakov ~9], and with our scheme of interpolation, we get our result (see the end of §4 for the formulation)°

~1..An. axiomatic

approach.t£ the Banach-RudinCarleson theorems

The Banach interpolation theorem was proved for the first time in E3]. It is the premise for the following definition. DEFINITION. Let _~ be a subset of ~ o It is called a Banach set if for every square-summable sequence~C~(~) _ ~CE~(_~) , there is a function ~ in C(T i ~ e ~ ~ satisfying A THEOREM (Banach E3] ). A finite union of gap subsetS of is a Banach set. A Banach subset ~ can also be described as a subset of generating the Riesz basis ( ~ ) ~ c ~ in the closed span of the family (~)~A in ~.P(T) , 0< ~<~ • Let ~Jk denote the set of all trigonometrical polynomials with frequencies in _~ . THEORE~ (Rudin ~6~ ). For every in ( 07 ~) the following conditions are equivalent: 1)/~is a Banach set; 2) there is a positive number ~ p such that

177

Here ( IIi ,I,p S~ Ij , j l ~ I P g , ~ stands for the usual LP -norm. For p~--~ this theorem is due to Banach [3]. It should be noted that after the paper ~] of Rudin the Banach sets were given the name of ~(~)-sets. Nevertheless we prefer to use the first term in honor of Banach. The following fundamental theorem describes interpolating sets for C A on the circle. THEOR_F2~ (Carleson - Rudin [ 2 ~ ) . Let E be a closed subset of T . Then CA I E : C ( ~ ) iff w~E-~-0It was proved in D 3 ] that i~ I ~ : ~ 07~,~,... then the concepts of Banach sets for ~A and C(T) are identical. Moreover, Vinogradov D ~ coupled Banach theorem and the theorem of C a r l e s o n - Rudin. Let G: CA---*-C(E)x~ ~ (JL) be defined by

GT=(# I E, #[A). THEOREM (Vinogradov B3]), The

identity(~(GA)=C(E)x~ (Jk)

holds if and only if ~E~ 0 and /~ is a Banach set. In this section we shall formulate simple axioms which imply the assertion of the above theorem for a large class of spaces ~ _ o The leading example will be the space ~ ~--- U A ° Let denote the aat of all polynomials. AXIOM 1. Let ~ be a Banach space of functions holomorphic in ~ such that A

~----~

II

=

I ,~

~< i

.

(A1)

It follows from this axiom that IT-'i < ~

• Indeed, co -

and the s~ries in the right-hand side of the equality converges in ~ (see (AI)). The space ~ satisfying the axiom I, it is possible to define the Cauchy transform

178 co ~=0 It

is

clear

space

~*

that

onto

Cauchy transform

is

a

one-to-one

map o f

the

~*

AXIO~ 2. There is a positive number

~

such that

~6

0<~<4 for every ~ > 0 and for every ~ in $w. Axioms I and 2 are valid for the spaces 0A and U A . ~or the space 0~ this follows from the Kolmogorov's theorem [26] and for ~ A from Vinogradov's theorem mentioned in § I. Here is another example . Let ( ~K ) k ~ ~ be an increasing sequence of positive integers with Hadamard's gaps: ~ ÷ ~ / ~ >/ ~ > ~ o Then

k-'-.- ~ o~

Clearly ~ACUA(~K)C CA and (A2) is a consequence of Kolmogorov' s theorem ( [28], ~h.XIII, I. 17). The following technical lemma reformulates the condition (A2). LEMNA 1.1. Let ~ be a Banach space satisfying (At) and (A2). Then ~ ~c ~ ~ P and moreover

p<1

~6

Proof. Let a number

%7 ~ > ~,

+o~

oo

0

0

be fixed and let

oo

p

P

p

0

It follows from this inequality that Therefore the radial limits %--" I - 0

~ * C

H P

if p <

179

exist for almost every

~

in

T

.

Let

E, and let % . { ~ . Then the set E4 is covered b y ( U O E $ , ) U ~ where ~ is a subset of T with zero Lebesgue measure. Therefore (2) is a consequence of the obvious inequality

mC.O Eto) ~ ~ Now we are in a position to formulate the main theorem of this section. THEORE~ 1.2. Let

~

,

~C

~

, be a Banach space endowed

with a norm stronger than the sup-norm and satisfyin~ axioms I and 2. Let ~ be a closed subset of ~ , ~E= 0 , and let _~

be a Banach set. Then G C ~ ) = ~(E)x~(A). it is interesting to note that known proofs of the CarlesonRudin theorem use the ~. and U ° Riesz's theorem. Our proof of theorem I°2 shows that this theorem may be avoided . The theorem 2,1 substitutes the theorem of ~. and ~.Riesz in our approach. This not only simplifies the proof but extends it in a more general setting. The idea of our approach appeared in connection with the recent paper [ 2 4 ] ° PROOF OP THEORE~ 1.2. It is clear that the operator ~ : ~ ~ C(E)× ~ ( ~ ) is bounded. By the Banaoh theorem [29] it is onto iff the conjugate operator ~* -is an isomorphism onto its image. Some words about the notation for the duality between the spaces we consider. The duality between C ( ~ ) ~ ( A ) is standard and is defined by and MCE,) * 4'3'C.A)

~ere E, 'k-* |$: t~E~*

~.A_

. The Cauchy transform defines the duality between ~ and ~ * on the dense subset of rational functions with poles outside the closed unit disc.It is easy now to compute the Cauchy transform of a functional ~ G* (p, ac) ~

( j ~ , ~ e MCE,) xOC~) : ~(~)=
~t,)'~, C { ~ , ~ ) > = < Q (4-~t7 ~, (ff,~:) > =

180

The space ~ being imbedded into the disc-algebra,every finite Borel measure ~ defines a continuous functional on ~ :

To simplify the notationT we denote this functional by the same symbol 5~ . Let Y-~= 5~ + ~5 be the Lebesgue decomposition of the measure 5~ with respect to the measure ~ . LEMMA 1.3. Let ~ be the same as in theorem 1.2. Then for 0 < p < ~ , there are positive constants 6b and ~ every P, such that for every finite Borel measure

5~

on

~

the follo-

wing inequalities hold

II ~1 H~ ~< ~ptl~ll~

~)

We shall give the proof of lemma later and now we shall show how to finish the proof of the theorem. Let 0 5 ~ ( 0 5 ~ ) ~ E ~ 0¢E 6 ~ ) ~ and let ~E M ( E~ ) . Then the formula

defines the Lebesg~e decomposition of ~ It is clear that for ~ -~- ~ ( ~ * ~) we have ~ ~ ~5 ~ . it follows from the inequality (4) that

11l,l,e~ ~ ~, ~ " IIP ~ ~p. I1q~I1~, The set J~

being a Banach set,there is

CA ~ 0

such that

~. I1~ ll~a, ,< I1, . ~ ~,, ~ ~ lip -< ~ tl* 11~ • On the other hand it follows from (3) that

II y IIM(E~) "< ~'' It * I1~,. This implies

II>~IM~E,)*It~tI~cA ~.< o~t t1¢1t~.

•

PROOF OF L E ~ 1.3. To prove the inequality (3) it is sufficient to compare the identity

181

with

(2).

T h i s i m p l i e s (3) w i t h 1.1. shows now t h a t

~-~-ge.~.

By Kolmogorov-Smirnov theorem [27] we have

II ~11~, -<

P

~_p

T h e r e f o r e u s i n g (3) we get

P

P

II x~ll~p .~ ~pI1~11~.+ _ II % IIM~, -< ~pII ~11~., oe II

ll~"

~2, An as,ymptotic formula for the distributiqn function of the boundar 2 values of Cauchy integrals In this section we prove theorem 2.1.,which was formulated in the introduction,and some of its generalizations. But we begin with a collection of proofs of the so-called Boole formulae (see formulae (5) s.ud (6) below). Let

T be the S~hwarz

integral for a positive measure ~

For a positive measure ~ on function

on ~

and let

we consider the distributi-

Unexpectedly enough there exist explicit formulae for A T and ~ *.)This fact has applications in ergodic theory and an interested reader may find the literature in [30] o The formula for was found independently in [31]. GoBoole has found the formula for ~ assuming that ~ is a finite sum of point masses [32], Boole's formulae can be written as follows:

*) The measure

is assumed to be singular.

182

The proof of (6) (accordingly to Boole and Levinson E33], p.69). Let

5 = I<---'I Z~ , ~ . ~

,

~ ~

o ,

cuing of the extended line ~ U { co} defined by the points ~K, k=~,...~ ~ . Then we have for every measurable non-negative function ~ :

R

k JK

B

Here ~K denotes the inverse function for the restriction ~ I~ . It should~.~ ! be notedt A.~+~that ~ increases on each ~ because ( ~ ) ( ~ ) ~ ~ ~ 0 . It is clear that the numbers 3C~(~))...~~C~(~) form a full collection of roots of the equation

~. H (:'r,,<-~)-Z] .~)~ H (.1;~-~)= o or

equivalently

of

L

,:,~

- K--=.'[ Y.., tK] +

"'"

=

o

"

It.

Vi~te theorem 3C~(~)+...+ and therefore

By

=~(V)=-~'~II~B+~ t~ =4

To prove (6) for any singular measure we use the following limit procedure. Without loss of generality we may assume that has a compact support ~ ,Skd3C = 0 • Choosing ~ complementary intervals of the set ~ , we put discrete masses at their ends so that the ~ - m e a s u r e of the closed segment connecting the neighboring points is equal to the sum masses at the end-points of this segment. Let ~ denote the discrete measure constructed above. Then ~ - ~ u n i~f o r m l y j on compacts separated from ~ . •

TO prove the formula (5) (see simplicity that

I1~11 = i

. Then

[2~, [30] ) assume f o r the

~gcz~ = 1+~z~Fgz)

183

and it is clear that every asymptotic formula for ~ implies an analogous formula for ~ . The measure ~ is non-negative and singular. So the Schwarz integral maps T onto and the unit disc into the right half-plane. It follows that

where I is an inner function, I ( 0 ) = 0 • The is measure preserving for the measure space ( T ~ )

T

. Indeed,

O, n~O,

~

and therefore

map$-~I($t Se?,

t¢~,o_~'4~

~

. Using this, we get

Our next proof works for the case of the disc and for the upper half-plane as well. Let

where i~ denotes a non-negative harmonic function in having zero boundary values a.e. on T . Let ~ 0 Then

D

, be fixed.

By the mean-value theorem the left-hand side of the identity is equal to ~ W ( O ) C ~ ( O ) ~ ) -I . The right-hand side is equal to ~iI+8~ ~ wheTrel°n T . Putting

6=

because II~II

i~(~)~ 0 almost every for the brevity, we get

co 6+ E

0

+&~

or after the change of variables ~ ~

@

I d ct)

~

~ ~i

~C :

184

~3)(.~)

(3]-~)

where =-~t~/~ denotes a positive measure on E 0 ~ ,oo ) . The integral transform ~(~)=~ is the well-know~ Stieltjes transform. The function ~ is holomorphic in ~ k E-0o, 0 ] and the Stieltjes inversion formula restores the measure ~ :

~--"0+ The equality holds in the sense of generalized functions

(see

D~I, p.zo). A simple calculation shows that

d~ Cx)=

qU

61,~+ :g

and therefore

THEOREN 2.1. Let ~ be a finite complex Borel measure on and let ~ 5 denote its singular part. Then

{~

,~-,,t:teT-Ic~(bl

Ul- 4.11~11

l~l

PRoo~ ~et denote the variation of the measure ~ LE~ 2.2. For every E > 0 there are smooth functions ~ in C oo (~) such that

II~-P Ij~lllMcvp e,

IlPlloo~,

Iju I-o~.~ IIM(V) ~

. ,

a,

[~too ~< I .

PROOF OF THE LEbNA. For a positive measure I)

, ~e~(T),

let

~n~ ~t W

~enoto @~o c ~ o o u ~

o~ t ~

~

~c~o{~).llstl~4 ~

~n t ~ sp~ce L ~ ( 0 ~ ) . I t i s easy to see t h a t B e W (and actually B = W ). Indeed, if D E ~°°(&$) and if II~ ~$I~ 4 for every ~ in W , then the identity

185

= 11q 4 IIMcT, = I I q°l 4v shows that I [ ~ ] ~< 4 ~ for f ~ . Putting ~ ) = Ij@I and -, we see that ~6h~(~I) and that I ~ I I~I-almost everywhere It follows that ~, ~ ~ and therefore for a given > 0 there are smooth functions p and ~ such that

The following lemma finishes the proof of the t~eorem. LF/~&~ 2.3. Let ~ ~(T) and let ~ ( ~ ) ae# ~:I~(~)I~}, ~ being a measurable function on ~ Then

.

PROOF O~ THE LEMMA. It is sufficient to consider the first inequality. The proof of the second one is analogous to the proof of the first. Let ~ be a fixed number in (0~4) . By lemma 2.2. we may find a function p in g ° ° C ~ ) such that

Now

where

~c~)=X(Pl~l)Czt-p('s~(14q)(z}=I ~--- ~(/4,-~aijl, l,l) . I t i s clear

Pinally,

11P I1~ .< 1

This implies

dj.(~)

~hat ~c C(~) an, By the Kolmogorov's theorem

and therefore .~ZI~ $ ~ ( ~ -- 0 we have ~6--~ oo ~ ) --

IIj~-pI~III

p(~t-p(~) -~-~_-~---

"

~

t ~ o.

and therefore

,

186 We

have

It remains only to make 8 tend to zero. @ REE~kRK. The assertion of lemma 2.3 may be sharpened. Let denote a measurable subset of ~ . Then

(7) The second theorem of this section finds its applications in delicate questions of the interpolation theory. Remind that a point ~ in the subset E of the circle is called a density point for E if

I £ being an arc of the circle centered at the point ~ with the arc length equal to ~. THEOREM 2.4. Let ~ be a compact on ~ and l e t ~ K > 0. Let E be a closed subset of the set of density points for K having a zero Lebesgue measure. Then for every measure supported on

II?II in

PROOF. The equality (7) shows that we may assume the measure to be positive. For every positive integer } and for every let E% = E%(~) denote the set of all points E such that the inequality

v~(In K) ~./g~(I) holds for every open a r c I

,

8

(8)

~el

,

~I

< j-1

The set E being a set of density points for ~ , it is clear that E = ~ I E} . The set E ~ is obviously closed. LEMNA 2.5. Let ( ~ ) ~ be a sequence of open sets o n T

187 satisfying b) fo~ each ~ the closure of every component of the set ~ contains a point of ~ •

Then "~,/'11,. I'1~( ~t'1,n k') ~ '1- ~. PROO~ OF THE LE~A. Let ~ ~ U ~ ~ , where (~ff~)~ is a sequence of components of the open set ~ . The condition a) of the lemma implies the inequality ~(~)~-~if ~ is sufficiently large. By the condition B) C~Q~n E~, =~ 0 and therefore (8) implies

~C~ n

K) ~ ( I -

~)~C~}

for every ~ if the number 14, is large. The proof is finished by adding the above inequalities. • For ~ 0 we may find ~ such that [1"~-~]~;[IM('~')~4- < ~ Put ~ ~ ~ ~ for the brevity and consider an increasing sequence ~ ~ + o~ . It is clear that the set mk

aJ

is open. It follows from theorem 2.1 that ~ ~(~)~0 and it is easy to see that the condition b) of le~na 2.5 is valid for ~ . Therefore

being large, and consequently

But it follows

that

from

188

(the last inequality is a consequence of Kolmogorov - Smirnov theorem), This means ( ~ is arbitrary positive number) that

4 "lff ,.-~- + O0

On the other hand by theorem 2.1

REMARK. For our applications the estimate

(9) will be sufficient, Therefore the assumptions on the set may be considerably weakened, If we assume that the set consists of points ~ satisfying ~t(In~) ~

F for

every interval ~ containing and having a sufficiently small diameter, then (9) holds with ~ = o". ~ -~" ~3, Interpolation in the space U A

This section is connected m~inly with the space U A it is convenient to consider two more spaces. Let U ( ~

but

stand for the space { ~ E C ( ~ ) : ~ o o l I D ~ ~-~IIo9 = 0 } . To define the second one let denote the partial ^ :DK,~ * ÷ . Then

~-*+OO and the norm in the space

~(?)

stf

(K,~) Clearly the map ~ ~ It is obvious that

~.

is defined by

is

an i s o m e t r y o f

,

A

ucT).

189 This section is opened by an analog of F. and M.Riesz's theorem for the space U(~)* . Its proof uses a construction due to Oberlin [24]. Let ~ be a Banach space imbedded continuously into the space C(~) . Then for every trigonometrical polynomial p we may define a continuous functional ~ E ~ by the formula
> =

~]

÷^( ~ ) p^( ~ )

.

Let

~ denote the set of all trigonometrical polynomials. DEFINITION. A functional ~ , ~ ~ , is called absolutely continuous (briefly ~ E }~ ) if

Using this term the classical F. and N.Riesz theorem may be formulated as follows. If ~ C ( ~ ) ~ and if ~ ---- 0 qF for ~=~... , then ~ C ( ~ ) ~ THEORE~ 3.1. Let ~U(~)* and let

Then

qD E U ~

(r~)

and moreover

So__me preparation is needed for the proof of this theorem. Let ~---{ 0~...~ co } denote the one-point compactification of the set N and let g -----T × ~ o For M.E let ~ denote the closed subset ~ X ~ I of the compact g . It is clear that the mapping

where Doo* ~ de~ ~ , is an isometrical imbedding of U ( T ) into C(~) - Let M(~) denote the Banach space of all finite Borel measures on g . Then U * : ~ (m(~))Therefore for every ~ , ~ ~U* , there is a measure on ~ such that ~)-----~j~ • Denoting fi~ d..~...... J ~ , ~ ~ , we get an identity

The following lemma is actually contained in ~24].

190 LF2~A 3 . 2 . Let ~ M (~) q:) ~-. t* (j~) satisfy

and let the functional

-~,...} Then the measure ~ o o is absolutely continuous with respect t o L e b e s g u e m e a s u r e on ~ . PROOF. A function ~ * ~ being a trigonometrical polynomial, we may r e p l a c e e v e r y m e a s u r e ~ K by i t s c o n v o l u t i o n with the Vallee-Poussin Kernel. So without loss of generality we may assume that ~ - ~ ~ ~ , ~ ~ . Let the sy'mbol @~ denote the restriction o f t h e m e a s u r e ~1, to the set K~O ~ and let ~ = ~* ( $ ~ ) • Clearly ~ is a b s o l u t e l y c o n t i n u o u s and

N

k=O

It follows from the condition of the lemma that

Too , I~l<

Multiplying these equalities by ~ over I~, ~ { ' [ , Z , . . . } we get

By V i n o g r a d o v ' s

, and summing

theorem

S T ing into account that

and that

~-'~ 4-O0 It now follows from theorem 2.1 that ~oo is absolutely continuous. @ Now we are in a position to prove theorem 3.1 o PROOF OF THEORE~ 3.1. Let ~ be a functional satisfying the conditions of the theorem. There is a measure ~ in M ( ~ ) such that ~ - - - ~*(~) and ~ k = ~ ~ ' J~K being a trigonometrical polynomial. It follows from lemma 3.3 that the

191

measure ~ is absolutely continuous. Therefore for every ~ ~ 0 there is a polynomial p satisfying <8

This implies

ilq~- %- ++~P'~+)IITu++¢V) +++ II++- % It-urn"+ and it is clear now that ~ + U ~ ( 7 ) . To prove the second part of the theorem, we note that D ~ is a contractive mapping of U ( T ) into itself. Duality arguments show

< * * ll.-u--+,-<- 11~ lu-',for q) ll-u* = o and therefore + +~+~,,, ~ oo II ,~+,,.,* ¢-¢ being orthogonal in C ~ ( ~ { % : ~ ~} o The functional , it is clear to the monomials ~ , that ~ * II THEOREM 3 •3. (An analog of the Lebesgue decomposition theorem. (respectively in U f ) there is a unique Por every ~ in U * , where j~ is a singular measure and pair , such that

(~e~]~A

~I,~[-'~-Z~... }

(~+, ¢~)

%~uj

PROOF. We s h a l l

give a proof only for

t h e space

and let (for U f it is analogous). Let $ C U ( ~ ) * be a measure on % satisfying the equality $---- t * ~ o If 7)_L ~ ( U ) , then ~ = 0 and by lemma 3.2 ~oo C M6 6 (7) . It is useful to note that

(~oQ)~ Let Then

denote the singular part of the measure ~

absolutely continuous. Denoting ~

~---{ ~ e O

is obviously we see that

.

192

IIq ltj *= olIM(T +II %ll t* The unicity of the above decomposition is a simple consequence of theorem 2.1. @ COROLLARY 3.4. Let E be a closed subset of ~ and let K ~ E ~- 0 . Then there is a linear interpolating operator

C(E)--~ U A such that II ~II = 4. The proof is a combination of theorem 3.1 with Bishop's theo~ rem ( ~ 5 3 , and with a theorem due to A .Pe2czynski (see E3~ , L:

E363 . In connection with this corollary a question arises whether a bounded linear interpolating operator in the Banach problem does exist. It appears that the answer is negative even in the case of the disc-algebra. THEOREM 3.5- Let ~

be

an infinite Banach set. Then

the restriction operator ~ : ~ ~l../'~_.. ~ ~ CA has no bounded right inverse operator. PROOF, Suppose that this were not the case. Then we could find a bounded operator ~ : -~(J~) ~ 0(~) , satisfying G ~ ~ ~(~i) . It follows that is a bounded projection in O(T) :

(, E G ) ~' =

P,,CQPJG

=

p

=

~

P~G

Using the famous Arens-Rudin trick ([23] the projection

) we get that

T is bounded. Simple computations show that

Here ~Jk denotes the indicator of the set ~ . It follows from the Banach theorem that for every sequence ~ ~bE~(_/~) ~ there is a function ~ in C ~ ) satisfying

The set of ~

~

being infinite,

there is an infinite subset

with Hadamard's gaps. Then

~0

is a

A 0

Sidon set (see

193 [37] ) and by the definition of

Sidon set we have a contradiOtion:

o

Up to now the problem of interpolation for subsetSof the circle has been considered. It is time to discuss that problem for subsets of the unit disc. The things are more complicated in this case. It turned out that there exists a subset of 6 ~ which is an interpolating set for the disc-algebra but fails to be interpolating for U A . The construction of the corresponding example is our second task in this section. The following theorem is a matter of common knowledge exercise 7 to ch.10 of [23]). THEOREm. A closed subset for

0A

(t)

. ~ ( E n T) =

of ~

(see, nevertheless,

~ o f clos D

the

is interpolating one

iff

0

En

D

}'- ~'1 >

0

~d

is discretesubset a

;

(2) ~ I[ XeEnD B~En(D,{A})

The c o n d i t i o n ( 2 ) o f t h e t h e o r e m i s t h e famous C a r l e s o n c o n 6ition. We r e f e r t h e r e a d e r t o t h e s u r v e y ~ e q , [ 3 9 ] , w h e r e one can find various reformulations of (2). Two conditions important for our construction are closely connected with the Carleson condition. These are Frostman condition

~P ~

~-I~~

and the sparseness condition >

o

It is easy to see that if a set satisfies (P) then it satisfies the Carleson condition too.

(s) and

(S)

both

The importance of the Frostman condition is emphasized by the followlng simple fact. Let ~ = ~ .A ~ E.e (nA~ - I~~ Ih~ ) ~ , where is the unit mass at the natural embedding ,4 ~O~A C ~ ( ~ ) is continuous iff F ~ (~) . ~oreover II

N II xqs I

194 Some technical preparations are needed for the construction of our example. From now on we shall consider the space U ~ as a closed subset of ~ U (~) with the induced norm. The space %U(T) is not an algebra but, the operator ~ ~ ~. being isometry, the algebra ~ g ( ~ ) = I ~eft) : E l~(~)l------ll~llm~< co} is contained in the algebra of all multipliers for ~U(~) and moreover

II

~p

I1 1I .,

Let ~ denote a function defined on the family of all finite subsets of ~ by the formula

For a finite subset ~A~ let ~lk with the zero set _~ • If ~ ~ j ~

denote the Blaschke pro~iuct we put for

brevity

B~ LENNA 3.6. Let

_]k

be a finite subset of ~

. Then

The proof is based on the explicit formula for ~ ( ~ )

where ~

de~

~Ik

for the sake of brevity. The equality

I]~11~ = ]] ~ ]lo~,~~

We have

:

and (]0) show that

by the Hahn-Banach theorem that

~A~ Ymw let ~

~-I~I~ ~lJ

be a fixed point of co

u~ ~)

In

~G~.~-I~I ~

and let

:

II,=d

~he no=m of th~ ~sual orthogon~l p=ojection of the space g U ( T ) onto U A being equal to eme, we have $tbpD 11@~11 ~< ~ • Clearly @ ~ I U A ~ 0 and<(~_~)-I ~ ~_ ( 4 _ ~ ) - I Hence

195

II:~I~-~,'~cA~~ p ~ p I ~ ~×~ ~-Ixl' l~l.<~ill~llc~.~,l BA(A) ~l~--,~ J -?,

~

'I-IX l~"

•

~,.

Let ~ denote the Hardy class of all bounded functions holomorphic in ~ and let

Our first example exhibits the difference between the spaoe~ -.~eO and from the interpolation theoretical point of view. EXANPLE I, Let (@K)K~ i be an increasing sequence of positive numbers such that b~ ~- ~K+~ ~ ~

--U~

4- a K ~ ~ - 4 And let integers satisfying

k--,,.. + oo

(61¢) K

~

be a sequence of positive

~ 'I

{/

has the following properties

FI°°IA-__ (~(j~) PROOF, It is clear that

, but u A j ~ ~ (~)

and it is easy to

checkE38~Thisthatimpliesthe m e a s u r e ~ A (~HOO~l~I~) i ~ o o (A)isa° weCarleS°nare goingmeasureto prove now that ~ t ~ ~ (~A~) ~ + oo which implies obK--~ viously UA°°l~ =~ 6e°(]k) , To do this we fix a positive int ~ teger K and note that

I+ P,,I~,l and

o~ o ~ f l ~ l l ~

~3

lerama 3,6. implies

t~T

~,eAK II~-Ai

To estimate the left-hand side of the above inequality we use a

196 simple but efficient estimate due to V.l. Vasjunin (see [40]):

; It follows that

"-K

The condition therefore

of the Iemma i m p l i e s

{{II~ ({--O~K)~(}(~Sk ~-- i- O0 d

~

~(A~)

:+co.

•

K-->.c<~ The n e x t lemma i s i m p o r t a n t f o r o u r s e c o n d e x a m p l e . LE~ 3.7- Let ~ be a positive integer, let integer dividing ~ and let

be

an

PROOF. It is well-known that the function

has non-negative Fourier coefficients.

Let

C--~--)

Therefore

, and l e t

Then

K----0 on ~ . We are going to build an analogous decomposition of the indicator of the set ~ . For this aim the functions ~K should be replaced by their Lagrange polynomials. Recall that the Lagrange polynomial of a function ~ ~ ~4 C~) , is defined by

~- I

K=o

~=__K(~}

We see that the restriction of

I ~ (}~ ~)

on the group of

197 roots of unity of the power function

~

~

is identical with that of the

o Clearly

Denoting PK(~) ~¢T I ~ ( ~ K ,

Z4/~)

, we have

j-=o Let E 0 = A ~ , 6

and l e t

EKe--- ~ .

,

It is easy to check that

K=O

For

36~ C(A~)

subset of

HA

let

( {e,

a~1:'"':'ac~-'l}

denote a

such that

The identity

being valid on the set

j~

, it follows that

Taking supremum in the above inequality, we get

THEOREM 3.8. There is an interpolating subset of C ~ for the disc-algebra which is not interpolating for the space

uA.

PROOP. Let

•

~cc. CL

pJK

be a sequence of arcs on the unit circle and let ~k~--set

is defined by

k-K

{ ,~'K = 4___ desired set E can

EK Then the

_~

be defined as follows:

. The

198

E = U K='1 E~ u{'~} C l e a r l y CAI E = C(E) . To prove U A I E = ~ C ( E ) it is obviously sufficient to check that ~ ~6(E~)=+oo. Applying le~na 3.7 to the set EK with ~,~-~K, , we have

~(~

) -<2,~<~~ ( E~ ) .

On the other hand

,'II~K,

~,~(,,I-(,T) -

~%

and therefor~

, ,I

'I

) t.,o-~.

,,-,

~(, F~ ) = + oo .

Tt i s n a t u r a l now to look f o r c o n d i t i o n s on the set E • Let sufficient for the equality U ~ I F = C ( E ) be a c l o s e d subset of C~5 ~ and let

E4= T~o~

ThE

,

E~=DnE.

3.9. The e~uality U A I E

----C ( E )

~olds iff

PRO0~o The necessity of the first of the above conditions is obvious and the necessity of the second one is a consequence of simple limit arguments. Let us assume now the fulfilment of the conditions of the theorem and let R E" ~ A ~ C ( F J denote the restriction operator. By the Banach theorem UAI E -iff the conjugate operator - 0(m) ~E is an isomorphism onto its own image in U Z • Every measure ~ , . ~ M ( E ) being decomposed in a stun ~ - ~ - ~LI * ~ ^ ' w~here U'de~ IJ'lE" ~= ~ the following formulae hold

<~,I~ E)~>

IE

These formulae show that the functional to the singular measure and that ~ E ~ tinuous, By theorem 3.2

~;

corresponds is absolutely con-

199 It follows from the identity

E

# llu#

U # J E~, =

~°°(F__$)

that

C'IIp I1M(E ) ,

C

being a positive constant. • The last theorem restricts our attention by the interpolation in U # D~INITION. A subset E of D satisfies the weak Newman condition (briefly E ~ ( w ~ ) ) if

p{(4-1AIf 4. THEOREM3.10. Let

C4-1 1)}

E e (~)

+

and let E = ~ U

~henU~l

a ,

where P ~ ( P ) , ~e (w~) . E= ~m(E). The proof is based on two applications of the famous Earl's theorem ~41], which were mentioned in ~39~ . For the sake of completeness of the exposition we reproduce the proofs here. The theorem of Earl asserts that if JlflOIloo is sufficiently small then there is a Blaschke product ~ such that $CI~=~I~. The zero set of ~ can be regarded as a small perturbation of the set ~ not destroying the property to be a Frostman set or a weak Newman set (see E38]). It remains to use two known results. The Blaschke product ~ / with the zero set satisfies

(see ~2], ~9], p.18) and therefore ~ # ~ ~ . It was shown in [50] that B ~ is a multiplier of the space of Cauchy integrals in the unit disc and that these multipliers are multipliers of the space H A cO (see also ~4 below). Hence =

B We are finishing this section by the discussion of an interesting phenomenon which connects the real variable interpolation theory with the complex one. The condition--oO~ ~ being satisfied for C '~ 0~ the necessity of Carleson-Rudin theorem is obvious. It appears that there are spaces X~ X ~ C(T) ~ containing functions vanishing on a set of positive measure, but nevertheless

~(~(~l~J6~

~A

~I E ~- C ( E ) The f ~ o ~

~

~enshov's t h e o r ~

YYGE ----- 0 yields such an example.

(11)

200 THEORF_+i~] (lClenshov [43] ). Let E : c ~ E c T let U(~)IE=C(E) . ~hen ~ E : 0.

and

Simpl~ p.oofs of thi~ ~eorom o~n be found in E441 = d

E24].

CAIT ¢ U(T)

It is clear that . Therefore the question arises whether the space X larger than disc-algebra and satisfying (11) does exist. We shall show that the space

has

the deslred properties ~ : P + ~ /-Ee~ " ~:~o

~(K)~ K ) . and let CHIE=¢(E).

~

TH-EOREi!~ 3.11. Let E = e C ~ E c Then .+E = O.

PROOF. There is a natural imbedding of the space into C ( T × { I,~}) defined by

=

s

C~

c,.

Then the annihilator of the subspace ~ C CH) the set of all pairs (~¢) satisfying

is equal to

~tti~ ~=~, ~4, in t~i~ f o ~ u l ~ , we get ~y*~= ~ d . ~ , & E ~4 . Let now ~ : = % ~ , rb~O , then the formula implies }~(~)+~(~) E ~01

:

0

, b~0

- It follows that

(~-N)d~,

. Therefore

Assuming E to be ~ interpolating set for 0 H with ~ E > 0 we get the existence of a positive constant 6 such that the following inequality holds:

being a summable function supported on E For every function ~ in L ~ (T) we put in the previous inequality ~-~+~IE, ~-~+~ , 9=-~_~ . Then we get .

IE

I I

T-E

I

If now ~ is the Poisson kernel at the point ~ - ~ then I P + E(N)I = t ~;- ~÷ P" I-+ for ~ in T

6 e (0~) . Denoting

201 ~_ e~'O

, we get

iOl+~

I01+~ e~0e E }

Let E* --~a~< O e (-~,~] : 0 is a density point for E* to the both parts of (3) we get

e.[-~c 101+~ I t is

clear now and therefore

[-~,~)~E

'~'

~

[_~>~]~ E~

and let

us assume that . Adding the integral C.~ l~t~Idm T~E

I01 *~-

"

that k(O) de~ !O ~l[_~,~Gj,E,(~)i~=O(O), 0_~_0 + , @

o

o

This implies the contradiction: ~7 -

II

-%

§ 4. Two applications o f the as,ymptotic f qrmula In this section we shall give two different applications of the results proved in ~2. The first one is connected with the interpolation theory in the space V(~) o Recall that denotes a compact of positive Lebesgue measure on the unit circle and that

The norm in the Banach

space

,°s ( i Clearly the space W ( K ) DEFINITION. A subset

V(K)

is defined by

i j~

is a closed subset of V ( ~ ) of ~ is c a l l e d ~ ( p ) - - 5 6 t

,

p > 0 , if the L% -norms are equivalent on the space of all trigonometrical polynomials with frequencies i m ~ for all

'~ ~ p . THEOREM 4.1. Let

E

be a closed subset of

~

contained

202 in the set of density points for K _A be a ~(6)-set for some 3 ~ fined by the formula

Then O,V(K)

.

and let ~ - ~ The operator

0 ~

.

Let is de-

C,(E ) ~?,~'(z).

=

The proof of theorem 4.1 is based on a description of the conjugate space V(K) ~ . Every functional ~ in v ( r ~ ) * gives rise to a pair of analytic functions eo t'1,= 0 oo

~

"=~

~

Let now denote the u s u a l Hardy c l a s s i n ~ and l e t be the Hardy c l a s s o u t s i d e the u n i t d i s c , By F a t o u ' s theorem these spaces may be c o n s i d e r e d as c l o s e d subspaces o f

°I.:,~(T)= H:mM ~ "

• Taking into account this agreement, we have

=

M_ * E

This __idemtitY implies ~ %U A __and % . ~ e M for ~ V (711)W . The r a ~ i a l l i m i t s of ~ exis%im~ the f u n c t i o n

~_

a.e.,

is defined almost e~erywhere on ~ . The correspondence~-*P~ is obviously one-to-one. Indead, it follows from ~ ( ~ ) = - ~ _ ~ ( ~ ) ~ ~T , that ~ C H ~ by Smirnov's theorem (see ~46~). Therefore ~ ~ - 0 o The space U ~ satisfying the axiom 2, for every p~ 0 < p < ~ ~ there is a positive constant Cp such that

] I

T

I

I1911PY(T)*

Let E. = { ~ET : ~E E ~ . The next lemma is a generalization of a statement in E17], p.140-141.

~,~A4.2.

~,et

K-----

c~

KcT

and let ~E V (T)* , ~ I V(K) P ~ ( ~ ) = 0 for almost all ~ in K , PROOF. The linear set

, ~-

~K~o 0

. Then

203

is weakly closed in V ( T } * " Indeed, the space V ( T ) is obviously separable and therefore by the Banach theorem it i~ sufficient to prove that ~(~) is a w~ak-, sequentially closed subspace. To do this let * - - ~ ? f 5 ~--- ~ ~ ( ~ ) . Then I [ ~ 1 1 ~ U ~ ~ 0('I) ~and ~ ~_~= ~_ in the weak topology of H ~_ o Closures of convex subsets of ~_~ in the weak and strong topologies being identical, we get a sequence ( ~ ) ~ 4 in V ( T ) * such that

K~ t,l,

~

c6k vl,

where ( o ~ k m , ) k ~ 4 are finite sequences of complex numbers satisfying {~ ~_~ ~ ~-~ in the norm-topology of M _~ In particular

1,1, > c o

and therefore

K>~I~

~/I~ (~)~

{$~

i t is clear also that

"~(~(~)

il~:~ll~

=

for

0(~)

~C

D •

. ~e have

~CK) "

by the definition of the space

which implies the convergence of the restrictions ~l~l,t,lK ,

i~ L? -metric. ~he space ~ U ~ ~eing conti~uously imbedded i-~o I-II/~, the ~inchin-Ostrovskii (see [4~I) theorem shows

~+

JC_q) =

o

a.e. on K ~ • Now we are in a position to finish the proof of the lemma. The duality arguments show that it is sufficient to prove that V(~) = ~ ( K ) ± . Let ~ V ( T ) n ~ ( ~ ) ~ [ . ;For every smooth function ~ with the support disjoint from K, let functional ~ $ Be defined by

q~ It is clear that particular, P Q ~

~e ~(~)

= o ~.e. and therefore

j" ~ ( ~ ( ~ a ~ = T The function

on

~

, ~ and in o This implies

o.

being arbitrary, we see that 3 t t p p ( ~ ) c K

204

[45]).

(see

•

The following lemma may be found in [47]. L ~ 4-3. Let ~ be a ~ ( 3 ) - set for some 5~ 5 ~ of the circle and let m ~ C0,$) . Then for every subset , Ht E ~ 0 , there is a positive constant C E such that

++ CJ l÷I+++) +/+~< ( T

for

every

_~

J I+I Pd~ E

)'/+

-polynomial,

PROO1~ O+ THEOREm 4.1. Let (0C, j ~ ) ~ ++(A~) x ~ ( ~ . ) . It is sufficient to check by the Banach theorem that

II It

++++¢ (II +

f o l l o w s f r o m lemma 4.1

Let s~bol ~

ll llm¢E,>) •

that

denote ~ { # , ~ )

. Then

E~

The c o m b i n a t i o n o f (12) f o r ~ = 4 / ~ the identity P ~ 'I ~ . ~--- 0

, the Smirnov's theorem,and a.e.,give

K, The space ~'U~

~a+isfies

the conditions of a~iom ~ ~ 1 ~ , = 0 .

Hence

It is clear that

tce_A. E, ~ - + + " The first sum in the right-hand side of the equality belongs obviously to L ~ CT) . By theorem 2.4 we get

and oonsequently

205

.11

IIv< )

Comparing the last inequality with (13) and with the inequality of len~na 4.3 we get the desired estimate. • In conclusion of the section we discuss some properties of the space

multipliers o f "LTA . This space is an interesting oh.leer for the investigation b e c a u s e the space ~r~(~UU ;~°~ is not an algebra. ) with two otherS, Our f i r s t t h e o r e m c o n j o i n s t h e space =&mely, w i t h of all

TM

and with

~eoall that CU~)~ stands fo: t~e closure of polynomials A

in the

norm-topology of ~-A THEOREM 4,4. The following identities hold @

PROOF. To begin with it is useful to observe that both spaces ~ C ~ A ~) and fl$(~;)~consist of bounded analytic functions (s~e a simple proof of this general fact in [5~ ). The first step is to prove the identity

If ~E~ ~ and if ~ E U A , then there is a sequence CP~)~7/0 of polynomials which converges to ~ in the w e a k * topology of U A . Clearly II ~ ' P ~ I I U * = 0(~) and by the trivial part of Banach-Steinhaus theoremwe may conclude that Let now ~EH~CU~) and let ~ C CU A * a sequence C ~ ) of polynomials satisfying

)~

o Again there is

It is clear that

i

A ~ U ~ Q I H ~ d~j_~ ~ p

0<~<4

I I~cP<%~>l ~ < ~ ) " T

206

and therefore

~'P~(U;)~

. The equality

shows that ~ . ~ (U~)~ • To finish the proof of the theorem we are going now to establish the identity

We shall use the natural duality between the spaces and C U ~ ) ~ and shall write it in an antilinear way:

< Then i t

is

U A c°

I clear

that

the Toeplitz

operator

T~ iS bounded on the space ~I~(U~) ° The space U ~A may be considered as a closed subspaoe of ~'[i~x~{r~) Clearly the shift operator ~ and its conjugate operator ~ are isometric in % U °° C T ) . Assuming the operator T ~ to be bounded, we have by a formula from [51] (for the non-Hilbertian case this formula appeared for the first time in ~52] ) that for a given ~ ~U(V) "

~oreover ~ -~oo { ~ ~ ~ ~T ~ p~ ~ ~ = ~ ~ in the weak ~ topology of ~oo C~} . Hence ~ ~ ~ ~ U ~x~ CT) and by the standard arguments we see that operator ~ - ~ - ~ maps the space ~ U ° ° C ~ ) into itself. In particular, ~" ~ ~ ~U~ and therefore ~ ~/~(~oa) . Assume now that ~ ~tC~) . Then it follows from the isometric property of the shift ~ that~-*~# is bounded on % U ° ° C ~ ) . Clearly the projection ~+ is also bounded on ~ U ° O C T ) . Combining these facts we see T ~ is a bounded operator. It appears that usually the space ~ U ~ is more convenient to deal with. Nevertheless we prefer a direct proof of the following theorem to one m a k i ~ u~e of theorem 4.4. THEOREM 4.5. Let ~ be a Blaschke product with the zero set satisfying the Frostman condition (see §3). Then ~ % ( U ~ ) . PROOF. It was shown in E50] that B ~ t ~ ( ~ C ~ ) . Therefore

207

where

#~ ~ M (T)

'

so that

Let now

S~

•

. It is clear al-

11~tl ~
Then

I

T~

~,o

Comparing the coefficients of the Taylor expansions in both sides of the previous identity, we get

LE~gkk

4.6o

Let

S~I~ (U~)

•

¢2~ ]ls'('~)le'<+°°

Then

'

0

PROOF. The p r o o f i s analogous to one g i v e n on ~0~ f o r the more narrow class. I I ~ ( ~ ) . Given a function } inm(U~ ) and a point ]~ in ~ it is possible to find a sequence of bounded Borel measures on ~ such that

To prove this assertion it is sufficient to note that the space ~ may be considered as the space of all holomorphic functions~ ~ in ~ having a representation

~o

k1~,O

Clearly

~/o

~

~o

G a t h e r i n g two obvious i n e q u a l i t i e s f

we have

T

208

4 i

•

0

~0

~>~0

4.4 imply the imbedding ~t4~C-~A)C~I~CU2) c ~t{~ ,where ~ d e n o t e s the Banach space of a l l holomorphie f u n c t i o n s f in ~ such t h a t The last lemma together with theorem

It was shown in [48] that for every subset ~i of ~ with Hadamard's gaps the transformation ~ ~ ~I/~ maps ~ I onto ~4(~) . This means that there is no analog for the Banach theorem in ~ ( U A) .

The n e x t theorem~being s i m i l a r to

one p r o v e d i n [53~,

gives a d scription of some inner functions in the space ~ ( U ~ ) . THEOREM 4.7. Let ~ denote an inner function in ~ C U ~ ) . Then ~ is a Blaschke product, the radial limits ~(~)=~Yt ~(%~) exist everywhere on T ' and I~(~)1-----~) ~CT ' %-~1- 0 PROOF. The existence of radial limits follows easily by lemma 4.6. To compute I ~(~)I for t6T we apply theorem 2.5. The function ~ being a multiplier, we see that ~(t-~5)-~U~ Therefore by theorem 3.3 we can conclude that there are a singular measure ~ in ~(F) and an absolutely continuous functional ~ such that

Let now E denote any closed arc on in its interior. Remembering that we get, obviously,

~ containing the point I ~(~)I = ~ a.e. on T

Clearly

~----> +o0 It follows now from (14) and by theorem 3.3 that the measure is supported by ~ and II~-~- 4 . The are ~ being chosen arbitrarily, it follows that ~ = c&. ~($) y I ~ ~ = ~ . Now we have

209

Our last application of the asymptotic formula deals with a generalization of a recent result due to de Leeuw, Katznelson, Kahane

~8~.

This result of three authors has given a positive

answer to a question posed by Sidon in [5 4 . Shortly after appearance of

~

S°Kisljakov has strengthened the theorem in D g ] .

(s. islj non-negative sequence in ~ A

ov

e ery square-s

(~)

~6~

able

there is a function

such that

The method of the present paper gives a possibility for the conjugation of the above theorem with the Banach theorem and with the theorem due to Carleson and Rudin. Let (see

~

be a Banach space satisfying the Axioms

I and 2

@3). THEORE~ 4.8. Let

~

be a closed subset of the circle having

zero Lebesgue measure and let for every

~

C(E)

there is a function

_A~

be a Banach set in ~

and for every ~

in

~

. Then

~---(~)~0E~(~)

satisfying

The proof of the theorem is based on 8]2 abstract scheme ori-

inating from D s ]

and on a metho

due to

. is j ov

Dg'

on the one hand a~ud on our approach on the other°

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Y. An introduction to harmonic analy-

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J.-P., K a t z n e 1 s o n

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R.E.A.C. A note on power series. J°London ~ t h . ,

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~,o~ui. AH CCCP, 1968, T.178, ~ 3, 5 1 1 -

- 514. 13. B E H 0 ~ p a A o B C.A. MHTepHoJI&HL~0HHNe Teope~i BaHaxa-Py~Ha-EapnecoHe z HOp~[ onepaTopoB ~ o a e H ~ a ~ HeX0TOp~X EaaCCOB aHa~ZTEqecEEX ~ y H E L ~ . 3all.HayqH. ceMHH.Z0~i, 1970,

T. 19, 6--54. 14. F O u r n i e r ficients of H N

J.J.F. An interpolation problem for coeffunctions. Proc.Amer.Math. Soc., 1974, v.42,

2.

15. B ~ H O r p a ~ o B C.A. CXOAm~OCTS nO~T~ BcD~y paZOB ~ p s e ~ y H E I ~ I~B ~ a H llOBe~eHEe I¢OS~C~ELI~eHTOB paBHOMepHo CXO~Hn~XC~ p ~ O B ~fp~e. ~oEa. AH CCCP, 1976, 230, ~ 3, 508-5II. 16. B ~ H o r p a ~ o B C.A. Yc~neH~e Teopem~ KoaMoropoBa o conpa~eHHo~ ~#HEI~m E ~HTepnonsl~OHH~e CBO~CTBa paBHOMepHo cxo~c~ CTeHeHHRX pHZOB. T p y ~ ~ a T e ~ T ~ e C E O P O nHCTETyTa m~. B. A. CTeF~oBa AH CCCP, ~. I55, Z., "HayEa, I980. ]7. X p y m g B C.B. Hpo6ae~a O~HOBpeMeHHOA annpoEcm~aku~ CT~paH~e oco6em~ocTe~ E H T e ~ O B ¢~na Kom~. T p y ~ MaTe~aT~qecEoro m~CT~TyTa HM.B.A.CTeFmoBa AH CCCP, T.I30, Z., "HayEa", 1978, cTp. I24-I95. 18. d e - L e e u w

K., K a t z n e 1 s o n

Y., K a h a n e

J.-P. Sur les coefficients de ?ourier des fonctions continues. C.R.Acad.Sci.Paris, 1977, 285, N 16, AI001-AIO03. 19. K H c ~ ~ E 0 B C ~ F o u r i e r coefficients of boundary values of analytic functions. Hpenpm~T ~ 0 ~ , E-3-78, ~., 1978. 20. K ~ c a ~ E o B C.B. K o s ~ e H T N ~yp~e r p a m ~ H K X s H a ~ e m ~ ~yHKU2~, saaaETH~ecKHx B Epyre ~ B 6 ~ E c E e . Tpy~N MaTeMaTE-qeoEoPo ~HCTHTyTa HM.B.A.CTeF~OBa AH CCCP, T.I55, ~., "HayEa"

211 I980.

21. E p ~ E E e

B. CpaBHeH~e ~?z~cmvsc~a MOAyn~ M H O r O ~ e H a CO BBBemeHHO~ Cym~o~ Moxyae~ e r o EOS~C~m~ZeHTOB. B e C T F ~ E Z I T , I978, ~ I, I42-I44. 22. C a r 1 e s o n L. On c o n v e r g e n c e and g r o w t h o f p a r t i a l sums of Fourier series. Acta Math.,1966,v. 116,N I-2,135-157. 23. Hoffman K. Banach spaces of analytic functions. Prentice-Hall, N.J., 1962. 24. 0 b e r 1 i n

D.M. Uniformly convergent Fourier series and

sets of measure zero. Preprint, 25.

Gamelin T.W. Uniform algebras

26. E 0 a M 0 r o p o B •

/

1978. , Prentice-Hall,

N.J.,

1962.

A.H. Sur les re_notions harmoniques #

consuguees et les series de Fourier. Fundamenta Mathematicae, 1925, v.7, 23-28. B.~. Sur les valeurs limites des fonctions

27. C M Z p H 0 B

regulieres a l'interieur d'un cercle. ~eHa-Hrp.~3-~aT.06--Ba, 1928, 2:2. 28. Zygmund A. Trigonometric Series, Cambridge, 29. K a H T 0 p 0 B ~ ~ H~

Z.B., A ~ ~ ~ o ~

aHazm3 B H o p ~ m p 0 B ~ U ~ X

30. A a r o n s o n

npocTpaHCTBaX.

1959.

r.n.

~y~oHa~-

M., "@.-M", 1959.

J. Ergodic theory for functions on the

upper half-plane. Ann. Inst.Henri Poincare, 233-253. 31. H e p e T e a ~

~rpHaJl

1978, v.14, N 3,

0.~. 0 conpazeHFmX ~yHEmULX. ABTope~epaT

ZmccepTan~HacoHc~aH~e

y~.cTeneH~ ~ O E T O p a ~ E S . - M a T . H a y E ,

T6ZaHCE, 1978. 32. B o o 1 e G. On the comparison of transcendents, with certain applications to the theory of definite integrals. Phil.Trans. of the Royal Soc., 1857, v. 147, 745-803. 33. L e v i n s o n N. Gap and density theorems.A~S Coll.Publ., 1940, v.26. 34. B p H ~ E O B D.A., H p y ~ H ~ E 0 B A.~. H H T e ~ B H H e npeoOpa3oBaHnaO6o6me~x~yHEU~. M., "Hay1{a", I977. 35. P e E c z y ~ s k i tinuous functions

A. On simultaneous extension of con-

(A generalization of theorems of Rudin-

-Carleson and Bishop). -304. 36. P e E c z y n s k i

Studia Math.,

1964, v. 24, N 3, 285-

A. Supplement to my paper "On simulta-

neous extension of continuous functions. v.25, 157-161. 37. L o p e z J.M,, R o s s

Studia Math.,

1964,

K.A. Sidon sets. Lecture notes

212 in pure and appl.math.,

v. 13, N.-Y.,

1975.

38. B ~ H O r p a ~ o B C.A., X a B z H B.H. CBO6O~j{a~ ]~HTepn o ~ B H ° ° ~ B He~oTop~x ~pyr~x 1~accax ~ y H ~ . 3an. HayqH. Ce~&H.Z0~7~, 1974, T.47, 15--54. 39. B z H 0 ~ p a A o B C.A., X a B z H B.~. CBOdO~J{a~ HHTepn 0 ~ B H c o B HeEoTopmx ~pyrzx ~ a c c a x @ylm~z9 H. San. Hay~H. C e ~ m H . Z G ~ ,

I976, T.54, I2-58.

40. B a c D H E H B.M. izpzya}rpm~e n p o e ~ E ~ o ~ e C T B , BcTpeqaDm~xc~ B T e o p ~ m{TepnozmpoBam,~. San.HayqH. CeM.ZOMM, I979, T.92, 51-59. 41. E a r 1 T.P, On the interpolation of bounded sequences by bounded analytic functions. J.London Math. Soc., 1970, v.2, 544-548. 42. N e w m a n D.J., S h a p i r o H.S. The Taylor coefficients of inner functions. Mich.Nath. J., 1962, v,9, N 3, 249-255. 43- M e H ~ m 0 B ~.E. 0 p ~ a x ¢,yp~e Henpep~-Bm~x ~ y H ~ . Y q e m ~ 3an~c~z NEG, I95I, BHn. I48, T.IY, I08-I32. 44. X p y m e B C.B. HpocToe ~0Ka3aTez~CTBO T e o p e ~ MeH~moBa o6 ~cnpa~neHz~. Hpenp~ET ZOMH, P-I-79, Z., I978. 45 Bremermsaun H.Distributions, complex variables and Fourier transforms, Addison-Wesley, 1 965. 46. ~ p H B a a o B M.M. I ~ a H m m ~ e CBO~CTBa a H a ~ T ~ m e c E ~ x ~yH~-n ~ . M.-Z., I ~ T Z , I950. 47. M ~ x e e B H.M. 0 pa~ax c ~ a ~ y H a ~ . H ~ , I975, T.98, }~ 4, 538-563.

~Te~aTHqecE~L~

cdop-

48. A H ~ p ~ a H 0 B a T.H. K O S { @ m ~ e H T ~ Tsflaopa c p e ~ m ~ { HoMepa~m ~ djy~, cy~mmpye~ux no r m o ~ z . San. H a y ~ . ce~. ZOMM, I976, T.65, I6I-I63. %" V ,* 49. H r u s c e v S.V., V i n o g r a d o v S.A. Inner functions and multipliers of Cauchy type integrals. LOMI preprints, E-I-80, Leningrad, 1980. 50. B ~ H O r p a ~I o B C.A. CBO~OT:Ba },~yJiBTEI~I~aTOpOB IIHTerpa-JIOB T~TA~ I
H.I{. 0 npooTpaHCTBaX

UeB~X ~,~Tpm~, A e ~ c T B y ~ m x T.6, ~ I, I46-I58. 53. H r u s ~ e v

B

~P

~ a~2edpax T~na~--

. C~d.~age~.Zyp~.,

S.V., V i n o g r a d o v

1966,

S.i. Inner fttncti-

213

ons and multipliers of Cauchy type integrals. LO~I preprints, E-I-80, 3-29 (to appear in Arkiv for N~thematik). 54° S i d o n S. Einige S~tze und Fragestellungen ~ber ~ourier-Koeffizienten. Math. Zeit., I932, b.34, N 4, 477-480. 55. K ~ c ~ ~ ~ o B C.B. KO~m~I~eHT1~ @yp~e ~ p a H z x ~ X 3Ha~eI~I~I @ y H E H ~ , aHa~i~Ti~ecEHx B Epyre ~ B 6~I~cEe. T p y ~ r ~ I , 1981, T. 155, 77-94.

S.V.Kisliakov

WHAT IS NEEDED FOR A

0 -ABSOLUTELY SUMMING

OPERATOR TO BE NUCLEAR?

O. Introduction, 1. Operators from the disk-algebra 2. An example: the spaces

C A to the space ~

~

3. Almost Euclidean subspaces and some finite dimensional examples. 4. Operators with an arbitrary range.

337

O. Introduction. A famous result of Grothendieck stating that each continuous linear operator from a space ~I(~) to a Hilbert space is ~ -absolutely summing is the origin of a considerable part of the operator ideals theory, and the present paper is also inspired by it. Besides the Grothendieck's paper Ill one can find proofs of this result e.g. in E2] or E3] , p.308. In symbols it can be written as follows: ~ ( ~ ( ~ ) ~ ~) = ----- ~ ( d ( ~ ) ~ M ) (for the notation and definitions see the end of this section), and it is known now that an analogous equality ~4 ( ~ M ) ~ ~ (X~ H ) (which we call the Grothendieck equality for the space X ) holds for some spaces ~ differing from ~ (~), cf. E2~ , [4] , [ ~ • It is still unknown, however, whether some particular spaces ~ that arise naturally in Analysis satisfy this equality. Of the greatest importance is probably the question whether we can take X = ~ , C~ being the disc algebra (cf. E6] , p.12-14 for a detailed discussion of this matter). If X = ~ * (in particular, if X~-- ~A~ ) then we can try to prove the equality ~ 4 ( X ~ M ) = ~ ( X ~ ~ ) in the following way. Using desiriptions of operator ideals adjoint to the ideals ~4 and ~ (of. [3] , p.269, theorem 19.2.3 or p.276, theorem I9.4.7; cf. also E7] ) we obtain that this equality is equivalent to the equality ~oo ( ~ X ) ~- I ~ ( H ~ X ) • The latter one holds if and only if q ( ~ ) ~ I 4 ( ~ H ) (here I~ and ~p , p=4,~ stand for the ideals of 4-integral and ~P -factorable operators respectively)*~ But since every ~) The fact that the conditions and F I ~ H ) = ~ i ( ~ ~) are equivalent can be easily derived by passing to conjugate operators provided one knows that the equality ~ H ) : I ~ ( ~ H ) implies t h a t ~ ( ~ H ) : I~ (~*~ ~ ) . This implication may be proved via the local reflexivity principle ( [3] , p.384) and theorem 8.7.4 at the page ll5 in [3] •

338 -integral operator with values in a reflexive space is nuclear we get finally that the conditions ~I(~M)= -~-~(~) and q (~)----- a i ( ~ H ) are equivalent. To check the second condition it is sufficient, of course, to check the inclusion ~i ( ~ M ) c N 4 ( ~ ) (for the reverse inclmsion is evident). But ~urey's improvement of Grothendieok theorem (cf. E8] , p. IlT) implies that ~ ( ~ ) C ~ 0 ( ~ ) ( ~ o denotes the ideal of 0 -absolutely summing operators). Therefore the space ~ satisfies the Grothendieck equality if posesses the following poperty: Every 0 -absolutely summing operator from ~ to a Hilbert space is - nuclear

(~P)

The notation (KP) is composed of the first letters of the names of the authors of [9] , because in this paper a similar scheme was applied to treat the case of translation - invariant 0-absooperators. For example, it is shown in LgJ that every lutely summing and transla$iQn-invariant operator from the disc algebra to the Hardy space H ~ is nuclear (of. also EIG] ). It seems to be very probable that the invariance assumption is superfluous. CONJECTURE. The space ~A has the property (K2). The next theorem is the strongest result which I was able to prove in support of this conjecture. Its proof inspired by the paper [9] is presented in Section I. THEOREM I. Every 0 -absolutely summing operator from the disc algebra to ~ is ~ -nuclear for each p ~ ~ . Note that if E and F are any Banach spaces then all classes ~ ( E ~ ) , 0~<~ coincide (cf. [8] or [3] , p.293, theorem 21.2.9), and hence ~ ( C A ~ % ) C N p ( C A ~ ~) if ~ > ~ and % < ~ . However for ~-----~ the last inclusion is nolonger true for the map ~---~- { ~ ( ~ ) } provides an example of an (invariant!) ~ -absolutely summing operator from the disc algebra onto ~i , cf. FII] , p.20, corollary 3.I. In section 2 we construct other operators defined in a similar manneramdpomeBmlngthesame properties, but also an additional property of being 0 -absolutely summing. Let, for L

example,

~- = e~o~Clv~<~j spar~

Lf

~

:

~ ~ 0

or

~----~,

339

Then (see Section 2) we have: ~ . ~ ) = ~

and,

moreover, ~ is translation invariant. So, by Theorem I, the space ~ is isomorphic to no factor-space of the disc algebra, though it is obtained from the ~atter one by adding a "small" space. We shall also discuss in Section 2 some other interesting properties of the space ~ . In section 3 we construct (among other things) a sequence of finite rank operators for which quantitative characteristics corresponding to some qualitative properties of the operator described above behave in an "extreme" way. The main idea of construction of these operators is similar to that for (both of them may be d~afted as follows: take a "known" space, "add" to it a "small" space and then "project" onto this small space). To exploit this idea in a finite dimensional setting we need (and prove) some facts about almost Euclidean subspaces of

~,!./ 7

being

"not very big" subspace of

s6~

(these facts are probably of independent interest). It is shown also in the same section that there is an infinite dimensional space 7. with ~ # ( Z t ~ ) = ~(~ ~ ) which does not have the property (KP). So the scheme of proving the Grothendieck equality described above does not work in general. Finally in Section 4 it is established that ~'~#-P]J~|~)7~z ~,

~''~'"'~'i~tC~J,'/,Jfore ~ e ~

Banach space Z

• #

ACKNOIg~EDGEIENT. I am grateful to A.Pelczynski for many stimulating discussions. Also I want to thank N.K.Nikolski~ and V.P.Havin for reading the manuscript. DEFINITIONS, NOTATION AND SO~E STANDARD FACTS. We denote by ~ ( ~ y ) the set of all continuous linear operators from a quasinormed space X to a quasinormed space ~ Let X be a Banach space, 0 < m < ~ , T E ~ (X~ Y ) . The operator ~ is said to be p-absolutel~ summing if

4'/p

H,

"i.,=-I

1trill P

for .ll

F ~

x

~ i,,,= 'f,

~,,...,i

<

The class of all p -absolutely summing operators is denoted by N p . We do not reproduce here the definition of 0 -absolutely summing operators (it may be found e.g. in E8] ), for it is known that if X and Y are Banach spaces then (as it

340

has already been mentioned) all classes ~ % ( X ~ ) ~ 0~%< coincide and it is more convenient for us to deal with ~%~ ~ > 0 than with ~o • It is well-known (cf. [3] , p.232) that T E ~ p ( X ~ ) if and only if there esists a probability Radon measure ~ supported by the unit ball ~ of the space X # (with the topology ~ ( X e ~ X) ) so that for every ~ in X we have

K (~oreover, if T E N p then ~ can be chosen in such a way that C = ~[~ ( T ) ) . This inequality and the measure ~ are called the Grothendieck-Pietsch inequalit~ and a GrothendieckPietsch measure for T . If ~ a n d A is a 6v(x*~ X) compact subset of K such that 6~o$ co~Ife~A=~(the closure in the topology ~( X*~ X) ) then a Grothendieck-Pietsch measure for T may be chosen to be concentrated on A . In particular if ~>~ ~ and X is a subspace of a space g(~) then we can assume that this measure is defined on ~ ° We need the following version of Grothendieck-Pietsch inequality for translation invariant operators (a more general result can be found in [12] ). THEOREM ON INVARIANT OPERATORS. Let ~ be a compact commutative topological group and let X be a translation-invariant subspace of ~(~) . Let ¢ be a homomorphism from ~ to the group of isometrics of the space ~ . Suppose that ~ >/ and let ~ be a p-absolutely summing operator from ~ to y satisfying ¢(~)(TO6) = T ( L ~ 36) for ~ , . DCE X ( [,@ is the rotation o p ~ r a t o r : ( ~ ) ( ~ ) = ~ ( ~ ) ) Then the Haar meadsure of ~ can be taken as a Grothendieck-Pietsch measure for ~ . Let ( ~ ) b e a measure space,~(~)
0 <

p

(X, Y;

341 Then T E ~ p ( ~ y) if and only if T = U ~ V where U and ~ are linear operators and R is a part of an operator

lp,

RENARE. Of course, any Grothendieck-Pietsch measure for T may serve as a " ~ " in this theorem. In particular if X is a subspace of a space C(~) and ~ 4 we can choose this to be defined on ~ . in this car,e the domain of the operator U coincides with the closure of the set X in ~P(~) . Note that the assumption p~/~ is essential here. Let X~ ~ be Banach spaces, p ~/~ . An operator T , T ~ ~(X~) is said to be p-integral if there exist a measure ~ and linear operators V from X to ~ ( ~ ) and U from ~P(~) to ~ so that @ 6 ~ = ~ the canonical imbedding operator of ~ the space of p-integral operators

p,~F~into ( ~6y~

stands for ). We denote and sup-

byIp Y)I/flI IIIIII

ply it with the norm %p~ Sp(~) ~''~5 ~ II~II (the infimum is taken over all factorizations as above). Again let p~/~ and T E ~ ( X~ ~) . Then T is said to be p -nuclear if ~ has a factorization of the form ~ -=~I~F~,~'~ V , where.~, is a ~discrete measure,VE~(X,~, eo(~)),~. U~ ~-(~P(~)~ ~ ). The norm V p in the space N-(X~ Y) of all p-nuclear operators is defined similarly to For the basic properties of the operator classes ~ p

Np

see

[3]

and

•

By ~ we shall denote the identity mapping of the complex plane ~ onto itself as well as restrictions of this mapping to certain subsets of ~ . Let ~ be the normalized Lebesgue measure on the unit circle ~ . The disc-algebra is the subalgebra of C (~) consisting of all continuous functions with ~(~) d¢~ I ~ p ~ b = 0 for I d , = - ~ - $ ~ . . . A function ~ in C(~) belongs to the disc-algebra if and only if { is a restriction to ~ of some function continuous in the closed unit disc ~ and analytic in the open disc (we reserve for this function the same notation ~ ; in general we often do not distinguish notationally between a harmonic function in the unit disc and its boundary values). The norm of a function ~ in the space LP(~) ( being a finite positive measure) will be denoted by

ll~IIp~fi_

nite measure on t

cle

~

then

m

I

stands for the

342

closure of the set CA in the space ~ P ( ~ ) . We write ~P for ~ P ( ~ ) and HP for ~ P ( ~ ) . The main facts about the classes ~P may be found in [13] . The symbol ~+ (resp. ~_ ) stands for the orthogonal projection of the space ~% onto H ~ (resp. onto ~ ). Note that ~ p_ ~$ . It is known that the operators ~t and ~_ act in fact from ~P to ~P , 4 < p < co and from ~I to ~ % ,

(cf. [z3] ). The space of all Radon (= Borel regular) measures on a compact space ~ will be denoted by M(~) and we identify this space with the space g ( ~ ) * . As usual, ~ denotes the group of integers, { ~e

Z t d¢~

The formulae in each particular section are numerated independently (if at all). I. Operat0rsfrom

the disc-algebra

C~ .......to ..... the space

.

As it has already been mentioned in the introduction, it is convlnient for us to deal with the classes ~ , 0<~< ~ , instead of ~0 . Fix (till the end of this section) a number with 0 < ~ < ~ and let T ~ (CA~%) . our aim will be to try to prove that T is ~ -nuclear (or at least p-nuclear for all p > ~ ). since ~ T c ~4 ( c f . [3] , p.235, theorem I7.3.9)~an application of Factor~zation theorem and of Remark after it gives so thatT admits the folloa measure ~ on the unit circle wing factorization:

). Let q) be the singular (with ~ 6 Z (~4(~), 6Z) part of the measure ~ with respect to ~ . Then~ = ~+ $~ ~ [,~ . We may assume without loss of generality that ~ (since the replacement of ~ by 6~+~ does not affect the factorization (I)). Now (cf. e.g. [II] , section 2)~1(~)---~1(~b)e L4(~),

CA÷Z ,Vl CA

and the operatorL,-

lea

) is (if considered as an operator from C A into lit(f) -integral. Since the space ~% is reflexive, the operator

343

~!4,~ I OA

i s nuclear ( c f . [ ~ , p.245, theorem 2 4 . 6 . 2 ) . On the other hand it is %-absolutely summing (this follows from ~urey's extension of Grothendieck theorem, cf. ~8] , p. IIT, th~oreme 94, for, ~ { ~ I C~ being nuclear, it is of the form ~ V with V ~ ~ ( ~ ~$) ). So the operator ~I{,@~vl ~A is also %-absolutely summing, and it is sufficient to prove that this operator is nuclear (or at least p-nuclear, p > { ). Now we have the following situation. There is a function , ~ ( ~ ) , @~ accompanied by an operator T ' T ~ ~ ( C A ~ ~ ~) admitting the following factorization

Ca It,,,.,..I C~, N ~ .........

~

(a.,'~)

,s" .f/. -'---

(2)

.

Let ~ * be the coordinate functionals of the space ~g . Consider for each fixed ~ a norm-preserving extension of the functional ~ from ~(~K~) to ~4(~H4) and let ~ , ~{~°°(R~) be the function corresponding to this extension after we identify the spaces ~(~K~) ~ and ~°O(~F~b) in the standard way. We have then the following formula for

~

:

101

Let ~ be an outer function satisfying = ~ a.e. (that is ~ ~(4+ ~) , where $b-----~ and is the harmonic conjugate of 14. ; the assumption ~ > ~ has been made to assure that such { exists, cf. ~I3] , p.24-25~ Set ~ = ~_(~ ~-1) (note that s~m II~ g~-lll0<:! < O0 and hence the functions ~g are uniformly bounded in ~P for every p < o o , ce. [13] ). LE~g~ I. In the situation just described the following inequality holds:

t~

where

Z'÷

G depends on ~ only. PROOF. For any (say, continuous)

~A -valued function #

344

on

~

we have

(To prove this note that if ~

=~_d~ k

with

~O~C~ and ~ ~ ~-# for all ~ then this inequality applied to ~ is just the inequality defining ~ -absolutely summing operators). Letting ~ ( ~ ) - - - ~(X~) 0~ ~ ~ and substituting ~(~j___~( ~ ( ~ ) ( ~ _ ~ ) ) - ~ with

~

fixed, 0 < ~

ct(~)

~< ~

we obtain:

4

I In view of the Kolmogorov-Smirnov theorem (cf. EI3J , p.57) the right-hand side of this inequality is less than or equal to the number o ~ q ~ ( T ) ~ v ~ { I I ~ f ~ I I : ~ ( T ) ~ ~ U ~ } ~ 6~(~) (since l~I ) ~ in the unit disc D not depend on ~ and # ). By Fatou lemma and the fact that ~ a.e. we can pass to limit as ~--~ a n d ~

; here

C~

does

~A(~)=~(~) ~

Since fo~ every ~ the function~v-->- C 4 - f ~ ) -t is in H ~ , the integral under the modulus sign coincides with the integral

J~(%)(~-f~)~1~FFb(~) . Now using the Cauchy formula and the fact that the function ~ (~) is analytic in we can rewrite the last inequality in the following way:

345

.< c ~ z ~ ( T ) .

To complete the proof it suffices to pass to limit as (applying the Fatou lemma once more). @ COROLLARY. Under the assumtions of Lemma I we have

II

(T).

PROOF. Apply Lemma I and the reverse Binkowski inequality in the space L ~/~ @ RENARK. Lemma I implies immediately the result of the paper [9] mentioned in the introduction. Indeed, let T , T ~ ~ ( C A ~ ~) be a translation-invariant operator, If we identify the spaces H ~ and ~ by means of the Fourier transform then by Theorem on invariant operators (see the introduction) the operator T admits a factorization of the form (2) with ~ and ~ 6~-~ ~ ( 6 ~ are some constants~ Lemma I gives now that ~ I C~ l ~ < o o ~ a n d this implies easily that the operator T is nuclear. PROOF OF THEORE~ I (for the statement see the introduction). Let V E ~ ( C ~ 6 ~) and suppose that ~ has a factorization of the form (2). Then for every p , ~ has also the following factorization:

I CA

~IC~ P'

~

14P(a~) ;J ~ H~(~m)

----~

~

~

( ~ is the formal identity operator). Fix (till the end of the proof) a number m~ p • ~ . By [II] , section 2 the operator I-~p ~ a ~ IC ', A is p-integral, hence it is sufficient to prove that the operator '%J is compact (of. [3] , p.346, Proposition 24.6.3). Suppose to the contrary that this operator is not compact. Then there exists a bounded sequence [ ~C~} in H P ( ~ ) (ll~llp~ ~ say ) as well as a positive number C So t h a t ~ 3 ~ ~ 0 weakly but tI~U~II>.C .

346 Pmssing to a subsequence we can assume that the s e q u e n c e ~ } is " equivalent to the unit vector basis of the space ~ . ~ Let X = C~5~ $ ~ ~ : ~C~'+ ~ and denote by P the orthogonal projection from ~$ onto X and by G the isomorphism between X and ~ which takes the vector ~ to the ~-th unit vector ~ ~ ~ 2 ' " . Finally, set T -~- ~ P V • This operator ~ clearly has a factorization of the form (2) (and from now on " ~ " will stand for the operator which arises in the factorization of ~ , but not of ~ ). ~oreover, the operator ~ has the following property: there is a sequence {~} in ~ I ( ~ ) such that in fact ~ H P ( ~ ) ~ I[~l[P~l~Me dsh o W for all ~ and the equality ~ ~-~ . shall Show that this is impossible. Given the factorization of T of the form (2) we construct the functions ~ ) ~ and ~ just in the same way as it was explained after the formula (2). Take a K ~ 0 and let ~k be the outer function with [ ~ K [ = ~ ( ~ 3 a.e. It is an easy consequence of well-known properties of the operator of harmonic conjugation that ~K ~ ~ in measure (the Lebesgue one is meant) as ~ ~ co . Since the functions ~ are uniformly bounded a.e., we obtain

r

,f/ r

) VPr because I~-t~K I ~ ~

and

~-~ ~K

the definition of the functions ~ = ~ ~0~... imply that

.-

~

in

measure.

But

and the equality ¢ ~ ~ 6 1 ~ ~ . Now the

preceding estimation shows that if ~ is a sufficiently large number (which from now on will be fixed) then for all ~ we have

347 Since ~ k ~ K ~

H

P

I k~ ~ ~~-'~ ~ = I ~

and ~p~ 5~ ~-'/~

~eO , we obtain that

.~-') ~ ~ d~ =~ ~. ~ ~ ~ .

He nce

ult lip, 1t Jtp K11 ttp, II~ lip,~

~ ~MIll,

lip,,

pr

. Fix now a number ~ with ~! < ~ < 0o . As it has been already mentioned we have the inequality 5 ~ [ ] ~ [[5 < co . Together with the inequality ~ll~f ~(~M~ ~his implies (in view of the fact that the function ~ ~]I~II~ is log-convex) that there exists a positive number- ~ ~such r that [ I ~ I[~ ~ ~ for all ~ . But this contradicts the Corollary to Lemma I. 2. An exemple..: t h e

spaces

The proof of Theorem I depends heavily on the fact that the Riesz projection ~. acts from L I to ~ , %~ I (see the proof of Lemma I). It should be noted that some analogues of Theorem I are probably valid for some spaces other than the disc algebra but sub, eared to the following condition: a projection is to be "assigned" to such a space and this projection must behave as the Riesz projection does. Of course the proof of Theorem I does not work in this general setting, for besides of t h e ( ~ - ~ ) continuity of ~. some very specific techniques of the theory of analytic functions in the unit disc have been involved (for example this proof does not work for the spaces C(~)(~ ~) , ~ ~ , ~ ~ ~ ). But if we restrict ourselves to translation invariant operators onl~ no additional arguments except those based on the - - - ( ~ ) -continuity mentioned above are needed (of. Eg~ ' ~ d ; see also Remark after Lemma I). In this section we show that a "small distorsion" of the space C A (which affects, however, this ( ~ I ~%) -continuity) may cause that no analogue of Theorem I is true for such a "distorted" space (even if we restrict ourselves to translation-in-

348 variant operators). Let _~ be an infinite Hadamard lacunary subset of ~ _ (i.e° inf I ~-~-il ~ ~ where ~ ~ } is the enumeration @~ of -~ according to the magnitudes of moduli of its terms). Denote by ~ii the closed linear span of the set ~ ~ : ~ E E-/~ U ~ + ~ in the space C(T) . Define ~n operator from ~_~ to ~(~i) by the f o r m u l a ~ = ~ ( ~ ) } ~ A • THEOREM 2. The operator ~ is 0 -absolutely summing and ~(~) = ~i(~i) (hence ~ is noncompact and therefore it cannot be ~-nuclear for any ~ )). REMARK. The equality ~ ( ~ ) ~ ~(2~ is probably known. However, I was not able to find an appropriate reference. PROOF. Pix a number %~ 0 < ~ < ~ . We shall show that ~ ~ (~ ~(A)) (then automatically ~ E E ~0 , as it was already mentioned). Let ~ be the closure of the set ~ in the space ~ . In the paper [14~ it is shown that ~ is the direct sum of the space ~ and the space ~ , ~ = C~@~L~ 8pa~ { ~ " M,~',.~} . Let P be the projection from X onto ~ whose kernel is ~ . It is well known (cf. e.g. EI5] ) that there exists an isomorphism between ~ and ~ ( ~ ) which takes the functions ~ , ~ to the unit vectors of ~(II) . Clearly ~ ~_~J~(i~,~l~&) and it remains to apply Factorization theorem from the Introduction, To prove the equality ~ ( ~ ) ~ - ~ ( ~ ) it is sufficient to check that the operator ~ is an isomorphism between the spaces ~$(~) and T~(~(J~)) . Let~_--_--~ M ( ~ ) : I~ = 0 for all ~ in ~ . By F. and M. Riesz theorem (cf. [I3] ) ~ ~ if and only if ~=~ with I ~ 4 and ~ ( g ) = 0 for ~ E (-~) and ~ - 0 (the set all such ~ s will be denoted by ~a ). Let us identify in the canonical way the spaces ~ nd ~ ( ~ ) / ~ . Then the fact we are to check may be restated as follows: if ~= ~~_~(_~i) and ~ then

where ~ is an absolute constant. But this is just the wellknown Paley's inequality, of. [15] , vol.2, Ch. I2, §7° • COROLLARY I. The space & is not isomorphic to any quotient space of the space C A . @

349

COROLLARY 2. The space CA is an uncomplemented subspace of gj~_ . PROOF. Let X = C~SO(T) ~ ~%~: ~E_~} Suppose to the contrary that there is a projection ~ from ~Jk onto defined by the formula Q ~ CA . Then the operator is the rotation operator, is a translation-invariant pro. Since the kernel of ~ is jection from ~ onto 0 A a n d ~-O
COROLLARY 3 (~aurey, [8] ). If

0< % < I

then

PROOF. Since ~ ( ~ ) = ~(A) , a n y operator from ~ to ~ ) can be factored through ~ • • We pass now to a discussion of some linear topological properties of the spaces ~Jk which are similar to properties of the disc-algebra. First of all we mention that it is not hard to verify that the most of the facts concerning the spaces ~A a n d ~; proved in Sections 4, 7, 8 of [II] and stated in purely linear-topological terms have natural analogues for the spaces ~j~ (To check that these analogues are true it is sufficient in some cases to repeat an appropriate proof from EII~ with minor changes; in some other cases one needs to combine a known result concerning the disc-algebra or its conjugate space with a simple arg~Lment based on the fact that ~Jk differs from C A by a "small" space). We state here explicitly only one proposition of this sort. PROPOSITION I. Every ~I -factorable operator from the space ~_fL to a separable conjugate space is compact. To prove this apply directly Remark at the page 27 in [II].i By this proposition ~ is an example of an operator of the class ~0 (and, hence of the class ~ ) which does not factor through an ~(~)-space. The question if such operators exist had u

350

been open not very long ago (now, however, many other examples are known). For some additional information on this subject see also Remark at the end of Section 3. The rest of this section will be devoted to a proof of the fact that (just as in the case of the disc algebra) for every Banach space~. ~ _ ~ and every p~ p ~ ~ the following equality holds: (~C^ ,~_~ y ) . In contrast to the facts mentioned in the preceding paragraphs the proof of a quantitative variant of this equality (as stated in Theorem 3 below) is rather involved compared with the proof of the analogous result for the disc algebra in [II] . Let X be a Banach space and p ~ I . set

IIp(~A~Y)~_Lp

~(X)=~pI~p(T):T~p(X~)~

%(~)~

y

arbitrary } .

These numeric characteristics of X were introduced in p. 56 and it is known that ~ ( C A) ~ ~ for p~/~

~p(OA)X(p-t)

"~

for

t
, cf.

[II] , and

[z6]

or D E ] __ p. I6, Theorem 2.4. The spaces ~ have the same property. THEORE~ 3. There are two constants ~/and ~ (depending on the set ~ A .°nly) so that C~p ~<.~'~ (~,) ~A ~< ~ for p ~/$ and ~ ( p _ ~ ) - 1 < ~ p ( ~ ) ~ W ~ ( p _ ~ )--- ~ . . . . for ~ < p ~ < ~ . In particular ~ p ( ~ ) ~ < cO if 4 < p< co . RE~d~RK. Let ~ and ~ be Banach spaces and let •

~e~

.

:

T~= '~t~ x } . In_~[Ii] , Section 9 some l o w e r e s t i m a t e s of the constants ~ C A ( ~ )_ )~ p < oo were obtained (they show in particular that these constants grow at least as fast as ~c8 as ~ tends to infinity, aL being a number dependent on p ). The unique property of tile space ~A used in [II] to prove these estimates was the fact that the constants ~ ( 0 A ) C ~ behave as described above. Hence the same estimates are valid for the spaces ~ A " To prove Theorem 3 we need a lemma which is probably of independent interest also. Its proof will be postponed till the end of this section. LE~b~A 2. Let j~ be a nonnegative finite measure on the unit circle ~ . Then there exists a finite measure ~ on T

~<

with the following properties:

cI12II

an operator

~

(~) i ~ ~

II It

c being an absolute constant; (2) there exists so that G projects },P(~) onto ~ P ( ~ )

351 s u t neousl

for all

and iI01rp,

for

i ~ is an absolute constant). ~oreover, the orthogonal projection from L$(~) onto ~i(~) may be taken to coincide with ~I ],~(~) . RE,LARK. A weakened version of lemma 2 was used in [II] (though it was not stated explicitly there) to prove that wp for pb~ and ~ p ( C A ) ~ ( p - ~ ) -~ for p~< ~ . To be more precise, a projection of UP(~) onto ~P(~) for some ~ > / ~ was also constructed in [II] , but in contrast to len~na 2 this projection "depended on p " (i.e. it was defined by different formulae for different ~ s ; this formulae did not agree even on ~co(~) ). However the proof of this fact involved only estimates of the Riesz projection ~+ in [,P , and to prove Lennna 2 analogous estimates in weighted spaces ~P(G~) are needed. If we argue similarly to the proof of Theorem 3 below but use only this weakened version of Lemma 2 we shall be able to prove a proposition differing from Theorem 3 only by a weaker estimate of ~p (~_A_) for ~ < ~ ~< (namely, ~p ( & ) ~ ~ (p-~)-3/~ instead of ~p ( ~ & ) ~<~(~_~)~ ). But to establish the "right" estimate Len~ma 2 in its full strength seems to be inavoidable. PROOF OF THEOREM 3. First of all note that the norm of the natural (i.e. translation-invariant) projection ~ of LP onto the space ¢~oshp % ~ & ~ b [ ~ : ~ j ~ } is less than or equal to

~p(CA)

c ( p - # ) -~t~ pends on ~

if

4
and cp 4/~

if p~z

( c de-

only), cf. [15] ,vol. 1, Ch.5, #8 . Evidently, the same numbers are metjorants :for the norms I/p ( ~ ~ • To begin with let us prove the lower estimates of the numbers ~p (~j~) . Let X p be the closure of the set ~jk in hP and let ~ be the formally identical operator from ~Jk to Xp (i.e. ~__ i p ~ l ~ l k ). Then ~ p ( ~ ) ~ . On the other hand it follows from the results of the paper [I23 that ~p (~) is equal to the norm of the translation invariant projection from ~P onto X p , that is to the number

U P++ -P IILP---~~P

• Hence

"~,p(&) ~ [I [])+llLp_>.~P- lip ]ILp_,_LP ,

and to get the desired lower bound for ~Pl ('I~ ~JM) it is sufficient to apply the above remark about IILp.__>.Lp and the known estimates from below for II ~+[I[,P__~[P

, of. [I3] ,

.

We pass to the upper estimates. Let ~

4
be a Banach space,

-~- ~

. Set

352

~='V'ICA. anf let~,~M(V) be

a

Grothendieck-Pietsch

measu-

for ~ ~]#Jl= ~ • Then the operator ~ admits a factorization ~ ~o(~ I~) with ~ ( ~ ) ~ y)~ II~II ~ ~ . Applying #~mma # to the measure ~ we get a measure ~ and an operator ~ with the properties listed in this lemma, and this enables us to factor the operator ~ in the following manner: re

here ~ and ~ are formal identity operators. Since the space ~oo(~) is metrically injective we can extend the operator to an operator ~ from ~ toL°°(~) withII~111 = II~II Let ~ - - - P~QIp, X]I . Then W E ~ ( ~ i i ~ y) is clearly an extension of ~ satisfying ~p ( W ) ~ <

and

~IIQp*p(~p,~)~llQIpllAu%IQIp d/PIl~"l "-< % C Vp ~O~llp is the constant from Semm~ ~). ~ince~,V-W) lq = @ (, we o hav~ V - I ¢ = ::D g with II ~ II -< k ( l l V - kT II)

( L depending on _A on~y). Consider first the case it) ~/ ~ . Then tp ( ~ - W )

'~(4+c'/P + ~,p ( V - k T ) If

IIQIIp)

~< @p,,

p~ ~

,

hence

~ ( , V ) ~ '~p(W)+

and we ,~re done.

we have to consider one more factorization of

with ~ being the formal identity operator. The crucial point is that ~ and P stand here for the same operators as in (I) and~moreover, P ~ Q T p ~ ~ ~ Q I ~ . Now the definition of the operator ~A~

(2) give ~< ~ ~

together with the factorization

thatll-Wl~IIPllli~Z llllOII~IIZA,~II7~ I~ an absolute constant. Of course the inequality

~p(W)~ d Ip IIQII~< ~d/~ (P-I)-~

obtained from the factorization (T) is also true. Now we have: ~p ( ~ )

•

353

PROOF OF LEI~&A 2. Let ~---~ 0 ~ + ~ , where ~ is singular with respect to ~ and ~4(~) . Without loss of generality we can assume that ~(~) 7/ ~ for some positive ~ . Since HP(~) = ~ P ( ~ ) ~ ~PC~) ( the ~ -direct sum is meant, cf. e.g. EII~ , p. I4) it is sufficient to treat the case ~--- G~t and to find for such a ~ a measure having the properties listed in lemma 2 and an additional property of being absolutely continuous with respect to the Lebesgue measure ~ . In the case j~ ~ ~ the operator Q will be defined in terms of the Riesz projection ~+ considered as an operator in a weighted space ~P(~ ~) . For conditions on 60 ensuring that ~÷ acts in such a space see EI7~ , EI8] , EIg~ For us the following special case of results Qbtained in EI7~ ,

bq

E

Let G0 be a nonnegative summable function on the circle. Then ~0 is said to satisfy the ~uckenhoupt condition (A~) if there exists a number O so that

( O0~

is the Hardy-Littlewood maximal function:

l THEORE~ ON WEIGHTED SPACES. if a function ~0 satisfies the condition (A~) the.~ the operator ~+ acts from ~P(~0~) to ~P(00~) if ~ < p < co and for ~ < p ~ < ~ we have the inequalityll ~+llp~0 ~ ~

04 (p_~)-~

with a constant

~

depending only on C from the condition ( A ~ ) • REd,ARK I. Let ~ > ~ ~ ~.~ ~ > 0 . Then it is well-known that there is a function ~ satisfying the conditions construction: co ~

2

~

I1 t1 .< set

~e=

II ~,

where ~

and ~.~

=

~

for

(a ~0

and

is a small positive number).

RENARK 2. If a function CO satisfies the condition (A~) and O < o~ < ~ then the function OO~ also satisfies the condition (Aw) with the same constant O :

354

T Now we continue the proof of 1,emma 2. Given a measure ~ , ~ ~4 as above, find a function ~ so that ~>/~,~ q/~ satisfies the condition (AT) with a constant independent on and

I~ ~

~ ~ ~ @~

,

~

also being an abso-

lute constant (this is possible in view of Remark I: s e t ~ C 5 ~/~, then S~[,~ ; starting by this S construct a G as in Remark I and set ~ ~-- G ~ ) . Let F be an outer function with ~ I = ~_ ~q/~ (such a t~ exists, for ~(~) ~ ~ ). Define an operator G by the formula ~ ~- i~M ~+ ( ~ # ) . This operator is evidently an orthogonal projection of the space I,~(61"~) onto ~(~4) since the mapping ~ ~-~ ~ is a unitary operator of ~ ( ~ ) onto [~ and it maps

IIQIIp, = Ilqll ,

~(#et~)_.4 onto ~$ .Hence p-t+oL--= ~ and it is sufficient llOllp~

for

~<~<

~

if

to estimate the number only. But for such a ~ we have

Since O ~ 4 - p / ~ / ~ i f 4
l lP d .)

@

3. Almost Euclidean subspaces and some finite dimensional examples. In this section we restrict ourselves to spaces over the reals, but the results are valid for complex spaces as well and only minor changes of the proofs are needed. We begin with the

355 statement of the main result of this section. THEOREM 4. There are three constants A4 , A~ and A S with the following properties. For each positive integer P~ there exist an ~-dimensional Banach space X ~ and an operator T ~ from ~ to ~ so that: (6) if ~ is an arbitrary operator from X ~ to ~ then ~K4 (~)~ A4 II~II ; (~) 5~p ~ 0 ( ~ ) ~ eo and # ~ 6 1 1 >/AS iI~jI for

every ~

in (T~(X~))* ~ (~u) "I)t (T~)~ A S - ~ •

REMARK. The second inequality in (6) means that the operators ~ are "uniformly with respect to ~b onto their images". So they are in a sense analogous to the operator ~ of the preceding section. Note also that the inequality (O) is the best (or rather the worst) possible, for if ~ is an operator of rank k~ then ~I ( ~ ) ~ ~ ~ (V) ~<~ ~o (V) (to prove this set ~l~bV , write ~ 4 ( V ) ~ q u ~ V ) ~ ( ~ y ) and use the fact that ~U¢($~Z ) = ~ provided ~ { # # ~ = k~ , cf. [3], p,286, Theorem 20.2.4 and p.385, Theorem 28.2.4). We need some facts concerning "almost Euclidean" subspaces of finite dimensional Banach spaces. Let ( E ~ l.I) be an ~ dimensional Euclidean space and let II°l[ be a norm on satisfying the inequality II~Cll<~ 13cI~ 0 ~ . Set ,

and

suppose that { 4~eeB4~ / ~ < 1] ved in

also

[20~

. The following result is pro-

, though it is not stated there explicitly

(cf.

[21] ).

THEOREM ON ALMOST EUCLIDEAN SUBSPACES. Let ~ ~ ~ < F~ Then there are two orthogonal (in the Euclidean space(E ~ I'l~ )

subspaces ~ 4 ' ~

of the space

~_

so that

~L~t~--k)~,m,(~t:

=ru-k. and f~r 0~ in ~ U C~, the following imequ~lity holds: II Here C ( ' , ") is a function of two real variables which does not de#mud on the triple ( E ~ I'I~ If'If) and is bounded on each of the sets { ( ~ ) : ~ < ~

(B I ; Let 0 < p < co . set ~ P = LP(~) where ~ is a measure on the set ~ 4 ~ ° . o ~ ~ ] defined by the formula LE~iA 3. Take in and choose the norm Then

l"Ithelsituati°n described aboveE : h ~ to be the norm of the space

~4

.

356

( B4 JI'U

and ~ are as above the balls corresponding to the norms and l'j )" PROOF. This fact follows easily from the known formulae for the volumes of the balls ~i and ~ e . See e.g. E20] . • Th s l e n a

shows that for the triples

L.,

IIIio ,,, II

the constant in the inequality from Theorem on almostEuclidean subspaces do~s not grow as k~ tends to infinity provided the ratio ~ is bounded away from 0 and from I . The next lemma is crucial for the proof of Theorem 4. It shows that the same is true for appropiate quotient spaces ~ / ~ with the induced norms. This fact seems to be of independent interest. Let 0 < ~ < ~ and let ~ be an ~-dimensional subspace of the space ~ with ~nt ~< ~ . Set ~ - - - ~ % / ~ and let I" j and Jl"Jl norms of the spaces ~¢

be the norms on ~ induced by the and ~ Set as above ~I =

~ - [ g c ~ E : li~II~<~ , ~ - - { ~ E ; 13oI~
,

where

~ depends on the constant ~ only. PROOF. Let ~ be the orthogonal complement of the space in ~ a~d let P be the orthogonal projection onto the subspace ~ . If we identify E and ~ in a natural way then the ball B ~ will "coincide" with the set U = { ~ : I 0CI~ , ~< ~ } and the ball ~I will "coincide" with the set ~ V = ~ ( { ~ E ~ : JIOCII1 ,,~<~}) • For every subset A of the set ~,...~ ~ deflne WA=~OCC~.JIDCIII u~l , ~ ( ~ ) = 0 for ~ ~ A } . We shall show that ,~.r-

Let us prove first that once the formula (I) is established, the lemma easily follows. Let vo~ be the ( ~ - ~ ) -dimensional volume defined by the Euclidean structure of the space ~,~ Then we have

357

where B = [//,~,..., I%-I~i}

. Consequently

4

~

Set U R = f0~"

I1~11~,~-~

, ~C(~) = 0

for ~B } . Then is an arbitrary vector satisfying then O 6 ~ W B if and only if

OC 96(~)= 0

for

4

OC4B ~-~

~-~

and

X~

U~

if and only if

Now Lemma 3 implies that

(~W~-~ - -

# C~W~]._

=~-~/

-

+i~

~i~

-<-A-~t--m-/'~'
and hence

#

B~

~-~ ( ~-~ ~ ~1~

We pass now to the proof of the formula4 (1). L e t { ~ } ~ < ~ < ~ be the unit vector basis of the space ~ :e~(~)~ 0 for ~=~ ~" , ~(_ ~') ~- ~ . For every O~ satisfyingiI~..,~.,~ there exist numbers ~ , ~:-~+ ~ so that OC lie~in the convex hull of the vecto~s~4~1~ 8 1 ~ . . . ~ 8 ~ C ~ @ • Let Since the range of ~ is ~ ) -dimensional, th~odo;y theorem (cf. E22] ) implies that we can choose ~ - ~ - ~ v e c t o r s ~ 1 ~ . . . ~ _ ~ + I from 7~... ~ 7~+4 so that POC----

358

~ow i f ~ :

o .

for some ~ uo

thenPz

o.ow

is c~early in o

.or

t.en

we can replace the above formula for ~ by another one a way that one of the vectors ~ in this new formula There are numbers ~i~,..~ ~ _ ~ + ~ at least one of which zero so that ~. ~ ; ~--- ~ . We may assume that .~

@

. Define ~a=~numb~r r

or4 + (£S~ ~" 0

by ~ :

5~

for all -@ ~ and set

~ ~:

~

0

~4 = O/x+ ~ 5 £

in such is ~ . is non~

?=4 # . Then

® and at least one of the numbers ~ is equal to zero. @ RE~iARK. Let ~ be an ~-dimensional subspace of the space ~ , I"14/11,~o~ ~ 0 ,~Since the intersection of ~ with the unit ball of the space ~ is contained in the orthogonal projection of this ball onto ~ , the same proof shows that

~C{ ~':

I1~ Ita>.~-<~ } } )

~" '

with

~o~ depending on o~ only. PROOF OF THEOREM 4. Theorem on almost Euclidean subspaces and lemma 3 imply that the space L ~ contains two subspaces ~ and ~k~ with the following properties: ( ~ ) d ~ k ~ = [ ~ 1 (~) H ~ = ~ ~ (~) the inequality

holds for every ~ in 6 ~ V Hpv with a constant C independent onto the of /I, and <3£ . Let ~ be the quotient map of L~~ space ~ /~ and let llI"Ill be the norm in this quotient space induced by the norm II'll4~ . using again Theorem on almost Euclidean subspaces and Lemma 4 we can find a subspace ~6 of the space ~ so that ~ t ~---~-[~] andll~ll~< ~iIII~(~)IIl for all 00 in ~ ( ~ being independent of k~ and 06 ). Let ~ k $ = G ~ _ ~ ( ~ 0 ~) and let ~ be the orthogonal projection from ~ onto ~ . Then for each ~ in ~ we have

359

Thus if we replace the L~ norm in E ~ by the L~ norm the projections P ~ remain bounded uniformly in ~ . From this it follows easily that the norms II'II~ and II'II~ are equivalent on ~ uniformly with respect to ~ , since they evidently are on G ~ and ~ separately. Using the log-convexity of ~ P norms we ~et the following: if ~ E E ~

then~OCll~< ~I 06114/%~ ~

~

does

(3)

not depend on ~ and ~l Now we are in a position to define the spaces X~ we want to construct. Let X~ be equal to E ~ as a set and endow it with co ~he ~ -norm Olearly ~ ×~= ~ . ~he inequality~(~)-
o is contained

(as a s e t )

in ~

and t h e r e f o r e

for

every ~

X~

the inequality (2) holds (and ~ in this inequality does not depend on 1!, ). Let ~ be the formal identity operator from X~ to E ~ (the latter being supplied with the norm II'II~~ ) and set ~-" T~=~, ~ . It follows from (3) that the prooections ~6 ar e ~fo~y~bounded by the constant C% in ~/%(~) -norms, so SU~/%(~) <~ O~ and hence ~lJ~p Suo(T~ ) ~ co " . To prove the inequalityll~ll~ A~ II~II ~ , ~E~-~- (T~(X~))* note that it is equivalent to the following one: -i

±

II + v But this inequality does hold for some A S independent of l since ~J_ X14, (in the space ~ ), s p a n ( ~ U ~ ) = ~-~-m 14, and for ~ in ~ the estimate (2) holds. It remains to show that ~ ( T ~ ) 7 / A 3 ~ . Let be the formal identity operator from ~ to ~ ( ~ is supplied as above with the ~(~)-norm). Then ilI~l14 " ~ and

and we are done. I COROLLARY. Let X =

(X~(~ X~ ~

...)co

. Then ~

do-

360

es n o t have t h e property (Kp) but flt(X*6~) ~_$~(X* ~). PROOF. The first assertion is evident. The second one is true in virtue of Remarque 8 in [~ , since the inequality (2) V~ L holds for all ~ in A ~ uniformly in ~ . This corollary shows that the scheme of proving the Grothendick equality for a space ~ * described in the introduction does not work in general. Note that it is possible to construct an example demonstrating this using only Lemma 3 and avoiding Lemma 4. ~ K . Let ~ be the standard norm in the ideal of L 4 factorable operators. Then ~I(T£) ~ C ~ (T~) , C being independent of ~ .(For if T ~ =

A

(hkA),

(B) FIIBII

then

dll A II

;

but

in iew of rot endieckt eorem

in view of the results

proved in

[4]

, [5]

) . Hence

in [23] a sequence of finite rank operators with analogous properties was constructed, but the operators ~ have an advantage of being "uniformly ont____ 9 their images". 4. 09erat0rs with an arbitrary range. Up to this section we discussed the problem of nuclearity of 0 -absolutely summing operators with values in a Hilbert space, but the analogous problem for operators with an arbitrary range might also be of interest. The next theorem answers a ques/ tion posed by A.Pe~czynski. THEORE~I 5. For every compact topological space ~ and every Banach space Z the following inclusion holds:

~e(C(~)~Z)c

c PROOF. We need a l e n a which is probably of independent interest. LE~Z~ 5- Let Z be a Banach space, ~ L a finite measure and X a subspace of ~4(~) . Denote by J ~ the closure of X J in the space h'(#) ( 0 < % < ~ ) and let ~ be the formal identity operator from hi(B) to ~ % ( ~ ) . If W is any continuous linear operator from y to Z then the eperat@r W o ( % I X ) is weakly compact. Theorem 5 follows easily from this lemma. Indeed, let V ~ ~H%(C(~)~Z) . Then faotorization theorem (see the introduction) shows that ~ ve diagram:

may be included into the following commutati-

361

E

e X

L,#t

#l×

y

v cd)

....

m ----. Z

Here ~ is a finite ~.easure, ~ and W are some continuous linear operators, E is a subspace of L°°(~) , X = c~O~L~(/~}E , i . the formal identity operator frgm to ~($~) . By Lemma 5 the operator ~ o (~I X) is weakly pact. Since the °perat°r (14'~ I ~)°~)C(~ is 1-absolutely summing and its domain is the space , this operator is lintegral (cf. E3] , p.233, Proposition 17.3.5). It remains to use the fact that if ~ is a weakly compact operator, B ~ I I and the composition A B is well-defined then A B is l-nuclear (cf. [3] , p.345, Theorem 24.6.2). • PROOF OF LE~L~ 5. Denote the composition ~ o (#I X) by T . Let ~ O ~ } be an arbitrary bounded sequence in X . We shall show that the sequence ~ T ~ } has a weak accumulation point. First of all we claim that there is a subsequence [ ~ } of the sequence ~T3cs.} accompanied by a sequence ~ek} of mutually disjoint ~-measurable sets so that the set { ~ ( ~ 6 ~ ) : ~=~... } is relatively weakly compact in~4(~) . To verify this claim one needs to repeat a part of the proof of Theorem 5 in [24J . Here is a sketch of the argument. Set ~ ( 6 ) = =St~II0C~l~:~e~<6~=~,~.,}and

let ~ = ~ ~(~) If ~ = 0 &-~O there is nothing to prove. If ~ 0 choose a sequence of positive integers ~ } and a sequence of sets ~e~} so that W < ~ < <~<'"' ~k ~ 0 and~ I~nk I ~ ~ f . S e t t i n g ~ = 06~ ek we get all we need except may be the disjointness of the sequence ~ C ~ , and so on. Now let tO~------~ ~ , , Ir~= ~ ( ~ - ~ 5 ~ ) . We shall assume (as we may) that ~ - - - ~ weakly in ~ (~,) a n d ~ 4 ~6 ~<~• By separation theorem there exists a sequence i ~ } in ~ (~) so that ~---*- tr in norm and ~Sg6CO~If[~r~ : ~ ~} . For each write %~ as E ~ g 1 ~ with {o6~} a finite sequence,

~>z0 ,~k=

4

and set 9J~-----~o6~~

. Then~(6~ipp ~f~)--~

362

and

.

and since ~ + ~ / ~

that T ~

)

X ~(iir)

Consequently¢llll=0,

it follows that } ~ weakly.

~

. Let us show

Note that tl~e above argument is applicable to any subsequence [ %r~ of the sequence [~f~} . It follows that for every such subsequence

Suppose that Then F ( T ~ ,

~(T

~)

/ >

~'l(W(o~'t)') )

) ----~ O~=~P(W(~r))

indices. Hence for

~t

for some

for some sequence

~

in

{~,}

y-J(" . of

sufficiently large we have

But this contradicts the formula (I). • It is natural to ask now whether Theorem I stated in the introducti6n and proved in Section I remains valid for operators from the disc algebra to some non-Hilbert spaces E . The answer is very likely to be positive (and, moreover, some parts of the proof may be repeated almost without changes for certain non-Hilbert spaces ~ ). However now when I am wwiting this I cannot formulate any satisfactory conjecture on this matter. The only thing I want to mention is that some facts concerning arbitrary spaces E can be derived from theorem I. Here is an example. THEORE~ 6. Let E be an infinite dimensional Banach space and let T ~ o ( C A ~ E ) . Then T ( C A ) = ~ = ~ FROOF. Suppose to the contrary that T ( C A ) = E . Since ~o c ~ , the operator T factors through a Hilbert space and hence ~ theorem I.

is also a •

Hilbert space. It remains to apply

References. f

f

I. A.G r o t h e n d i e c k. Resume de la theorie metrique des produits tensorie]stopologiques. Bol. S o c . ~ t . SHo Paulo, 1956, 8, 1-79. ! 2. J.L i n d e n s t r a u s s, A.P e ~ c z y n s k i. Absolutely summing operators in ~ p -spaces and their applications. Studia Math., 1968, 29, 275-326. 3. A.P i e t s c h. Operator ideals. Berlin, 1978. 4. C.B.K H c x ~ E o B. 0 HpOCTpSHCTBaX C "t~J~Et~" aH~yJ~TOpOm.

363

8an. ~ay~H. c e ~ H .

~0~,

1976, 65, I92-I95.

5. G.P i s i e r. Une nouvelle fiant le theoreme

t de Banach veri-

classe d'espaces

de Grothendieck.

Ann. Inst. Fourier,

1978,

28,

N 1, 69-90. 6. H.K.H H E o ~ ~ c E H ~, B.H.X a B H H, C.B.X p y m e" B

(COCTaB~TeZa ~ p e ~ T 0 p ~ ) . 99 sepem~Hsax sa~a~ J m H e ~ o r o EoMn~eEcsoro a s a ~ s a . San. Hayes. c e ~ H . 20~g, I978, 81. 7. Y.G o r d o n, D.R.L e w i s, J.R.R e t h e r f o r d. Banach ideals of operators with applications. J.Funct.Anal., 1973, 14, 85-129. 8. B.N a u r e y. Theoremes lineaires 11.

a valeurs

de factorisation

pour les op~rateurs

dans les espaces L P . Asterisque,

/ / 9, S.K w a p i e n, A.P e ~ c z y n s k i° Remarks summing translation

inva~iant

ra and its dual. Michigan

operators

J.Math.,

on absolutely

from the disc algeb-

1978,

25,

173-181.

10.S.K w a p i e ~, A.P e & c z y ~ s k i. Absolutely operators compact wa,

and translation

abelian

groups.

invariant

spaces

Inst.Mat.PAN,

1974,

summing

of functions

preprint

on

N 162, Warsza-

1978.

/ 11.A.P e ~ c z y n s k i. Banach spaces and absolutely

summing operators.

of analytic

AMS Regional

ries in Nathematics, 30, Providence, 1977. / 12.A.P e ~ c z y n s k i. p - i n t e g r a l operators

functions

Conference

Se-

commuting with

group representations

and examples

of quasi

p -integral

rators which are not

p-integral.

Studia Y~th.,

1969,

ope-

33,

63-7O. 13.P.L.D u r e n. Theory of H P and London,

spaces.

Academic

Press,

N e w ~ork

1970.

I¢. C.B.E ~ c ~ ~ E o B. 0 p e ~ e ~ c m s ~ x ~o~HpooTpaHOTBaX npocTpa~. CTBa CA . ~yHE~. a~aJm~ ~ ero npH~o~., I979, I3, ~ I,

21-30. 15.A.Z y g m u n d .

Trigonimetric

series.

Vol.

1,2. Cambridge,

1959.

/ 16.B.S.M i t j a g i n, A.P e T c z y n s k i. On the nonexistence of linear isomorphisms functions 1975,

between Banach

of one and several

spaces

complex variables.

of analytic Studia }~ath.,

56, 85-96.

17.B.M u c k e n h o u p t. Weighted norm inequalities

for Hardy

364

maximal function.

T r a n s . A m e r . ~ t h . Soc.,

1972,

165, 207-226.

1 8 . R . A . H u n t, B . M u c k e n h o u p t, R . L . W h e e d e n. Weighted norm inequalities Hilbert

transform.

for the c o n j u g a t e

Trans.Amer.Nath.Soc.,

f u n c t i o n and

1973,

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227-251.

1 9 . R . R . C o i f m a n, C . F e f f e r m a n. W e i g h t e d n o r m i n e q u a l i t i e s for m a x i m a l f u n c t i o n s a n d s i n g u l a r i n t e g r a l s . ~ath.,

1974,

51, N 3, 241-250.

2 0 . S . S z a r e k. On K a s h i n ' s a l m o s t

euclidean orthogonal

p o s i t i o n of

. B u l l . A c a d . P o l o n . Sci.,

et phys.,

26 N 8, 691-694.

21.S.S

Studia

1978,

ser. sci.math.,

decoma~tr.

z a r e k, N . T o m c z a k - J a e g e r m a n n. On n e a r -

ly e u c l i d e a n d e c o m p o s i t i o n s Compos.r~th.,

1980,

for some c l a s s e s of B a n a c h spaces.

40, N 3, 367-385.

2 2 . L . D a n z e r, B.G r u n b a u m, V . K 1 e e. H e l l y ' s t h e o r e m and its r e l a t i v e s . Amer.N~th.Soc.,

"Convexity",

P r o c . s y m p o s . p u r e math.,

vol.7,

Providence,

1963. z f 2 3 . T . F i g i e l, S . K w a p i e n, A . P e ~ c z y n s k i. S h a r p e s t i m a t e s f o r the c o n s t a n t s of N i n k o w s k i astr.

spaces.

et phys.,

of l o c a l u n c o n d i t i o n a l

B u l l . A c a d . P o l o n . Sci.,

ser.

25, N 11, 1221-1226. r 2 4 . M . I . K a d e c, A . P e ~ c z y n s k i. Bases,

1861/62,

sci math.,

1977,

ces a n d c o m p l e m e n t e d s u b s p a c e s Math.,

structure

21,

161-176.

in the s p a c e s

lacunary sequenLv

. Studia

S.V.Kisliakov

WHAT IS NEEDED FOR A

0 -ABSOLUTELY SUMMING

OPERATOR TO BE NUCLEAR?

O. Introduction, 1. Operators from the disk-algebra 2. An example: the spaces

C A to the space ~

~

3. Almost Euclidean subspaces and some finite dimensional examples. 4. Operators with an arbitrary range.

337

O. Introduction. A famous result of Grothendieck stating that each continuous linear operator from a space ~I(~) to a Hilbert space is ~ -absolutely summing is the origin of a considerable part of the operator ideals theory, and the present paper is also inspired by it. Besides the Grothendieck's paper Ill one can find proofs of this result e.g. in E2] or E3] , p.308. In symbols it can be written as follows: ~ ( ~ ( ~ ) ~ ~) = ----- ~ ( d ( ~ ) ~ M ) (for the notation and definitions see the end of this section), and it is known now that an analogous equality ~4 ( ~ M ) ~ ~ (X~ H ) (which we call the Grothendieck equality for the space X ) holds for some spaces ~ differing from ~ (~), cf. E2~ , [4] , [ ~ • It is still unknown, however, whether some particular spaces ~ that arise naturally in Analysis satisfy this equality. Of the greatest importance is probably the question whether we can take X = ~ , C~ being the disc algebra (cf. E6] , p.12-14 for a detailed discussion of this matter). If X = ~ * (in particular, if X~-- ~A~ ) then we can try to prove the equality ~ 4 ( X ~ M ) = ~ ( X ~ ~ ) in the following way. Using desiriptions of operator ideals adjoint to the ideals ~4 and ~ (of. [3] , p.269, theorem 19.2.3 or p.276, theorem I9.4.7; cf. also E7] ) we obtain that this equality is equivalent to the equality ~oo ( ~ X ) ~- I ~ ( H ~ X ) • The latter one holds if and only if q ( ~ ) ~ I 4 ( ~ H ) (here I~ and ~p , p=4,~ stand for the ideals of 4-integral and ~P -factorable operators respectively)*~ But since every ~) The fact that the conditions and F I ~ H ) = ~ i ( ~ ~) are equivalent can be easily derived by passing to conjugate operators provided one knows that the equality ~ H ) : I ~ ( ~ H ) implies t h a t ~ ( ~ H ) : I~ (~*~ ~ ) . This implication may be proved via the local reflexivity principle ( [3] , p.384) and theorem 8.7.4 at the page ll5 in [3] •

338 -integral operator with values in a reflexive space is nuclear we get finally that the conditions ~I(~M)= -~-~(~) and q (~)----- a i ( ~ H ) are equivalent. To check the second condition it is sufficient, of course, to check the inclusion ~i ( ~ M ) c N 4 ( ~ ) (for the reverse inclmsion is evident). But ~urey's improvement of Grothendieok theorem (cf. E8] , p. IlT) implies that ~ ( ~ ) C ~ 0 ( ~ ) ( ~ o denotes the ideal of 0 -absolutely summing operators). Therefore the space ~ satisfies the Grothendieck equality if posesses the following poperty: Every 0 -absolutely summing operator from ~ to a Hilbert space is - nuclear

(~P)

The notation (KP) is composed of the first letters of the names of the authors of [9] , because in this paper a similar scheme was applied to treat the case of translation - invariant 0-absooperators. For example, it is shown in LgJ that every lutely summing and transla$iQn-invariant operator from the disc algebra to the Hardy space H ~ is nuclear (of. also EIG] ). It seems to be very probable that the invariance assumption is superfluous. CONJECTURE. The space ~A has the property (K2). The next theorem is the strongest result which I was able to prove in support of this conjecture. Its proof inspired by the paper [9] is presented in Section I. THEOREM I. Every 0 -absolutely summing operator from the disc algebra to ~ is ~ -nuclear for each p ~ ~ . Note that if E and F are any Banach spaces then all classes ~ ( E ~ ) , 0~<~ coincide (cf. [8] or [3] , p.293, theorem 21.2.9), and hence ~ ( C A ~ % ) C N p ( C A ~ ~) if ~ > ~ and % < ~ . However for ~-----~ the last inclusion is nolonger true for the map ~---~- { ~ ( ~ ) } provides an example of an (invariant!) ~ -absolutely summing operator from the disc algebra onto ~i , cf. FII] , p.20, corollary 3.I. In section 2 we construct other operators defined in a similar manneramdpomeBmlngthesame properties, but also an additional property of being 0 -absolutely summing. Let, for L

example,

~- = e~o~Clv~<~j spar~

Lf

~

:

~ ~ 0

or

~----~,

339

Then (see Section 2) we have: ~ . ~ ) = ~

and,

moreover, ~ is translation invariant. So, by Theorem I, the space ~ is isomorphic to no factor-space of the disc algebra, though it is obtained from the ~atter one by adding a "small" space. We shall also discuss in Section 2 some other interesting properties of the space ~ . In section 3 we construct (among other things) a sequence of finite rank operators for which quantitative characteristics corresponding to some qualitative properties of the operator described above behave in an "extreme" way. The main idea of construction of these operators is similar to that for (both of them may be d~afted as follows: take a "known" space, "add" to it a "small" space and then "project" onto this small space). To exploit this idea in a finite dimensional setting we need (and prove) some facts about almost Euclidean subspaces of

~,!./ 7

being

"not very big" subspace of

s6~

(these facts are probably of independent interest). It is shown also in the same section that there is an infinite dimensional space 7. with ~ # ( Z t ~ ) = ~(~ ~ ) which does not have the property (KP). So the scheme of proving the Grothendieck equality described above does not work in general. Finally in Section 4 it is established that ~'~#-P]J~|~)7~z ~,

~''~'"'~'i~tC~J,'/,Jfore ~ e ~

Banach space Z

• #

ACKNOIg~EDGEIENT. I am grateful to A.Pelczynski for many stimulating discussions. Also I want to thank N.K.Nikolski~ and V.P.Havin for reading the manuscript. DEFINITIONS, NOTATION AND SO~E STANDARD FACTS. We denote by ~ ( ~ y ) the set of all continuous linear operators from a quasinormed space X to a quasinormed space ~ Let X be a Banach space, 0 < m < ~ , T E ~ (X~ Y ) . The operator ~ is said to be p-absolutel~ summing if

4'/p

H,

"i.,=-I

1trill P

for .ll

F ~

x

~ i,,,= 'f,

~,,...,i

<

The class of all p -absolutely summing operators is denoted by N p . We do not reproduce here the definition of 0 -absolutely summing operators (it may be found e.g. in E8] ), for it is known that if X and Y are Banach spaces then (as it

340

has already been mentioned) all classes ~ % ( X ~ ) ~ 0~%< coincide and it is more convenient for us to deal with ~%~ ~ > 0 than with ~o • It is well-known (cf. [3] , p.232) that T E ~ p ( X ~ ) if and only if there esists a probability Radon measure ~ supported by the unit ball ~ of the space X # (with the topology ~ ( X e ~ X) ) so that for every ~ in X we have

K (~oreover, if T E N p then ~ can be chosen in such a way that C = ~[~ ( T ) ) . This inequality and the measure ~ are called the Grothendieck-Pietsch inequalit~ and a GrothendieckPietsch measure for T . If ~ a n d A is a 6v(x*~ X) compact subset of K such that 6~o$ co~Ife~A=~(the closure in the topology ~( X*~ X) ) then a Grothendieck-Pietsch measure for T may be chosen to be concentrated on A . In particular if ~>~ ~ and X is a subspace of a space g(~) then we can assume that this measure is defined on ~ ° We need the following version of Grothendieck-Pietsch inequality for translation invariant operators (a more general result can be found in [12] ). THEOREM ON INVARIANT OPERATORS. Let ~ be a compact commutative topological group and let X be a translation-invariant subspace of ~(~) . Let ¢ be a homomorphism from ~ to the group of isometrics of the space ~ . Suppose that ~ >/ and let ~ be a p-absolutely summing operator from ~ to y satisfying ¢(~)(TO6) = T ( L ~ 36) for ~ , . DCE X ( [,@ is the rotation o p ~ r a t o r : ( ~ ) ( ~ ) = ~ ( ~ ) ) Then the Haar meadsure of ~ can be taken as a Grothendieck-Pietsch measure for ~ . Let ( ~ ) b e a measure space,~(~)
0 <

p

(X, Y;

341 Then T E ~ p ( ~ y) if and only if T = U ~ V where U and ~ are linear operators and R is a part of an operator

lp,

RENARE. Of course, any Grothendieck-Pietsch measure for T may serve as a " ~ " in this theorem. In particular if X is a subspace of a space C(~) and ~ 4 we can choose this to be defined on ~ . in this car,e the domain of the operator U coincides with the closure of the set X in ~P(~) . Note that the assumption p~/~ is essential here. Let X~ ~ be Banach spaces, p ~/~ . An operator T , T ~ ~(X~) is said to be p-integral if there exist a measure ~ and linear operators V from X to ~ ( ~ ) and U from ~P(~) to ~ so that @ 6 ~ = ~ the canonical imbedding operator of ~ the space of p-integral operators

p,~F~into ( ~6y~

stands for ). We denote and sup-

byIp Y)I/flI IIIIII

ply it with the norm %p~ Sp(~) ~''~5 ~ II~II (the infimum is taken over all factorizations as above). Again let p~/~ and T E ~ ( X~ ~) . Then T is said to be p -nuclear if ~ has a factorization of the form ~ -=~I~F~,~'~ V , where.~, is a ~discrete measure,VE~(X,~, eo(~)),~. U~ ~-(~P(~)~ ~ ). The norm V p in the space N-(X~ Y) of all p-nuclear operators is defined similarly to For the basic properties of the operator classes ~ p

Np

see

[3]

and

•

By ~ we shall denote the identity mapping of the complex plane ~ onto itself as well as restrictions of this mapping to certain subsets of ~ . Let ~ be the normalized Lebesgue measure on the unit circle ~ . The disc-algebra is the subalgebra of C (~) consisting of all continuous functions with ~(~) d¢~ I ~ p ~ b = 0 for I d , = - ~ - $ ~ . . . A function ~ in C(~) belongs to the disc-algebra if and only if { is a restriction to ~ of some function continuous in the closed unit disc ~ and analytic in the open disc (we reserve for this function the same notation ~ ; in general we often do not distinguish notationally between a harmonic function in the unit disc and its boundary values). The norm of a function ~ in the space LP(~) ( being a finite positive measure) will be denoted by

ll~IIp~fi_

nite measure on t

cle

~

then

m

I

stands for the

342

closure of the set CA in the space ~ P ( ~ ) . We write ~P for ~ P ( ~ ) and HP for ~ P ( ~ ) . The main facts about the classes ~P may be found in [13] . The symbol ~+ (resp. ~_ ) stands for the orthogonal projection of the space ~% onto H ~ (resp. onto ~ ). Note that ~ p_ ~$ . It is known that the operators ~t and ~_ act in fact from ~P to ~P , 4 < p < co and from ~I to ~ % ,

(cf. [z3] ). The space of all Radon (= Borel regular) measures on a compact space ~ will be denoted by M(~) and we identify this space with the space g ( ~ ) * . As usual, ~ denotes the group of integers, { ~e

Z t d¢~

The formulae in each particular section are numerated independently (if at all). I. Operat0rsfrom

the disc-algebra

C~ .......to ..... the space

.

As it has already been mentioned in the introduction, it is convlnient for us to deal with the classes ~ , 0<~< ~ , instead of ~0 . Fix (till the end of this section) a number with 0 < ~ < ~ and let T ~ (CA~%) . our aim will be to try to prove that T is ~ -nuclear (or at least p-nuclear for all p > ~ ). since ~ T c ~4 ( c f . [3] , p.235, theorem I7.3.9)~an application of Factor~zation theorem and of Remark after it gives so thatT admits the folloa measure ~ on the unit circle wing factorization:

). Let q) be the singular (with ~ 6 Z (~4(~), 6Z) part of the measure ~ with respect to ~ . Then~ = ~+ $~ ~ [,~ . We may assume without loss of generality that ~ (since the replacement of ~ by 6~+~ does not affect the factorization (I)). Now (cf. e.g. [II] , section 2)~1(~)---~1(~b)e L4(~),

CA÷Z ,Vl CA

and the operatorL,-

lea

) is (if considered as an operator from C A into lit(f) -integral. Since the space ~% is reflexive, the operator

343

~!4,~ I OA

i s nuclear ( c f . [ ~ , p.245, theorem 2 4 . 6 . 2 ) . On the other hand it is %-absolutely summing (this follows from ~urey's extension of Grothendieck theorem, cf. ~8] , p. IIT, th~oreme 94, for, ~ { ~ I C~ being nuclear, it is of the form ~ V with V ~ ~ ( ~ ~$) ). So the operator ~I{,@~vl ~A is also %-absolutely summing, and it is sufficient to prove that this operator is nuclear (or at least p-nuclear, p > { ). Now we have the following situation. There is a function , ~ ( ~ ) , @~ accompanied by an operator T ' T ~ ~ ( C A ~ ~ ~) admitting the following factorization

Ca It,,,.,..I C~, N ~ .........

~

(a.,'~)

,s" .f/. -'---

(2)

.

Let ~ * be the coordinate functionals of the space ~g . Consider for each fixed ~ a norm-preserving extension of the functional ~ from ~(~K~) to ~4(~H4) and let ~ , ~{~°°(R~) be the function corresponding to this extension after we identify the spaces ~(~K~) ~ and ~°O(~F~b) in the standard way. We have then the following formula for

~

:

101

Let ~ be an outer function satisfying = ~ a.e. (that is ~ ~(4+ ~) , where $b-----~ and is the harmonic conjugate of 14. ; the assumption ~ > ~ has been made to assure that such { exists, cf. ~I3] , p.24-25~ Set ~ = ~_(~ ~-1) (note that s~m II~ g~-lll0<:! < O0 and hence the functions ~g are uniformly bounded in ~P for every p < o o , ce. [13] ). LE~g~ I. In the situation just described the following inequality holds:

t~

where

Z'÷

G depends on ~ only. PROOF. For any (say, continuous)

~A -valued function #

344

on

~

we have

(To prove this note that if ~

=~_d~ k

with

~O~C~ and ~ ~ ~-# for all ~ then this inequality applied to ~ is just the inequality defining ~ -absolutely summing operators). Letting ~ ( ~ ) - - - ~(X~) 0~ ~ ~ and substituting ~(~j___~( ~ ( ~ ) ( ~ _ ~ ) ) - ~ with

~

fixed, 0 < ~

ct(~)

~< ~

we obtain:

4

I In view of the Kolmogorov-Smirnov theorem (cf. EI3J , p.57) the right-hand side of this inequality is less than or equal to the number o ~ q ~ ( T ) ~ v ~ { I I ~ f ~ I I : ~ ( T ) ~ ~ U ~ } ~ 6~(~) (since l~I ) ~ in the unit disc D not depend on ~ and # ). By Fatou lemma and the fact that ~ a.e. we can pass to limit as ~--~ a n d ~

; here

C~

does

~A(~)=~(~) ~

Since fo~ every ~ the function~v-->- C 4 - f ~ ) -t is in H ~ , the integral under the modulus sign coincides with the integral

J~(%)(~-f~)~1~FFb(~) . Now using the Cauchy formula and the fact that the function ~ (~) is analytic in we can rewrite the last inequality in the following way:

345

.< c ~ z ~ ( T ) .

To complete the proof it suffices to pass to limit as (applying the Fatou lemma once more). @ COROLLARY. Under the assumtions of Lemma I we have

II

(T).

PROOF. Apply Lemma I and the reverse Binkowski inequality in the space L ~/~ @ RENARK. Lemma I implies immediately the result of the paper [9] mentioned in the introduction. Indeed, let T , T ~ ~ ( C A ~ ~) be a translation-invariant operator, If we identify the spaces H ~ and ~ by means of the Fourier transform then by Theorem on invariant operators (see the introduction) the operator T admits a factorization of the form (2) with ~ and ~ 6~-~ ~ ( 6 ~ are some constants~ Lemma I gives now that ~ I C~ l ~ < o o ~ a n d this implies easily that the operator T is nuclear. PROOF OF THEORE~ I (for the statement see the introduction). Let V E ~ ( C ~ 6 ~) and suppose that ~ has a factorization of the form (2). Then for every p , ~ has also the following factorization:

I CA

~IC~ P'

~

14P(a~) ;J ~ H~(~m)

----~

~

~

( ~ is the formal identity operator). Fix (till the end of the proof) a number m~ p • ~ . By [II] , section 2 the operator I-~p ~ a ~ IC ', A is p-integral, hence it is sufficient to prove that the operator '%J is compact (of. [3] , p.346, Proposition 24.6.3). Suppose to the contrary that this operator is not compact. Then there exists a bounded sequence [ ~C~} in H P ( ~ ) (ll~llp~ ~ say ) as well as a positive number C So t h a t ~ 3 ~ ~ 0 weakly but tI~U~II>.C .

346 Pmssing to a subsequence we can assume that the s e q u e n c e ~ } is " equivalent to the unit vector basis of the space ~ . ~ Let X = C~5~ $ ~ ~ : ~C~'+ ~ and denote by P the orthogonal projection from ~$ onto X and by G the isomorphism between X and ~ which takes the vector ~ to the ~-th unit vector ~ ~ ~ 2 ' " . Finally, set T -~- ~ P V • This operator ~ clearly has a factorization of the form (2) (and from now on " ~ " will stand for the operator which arises in the factorization of ~ , but not of ~ ). ~oreover, the operator ~ has the following property: there is a sequence {~} in ~ I ( ~ ) such that in fact ~ H P ( ~ ) ~ I[~l[P~l~Me dsh o W for all ~ and the equality ~ ~-~ . shall Show that this is impossible. Given the factorization of T of the form (2) we construct the functions ~ ) ~ and ~ just in the same way as it was explained after the formula (2). Take a K ~ 0 and let ~k be the outer function with [ ~ K [ = ~ ( ~ 3 a.e. It is an easy consequence of well-known properties of the operator of harmonic conjugation that ~K ~ ~ in measure (the Lebesgue one is meant) as ~ ~ co . Since the functions ~ are uniformly bounded a.e., we obtain

r

,f/ r

) VPr because I~-t~K I ~ ~

and

~-~ ~K

the definition of the functions ~ = ~ ~0~... imply that

.-

~

in

measure.

But

and the equality ¢ ~ ~ 6 1 ~ ~ . Now the

preceding estimation shows that if ~ is a sufficiently large number (which from now on will be fixed) then for all ~ we have

347 Since ~ k ~ K ~

H

P

I k~ ~ ~~-'~ ~ = I ~

and ~p~ 5~ ~-'/~

~eO , we obtain that

.~-') ~ ~ d~ =~ ~. ~ ~ ~ .

He nce

ult lip, 1t Jtp K11 ttp, II~ lip,~

~ ~MIll,

lip,,

pr

. Fix now a number ~ with ~! < ~ < 0o . As it has been already mentioned we have the inequality 5 ~ [ ] ~ [[5 < co . Together with the inequality ~ll~f ~(~M~ ~his implies (in view of the fact that the function ~ ~]I~II~ is log-convex) that there exists a positive number- ~ ~such r that [ I ~ I[~ ~ ~ for all ~ . But this contradicts the Corollary to Lemma I. 2. An exemple..: t h e

spaces

The proof of Theorem I depends heavily on the fact that the Riesz projection ~. acts from L I to ~ , %~ I (see the proof of Lemma I). It should be noted that some analogues of Theorem I are probably valid for some spaces other than the disc algebra but sub, eared to the following condition: a projection is to be "assigned" to such a space and this projection must behave as the Riesz projection does. Of course the proof of Theorem I does not work in this general setting, for besides of t h e ( ~ - ~ ) continuity of ~. some very specific techniques of the theory of analytic functions in the unit disc have been involved (for example this proof does not work for the spaces C(~)(~ ~) , ~ ~ , ~ ~ ~ ). But if we restrict ourselves to translation invariant operators onl~ no additional arguments except those based on the - - - ( ~ ) -continuity mentioned above are needed (of. Eg~ ' ~ d ; see also Remark after Lemma I). In this section we show that a "small distorsion" of the space C A (which affects, however, this ( ~ I ~%) -continuity) may cause that no analogue of Theorem I is true for such a "distorted" space (even if we restrict ourselves to translation-in-

348 variant operators). Let _~ be an infinite Hadamard lacunary subset of ~ _ (i.e° inf I ~-~-il ~ ~ where ~ ~ } is the enumeration @~ of -~ according to the magnitudes of moduli of its terms). Denote by ~ii the closed linear span of the set ~ ~ : ~ E E-/~ U ~ + ~ in the space C(T) . Define ~n operator from ~_~ to ~(~i) by the f o r m u l a ~ = ~ ( ~ ) } ~ A • THEOREM 2. The operator ~ is 0 -absolutely summing and ~(~) = ~i(~i) (hence ~ is noncompact and therefore it cannot be ~-nuclear for any ~ )). REMARK. The equality ~ ( ~ ) ~ ~(2~ is probably known. However, I was not able to find an appropriate reference. PROOF. Pix a number %~ 0 < ~ < ~ . We shall show that ~ ~ (~ ~(A)) (then automatically ~ E E ~0 , as it was already mentioned). Let ~ be the closure of the set ~ in the space ~ . In the paper [14~ it is shown that ~ is the direct sum of the space ~ and the space ~ , ~ = C~@~L~ 8pa~ { ~ " M,~',.~} . Let P be the projection from X onto ~ whose kernel is ~ . It is well known (cf. e.g. EI5] ) that there exists an isomorphism between ~ and ~ ( ~ ) which takes the functions ~ , ~ to the unit vectors of ~(II) . Clearly ~ ~_~J~(i~,~l~&) and it remains to apply Factorization theorem from the Introduction, To prove the equality ~ ( ~ ) ~ - ~ ( ~ ) it is sufficient to check that the operator ~ is an isomorphism between the spaces ~$(~) and T~(~(J~)) . Let~_--_--~ M ( ~ ) : I~ = 0 for all ~ in ~ . By F. and M. Riesz theorem (cf. [I3] ) ~ ~ if and only if ~=~ with I ~ 4 and ~ ( g ) = 0 for ~ E (-~) and ~ - 0 (the set all such ~ s will be denoted by ~a ). Let us identify in the canonical way the spaces ~ nd ~ ( ~ ) / ~ . Then the fact we are to check may be restated as follows: if ~= ~~_~(_~i) and ~ then

where ~ is an absolute constant. But this is just the wellknown Paley's inequality, of. [15] , vol.2, Ch. I2, §7° • COROLLARY I. The space & is not isomorphic to any quotient space of the space C A . @

349

COROLLARY 2. The space CA is an uncomplemented subspace of gj~_ . PROOF. Let X = C~SO(T) ~ ~%~: ~E_~} Suppose to the contrary that there is a projection ~ from ~Jk onto defined by the formula Q ~ CA . Then the operator is the rotation operator, is a translation-invariant pro. Since the kernel of ~ is jection from ~ onto 0 A a n d ~-O
COROLLARY 3 (~aurey, [8] ). If

0< % < I

then

PROOF. Since ~ ( ~ ) = ~(A) , a n y operator from ~ to ~ ) can be factored through ~ • • We pass now to a discussion of some linear topological properties of the spaces ~Jk which are similar to properties of the disc-algebra. First of all we mention that it is not hard to verify that the most of the facts concerning the spaces ~A a n d ~; proved in Sections 4, 7, 8 of [II] and stated in purely linear-topological terms have natural analogues for the spaces ~j~ (To check that these analogues are true it is sufficient in some cases to repeat an appropriate proof from EII~ with minor changes; in some other cases one needs to combine a known result concerning the disc-algebra or its conjugate space with a simple arg~Lment based on the fact that ~Jk differs from C A by a "small" space). We state here explicitly only one proposition of this sort. PROPOSITION I. Every ~I -factorable operator from the space ~_fL to a separable conjugate space is compact. To prove this apply directly Remark at the page 27 in [II].i By this proposition ~ is an example of an operator of the class ~0 (and, hence of the class ~ ) which does not factor through an ~(~)-space. The question if such operators exist had u

350

been open not very long ago (now, however, many other examples are known). For some additional information on this subject see also Remark at the end of Section 3. The rest of this section will be devoted to a proof of the fact that (just as in the case of the disc algebra) for every Banach space~. ~ _ ~ and every p~ p ~ ~ the following equality holds: (~C^ ,~_~ y ) . In contrast to the facts mentioned in the preceding paragraphs the proof of a quantitative variant of this equality (as stated in Theorem 3 below) is rather involved compared with the proof of the analogous result for the disc algebra in [II] . Let X be a Banach space and p ~ I . set

IIp(~A~Y)~_Lp

~(X)=~pI~p(T):T~p(X~)~

%(~)~

y

arbitrary } .

These numeric characteristics of X were introduced in p. 56 and it is known that ~ ( C A) ~ ~ for p~/~

~p(OA)X(p-t)

"~

for

t
, cf.

[II] , and

[z6]

or D E ] __ p. I6, Theorem 2.4. The spaces ~ have the same property. THEORE~ 3. There are two constants ~/and ~ (depending on the set ~ A .°nly) so that C~p ~<.~'~ (~,) ~A ~< ~ for p ~/$ and ~ ( p _ ~ ) - 1 < ~ p ( ~ ) ~ W ~ ( p _ ~ )--- ~ . . . . for ~ < p ~ < ~ . In particular ~ p ( ~ ) ~ < cO if 4 < p< co . RE~d~RK. Let ~ and ~ be Banach spaces and let •

~e~

.

:

T~= '~t~ x } . In_~[Ii] , Section 9 some l o w e r e s t i m a t e s of the constants ~ C A ( ~ )_ )~ p < oo were obtained (they show in particular that these constants grow at least as fast as ~c8 as ~ tends to infinity, aL being a number dependent on p ). The unique property of tile space ~A used in [II] to prove these estimates was the fact that the constants ~ ( 0 A ) C ~ behave as described above. Hence the same estimates are valid for the spaces ~ A " To prove Theorem 3 we need a lemma which is probably of independent interest also. Its proof will be postponed till the end of this section. LE~b~A 2. Let j~ be a nonnegative finite measure on the unit circle ~ . Then there exists a finite measure ~ on T

~<

with the following properties:

cI12II

an operator

~

(~) i ~ ~

II It

c being an absolute constant; (2) there exists so that G projects },P(~) onto ~ P ( ~ )

351 s u t neousl

for all

and iI01rp,

for

i ~ is an absolute constant). ~oreover, the orthogonal projection from L$(~) onto ~i(~) may be taken to coincide with ~I ],~(~) . RE,LARK. A weakened version of lemma 2 was used in [II] (though it was not stated explicitly there) to prove that wp for pb~ and ~ p ( C A ) ~ ( p - ~ ) -~ for p~< ~ . To be more precise, a projection of UP(~) onto ~P(~) for some ~ > / ~ was also constructed in [II] , but in contrast to len~na 2 this projection "depended on p " (i.e. it was defined by different formulae for different ~ s ; this formulae did not agree even on ~co(~) ). However the proof of this fact involved only estimates of the Riesz projection ~+ in [,P , and to prove Lennna 2 analogous estimates in weighted spaces ~P(G~) are needed. If we argue similarly to the proof of Theorem 3 below but use only this weakened version of Lemma 2 we shall be able to prove a proposition differing from Theorem 3 only by a weaker estimate of ~p (~_A_) for ~ < ~ ~< (namely, ~p ( & ) ~ ~ (p-~)-3/~ instead of ~p ( ~ & ) ~<~(~_~)~ ). But to establish the "right" estimate Len~ma 2 in its full strength seems to be inavoidable. PROOF OF THEOREM 3. First of all note that the norm of the natural (i.e. translation-invariant) projection ~ of LP onto the space ¢~oshp % ~ & ~ b [ ~ : ~ j ~ } is less than or equal to

~p(CA)

c ( p - # ) -~t~ pends on ~

if

4
and cp 4/~

if p~z

( c de-

only), cf. [15] ,vol. 1, Ch.5, #8 . Evidently, the same numbers are metjorants :for the norms I/p ( ~ ~ • To begin with let us prove the lower estimates of the numbers ~p (~j~) . Let X p be the closure of the set ~jk in hP and let ~ be the formally identical operator from ~Jk to Xp (i.e. ~__ i p ~ l ~ l k ). Then ~ p ( ~ ) ~ . On the other hand it follows from the results of the paper [I23 that ~p (~) is equal to the norm of the translation invariant projection from ~P onto X p , that is to the number

U P++ -P IILP---~~P

• Hence

"~,p(&) ~ [I [])+llLp_>.~P- lip ]ILp_,_LP ,

and to get the desired lower bound for ~Pl ('I~ ~JM) it is sufficient to apply the above remark about IILp.__>.Lp and the known estimates from below for II ~+[I[,P__~[P

, of. [I3] ,

.

We pass to the upper estimates. Let ~

4
be a Banach space,

-~- ~

. Set

352

~='V'ICA. anf let~,~M(V) be

a

Grothendieck-Pietsch

measu-

for ~ ~]#Jl= ~ • Then the operator ~ admits a factorization ~ ~o(~ I~) with ~ ( ~ ) ~ y)~ II~II ~ ~ . Applying #~mma # to the measure ~ we get a measure ~ and an operator ~ with the properties listed in this lemma, and this enables us to factor the operator ~ in the following manner: re

here ~ and ~ are formal identity operators. Since the space ~oo(~) is metrically injective we can extend the operator to an operator ~ from ~ toL°°(~) withII~111 = II~II Let ~ - - - P~QIp, X]I . Then W E ~ ( ~ i i ~ y) is clearly an extension of ~ satisfying ~p ( W ) ~ <

and

~IIQp*p(~p,~)~llQIpllAu%IQIp d/PIl~"l "-< % C Vp ~O~llp is the constant from Semm~ ~). ~ince~,V-W) lq = @ (, we o hav~ V - I ¢ = ::D g with II ~ II -< k ( l l V - kT II)

( L depending on _A on~y). Consider first the case it) ~/ ~ . Then tp ( ~ - W )

'~(4+c'/P + ~,p ( V - k T ) If

IIQIIp)

~< @p,,

p~ ~

,

hence

~ ( , V ) ~ '~p(W)+

and we ,~re done.

we have to consider one more factorization of

with ~ being the formal identity operator. The crucial point is that ~ and P stand here for the same operators as in (I) and~moreover, P ~ Q T p ~ ~ ~ Q I ~ . Now the definition of the operator ~A~

(2) give ~< ~ ~

together with the factorization

thatll-Wl~IIPllli~Z llllOII~IIZA,~II7~ I~ an absolute constant. Of course the inequality

~p(W)~ d Ip IIQII~< ~d/~ (P-I)-~

obtained from the factorization (T) is also true. Now we have: ~p ( ~ )

•

353

PROOF OF LEI~&A 2. Let ~---~ 0 ~ + ~ , where ~ is singular with respect to ~ and ~4(~) . Without loss of generality we can assume that ~(~) 7/ ~ for some positive ~ . Since HP(~) = ~ P ( ~ ) ~ ~PC~) ( the ~ -direct sum is meant, cf. e.g. EII~ , p. I4) it is sufficient to treat the case ~--- G~t and to find for such a ~ a measure having the properties listed in lemma 2 and an additional property of being absolutely continuous with respect to the Lebesgue measure ~ . In the case j~ ~ ~ the operator Q will be defined in terms of the Riesz projection ~+ considered as an operator in a weighted space ~P(~ ~) . For conditions on 60 ensuring that ~÷ acts in such a space see EI7~ , EI8] , EIg~ For us the following special case of results Qbtained in EI7~ ,

bq

E

Let G0 be a nonnegative summable function on the circle. Then ~0 is said to satisfy the ~uckenhoupt condition (A~) if there exists a number O so that

( O0~

is the Hardy-Littlewood maximal function:

l THEORE~ ON WEIGHTED SPACES. if a function ~0 satisfies the condition (A~) the.~ the operator ~+ acts from ~P(~0~) to ~P(00~) if ~ < p < co and for ~ < p ~ < ~ we have the inequalityll ~+llp~0 ~ ~

04 (p_~)-~

with a constant

~

depending only on C from the condition ( A ~ ) • REd,ARK I. Let ~ > ~ ~ ~.~ ~ > 0 . Then it is well-known that there is a function ~ satisfying the conditions construction: co ~

2

~

I1 t1 .< set

~e=

II ~,

where ~

and ~.~

=

~

for

(a ~0

and

is a small positive number).

RENARK 2. If a function CO satisfies the condition (A~) and O < o~ < ~ then the function OO~ also satisfies the condition (Aw) with the same constant O :

354

T Now we continue the proof of 1,emma 2. Given a measure ~ , ~ ~4 as above, find a function ~ so that ~>/~,~ q/~ satisfies the condition (AT) with a constant independent on and

I~ ~

~ ~ ~ @~

,

~

also being an abso-

lute constant (this is possible in view of Remark I: s e t ~ C 5 ~/~, then S~[,~ ; starting by this S construct a G as in Remark I and set ~ ~-- G ~ ) . Let F be an outer function with ~ I = ~_ ~q/~ (such a t~ exists, for ~(~) ~ ~ ). Define an operator G by the formula ~ ~- i~M ~+ ( ~ # ) . This operator is evidently an orthogonal projection of the space I,~(61"~) onto ~(~4) since the mapping ~ ~-~ ~ is a unitary operator of ~ ( ~ ) onto [~ and it maps

IIQIIp, = Ilqll ,

~(#et~)_.4 onto ~$ .Hence p-t+oL--= ~ and it is sufficient llOllp~

for

~<~<

~

if

to estimate the number only. But for such a ~ we have

Since O ~ 4 - p / ~ / ~ i f 4
l lP d .)

@

3. Almost Euclidean subspaces and some finite dimensional examples. In this section we restrict ourselves to spaces over the reals, but the results are valid for complex spaces as well and only minor changes of the proofs are needed. We begin with the

355 statement of the main result of this section. THEOREM 4. There are three constants A4 , A~ and A S with the following properties. For each positive integer P~ there exist an ~-dimensional Banach space X ~ and an operator T ~ from ~ to ~ so that: (6) if ~ is an arbitrary operator from X ~ to ~ then ~K4 (~)~ A4 II~II ; (~) 5~p ~ 0 ( ~ ) ~ eo and # ~ 6 1 1 >/AS iI~jI for

every ~

in (T~(X~))* ~ (~u) "I)t (T~)~ A S - ~ •

REMARK. The second inequality in (6) means that the operators ~ are "uniformly with respect to ~b onto their images". So they are in a sense analogous to the operator ~ of the preceding section. Note also that the inequality (O) is the best (or rather the worst) possible, for if ~ is an operator of rank k~ then ~I ( ~ ) ~ ~ ~ (V) ~<~ ~o (V) (to prove this set ~l~bV , write ~ 4 ( V ) ~ q u ~ V ) ~ ( ~ y ) and use the fact that ~U¢($~Z ) = ~ provided ~ { # # ~ = k~ , cf. [3], p,286, Theorem 20.2.4 and p.385, Theorem 28.2.4). We need some facts concerning "almost Euclidean" subspaces of finite dimensional Banach spaces. Let ( E ~ l.I) be an ~ dimensional Euclidean space and let II°l[ be a norm on satisfying the inequality II~Cll<~ 13cI~ 0 ~ . Set ,

and

suppose that { 4~eeB4~ / ~ < 1] ved in

also

[20~

. The following result is pro-

, though it is not stated there explicitly

(cf.

[21] ).

THEOREM ON ALMOST EUCLIDEAN SUBSPACES. Let ~ ~ ~ < F~ Then there are two orthogonal (in the Euclidean space(E ~ I'l~ )

subspaces ~ 4 ' ~

of the space

~_

so that

~L~t~--k)~,m,(~t:

=ru-k. and f~r 0~ in ~ U C~, the following imequ~lity holds: II Here C ( ' , ") is a function of two real variables which does not de#mud on the triple ( E ~ I'I~ If'If) and is bounded on each of the sets { ( ~ ) : ~ < ~

(B I ; Let 0 < p < co . set ~ P = LP(~) where ~ is a measure on the set ~ 4 ~ ° . o ~ ~ ] defined by the formula LE~iA 3. Take in and choose the norm Then

l"Ithelsituati°n described aboveE : h ~ to be the norm of the space

~4

.

356

( B4 JI'U

and ~ are as above the balls corresponding to the norms and l'j )" PROOF. This fact follows easily from the known formulae for the volumes of the balls ~i and ~ e . See e.g. E20] . • Th s l e n a

shows that for the triples

L.,

IIIio ,,, II

the constant in the inequality from Theorem on almostEuclidean subspaces do~s not grow as k~ tends to infinity provided the ratio ~ is bounded away from 0 and from I . The next lemma is crucial for the proof of Theorem 4. It shows that the same is true for appropiate quotient spaces ~ / ~ with the induced norms. This fact seems to be of independent interest. Let 0 < ~ < ~ and let ~ be an ~-dimensional subspace of the space ~ with ~nt ~< ~ . Set ~ - - - ~ % / ~ and let I" j and Jl"Jl norms of the spaces ~¢

be the norms on ~ induced by the and ~ Set as above ~I =

~ - [ g c ~ E : li~II~<~ , ~ - - { ~ E ; 13oI~
,

where

~ depends on the constant ~ only. PROOF. Let ~ be the orthogonal complement of the space in ~ a~d let P be the orthogonal projection onto the subspace ~ . If we identify E and ~ in a natural way then the ball B ~ will "coincide" with the set U = { ~ : I 0CI~ , ~< ~ } and the ball ~I will "coincide" with the set ~ V = ~ ( { ~ E ~ : JIOCII1 ,,~<~}) • For every subset A of the set ~,...~ ~ deflne WA=~OCC~.JIDCIII u~l , ~ ( ~ ) = 0 for ~ ~ A } . We shall show that ,~.r-

Let us prove first that once the formula (I) is established, the lemma easily follows. Let vo~ be the ( ~ - ~ ) -dimensional volume defined by the Euclidean structure of the space ~,~ Then we have

357

where B = [//,~,..., I%-I~i}

. Consequently

4

~

Set U R = f0~"

I1~11~,~-~

, ~C(~) = 0

for ~B } . Then is an arbitrary vector satisfying then O 6 ~ W B if and only if

OC 96(~)= 0

for

4

OC4B ~-~

~-~

and

X~

U~

if and only if

Now Lemma 3 implies that

(~W~-~ - -

# C~W~]._

=~-~/

-

+i~

~i~

-<-A-~t--m-/'~'
and hence

#

B~

~-~ ( ~-~ ~ ~1~

We pass now to the proof of the formula4 (1). L e t { ~ } ~ < ~ < ~ be the unit vector basis of the space ~ :e~(~)~ 0 for ~=~ ~" , ~(_ ~') ~- ~ . For every O~ satisfyingiI~..,~.,~ there exist numbers ~ , ~:-~+ ~ so that OC lie~in the convex hull of the vecto~s~4~1~ 8 1 ~ . . . ~ 8 ~ C ~ @ • Let Since the range of ~ is ~ ) -dimensional, th~odo;y theorem (cf. E22] ) implies that we can choose ~ - ~ - ~ v e c t o r s ~ 1 ~ . . . ~ _ ~ + I from 7~... ~ 7~+4 so that POC----

358

~ow i f ~ :

o .

for some ~ uo

thenPz

o.ow

is c~early in o

.or

t.en

we can replace the above formula for ~ by another one a way that one of the vectors ~ in this new formula There are numbers ~i~,..~ ~ _ ~ + ~ at least one of which zero so that ~. ~ ; ~--- ~ . We may assume that .~

@

. Define ~a=~numb~r r

or4 + (£S~ ~" 0

by ~ :

5~

for all -@ ~ and set

~ ~:

~

0

~4 = O/x+ ~ 5 £

in such is ~ . is non~

?=4 # . Then

® and at least one of the numbers ~ is equal to zero. @ RE~iARK. Let ~ be an ~-dimensional subspace of the space ~ , I"14/11,~o~ ~ 0 ,~Since the intersection of ~ with the unit ball of the space ~ is contained in the orthogonal projection of this ball onto ~ , the same proof shows that

~C{ ~':

I1~ Ita>.~-<~ } } )

~" '

with

~o~ depending on o~ only. PROOF OF THEOREM 4. Theorem on almost Euclidean subspaces and lemma 3 imply that the space L ~ contains two subspaces ~ and ~k~ with the following properties: ( ~ ) d ~ k ~ = [ ~ 1 (~) H ~ = ~ ~ (~) the inequality

holds for every ~ in 6 ~ V Hpv with a constant C independent onto the of /I, and <3£ . Let ~ be the quotient map of L~~ space ~ /~ and let llI"Ill be the norm in this quotient space induced by the norm II'll4~ . using again Theorem on almost Euclidean subspaces and Lemma 4 we can find a subspace ~6 of the space ~ so that ~ t ~---~-[~] andll~ll~< ~iIII~(~)IIl for all 00 in ~ ( ~ being independent of k~ and 06 ). Let ~ k $ = G ~ _ ~ ( ~ 0 ~) and let ~ be the orthogonal projection from ~ onto ~ . Then for each ~ in ~ we have

359

Thus if we replace the L~ norm in E ~ by the L~ norm the projections P ~ remain bounded uniformly in ~ . From this it follows easily that the norms II'II~ and II'II~ are equivalent on ~ uniformly with respect to ~ , since they evidently are on G ~ and ~ separately. Using the log-convexity of ~ P norms we ~et the following: if ~ E E ~

then~OCll~< ~I 06114/%~ ~

~

does

(3)

not depend on ~ and ~l Now we are in a position to define the spaces X~ we want to construct. Let X~ be equal to E ~ as a set and endow it with co ~he ~ -norm Olearly ~ ×~= ~ . ~he inequality~(~)-
o is contained

(as a s e t )

in ~

and t h e r e f o r e

for

every ~

X~

the inequality (2) holds (and ~ in this inequality does not depend on 1!, ). Let ~ be the formal identity operator from X~ to E ~ (the latter being supplied with the norm II'II~~ ) and set ~-" T~=~, ~ . It follows from (3) that the prooections ~6 ar e ~fo~y~bounded by the constant C% in ~/%(~) -norms, so SU~/%(~) <~ O~ and hence ~lJ~p Suo(T~ ) ~ co " . To prove the inequalityll~ll~ A~ II~II ~ , ~E~-~- (T~(X~))* note that it is equivalent to the following one: -i

±

II + v But this inequality does hold for some A S independent of l since ~J_ X14, (in the space ~ ), s p a n ( ~ U ~ ) = ~-~-m 14, and for ~ in ~ the estimate (2) holds. It remains to show that ~ ( T ~ ) 7 / A 3 ~ . Let be the formal identity operator from ~ to ~ ( ~ is supplied as above with the ~(~)-norm). Then ilI~l14 " ~ and

and we are done. I COROLLARY. Let X =

(X~(~ X~ ~

...)co

. Then ~

do-

360

es n o t have t h e property (Kp) but flt(X*6~) ~_$~(X* ~). PROOF. The first assertion is evident. The second one is true in virtue of Remarque 8 in [~ , since the inequality (2) V~ L holds for all ~ in A ~ uniformly in ~ . This corollary shows that the scheme of proving the Grothendick equality for a space ~ * described in the introduction does not work in general. Note that it is possible to construct an example demonstrating this using only Lemma 3 and avoiding Lemma 4. ~ K . Let ~ be the standard norm in the ideal of L 4 factorable operators. Then ~I(T£) ~ C ~ (T~) , C being independent of ~ .(For if T ~ =

A

(hkA),

(B) FIIBII

then

dll A II

;

but

in iew of rot endieckt eorem

in view of the results

proved in

[4]

, [5]

) . Hence

in [23] a sequence of finite rank operators with analogous properties was constructed, but the operators ~ have an advantage of being "uniformly ont____ 9 their images". 4. 09erat0rs with an arbitrary range. Up to this section we discussed the problem of nuclearity of 0 -absolutely summing operators with values in a Hilbert space, but the analogous problem for operators with an arbitrary range might also be of interest. The next theorem answers a ques/ tion posed by A.Pe~czynski. THEORE~I 5. For every compact topological space ~ and every Banach space Z the following inclusion holds:

~e(C(~)~Z)c

c PROOF. We need a l e n a which is probably of independent interest. LE~Z~ 5- Let Z be a Banach space, ~ L a finite measure and X a subspace of ~4(~) . Denote by J ~ the closure of X J in the space h'(#) ( 0 < % < ~ ) and let ~ be the formal identity operator from hi(B) to ~ % ( ~ ) . If W is any continuous linear operator from y to Z then the eperat@r W o ( % I X ) is weakly compact. Theorem 5 follows easily from this lemma. Indeed, let V ~ ~H%(C(~)~Z) . Then faotorization theorem (see the introduction) shows that ~ ve diagram:

may be included into the following commutati-

361

E

e X

L,#t

#l×

y

v cd)

....

m ----. Z

Here ~ is a finite ~.easure, ~ and W are some continuous linear operators, E is a subspace of L°°(~) , X = c~O~L~(/~}E , i . the formal identity operator frgm to ~($~) . By Lemma 5 the operator ~ o (~I X) is weakly pact. Since the °perat°r (14'~ I ~)°~)C(~ is 1-absolutely summing and its domain is the space , this operator is lintegral (cf. E3] , p.233, Proposition 17.3.5). It remains to use the fact that if ~ is a weakly compact operator, B ~ I I and the composition A B is well-defined then A B is l-nuclear (cf. [3] , p.345, Theorem 24.6.2). • PROOF OF LE~L~ 5. Denote the composition ~ o (#I X) by T . Let ~ O ~ } be an arbitrary bounded sequence in X . We shall show that the sequence ~ T ~ } has a weak accumulation point. First of all we claim that there is a subsequence [ ~ } of the sequence ~T3cs.} accompanied by a sequence ~ek} of mutually disjoint ~-measurable sets so that the set { ~ ( ~ 6 ~ ) : ~=~... } is relatively weakly compact in~4(~) . To verify this claim one needs to repeat a part of the proof of Theorem 5 in [24J . Here is a sketch of the argument. Set ~ ( 6 ) = =St~II0C~l~:~e~<6~=~,~.,}and

let ~ = ~ ~(~) If ~ = 0 &-~O there is nothing to prove. If ~ 0 choose a sequence of positive integers ~ } and a sequence of sets ~e~} so that W < ~ < <~<'"' ~k ~ 0 and~ I~nk I ~ ~ f . S e t t i n g ~ = 06~ ek we get all we need except may be the disjointness of the sequence ~ C ~ , and so on. Now let tO~------~ ~ , , Ir~= ~ ( ~ - ~ 5 ~ ) . We shall assume (as we may) that ~ - - - ~ weakly in ~ (~,) a n d ~ 4 ~6 ~<~• By separation theorem there exists a sequence i ~ } in ~ (~) so that ~---*- tr in norm and ~Sg6CO~If[~r~ : ~ ~} . For each write %~ as E ~ g 1 ~ with {o6~} a finite sequence,

~>z0 ,~k=

4

and set 9J~-----~o6~~

. Then~(6~ipp ~f~)--~

362

and

.

and since ~ + ~ / ~

that T ~

)

X ~(iir)

Consequently¢llll=0,

it follows that } ~ weakly.

~

. Let us show

Note that tl~e above argument is applicable to any subsequence [ %r~ of the sequence [~f~} . It follows that for every such subsequence

Suppose that Then F ( T ~ ,

~(T

~)

/ >

~'l(W(o~'t)') )

) ----~ O~=~P(W(~r))

indices. Hence for

~t

for some

for some sequence

~

in

{~,}

y-J(" . of

sufficiently large we have

But this contradicts the formula (I). • It is natural to ask now whether Theorem I stated in the introducti6n and proved in Section I remains valid for operators from the disc algebra to some non-Hilbert spaces E . The answer is very likely to be positive (and, moreover, some parts of the proof may be repeated almost without changes for certain non-Hilbert spaces ~ ). However now when I am wwiting this I cannot formulate any satisfactory conjecture on this matter. The only thing I want to mention is that some facts concerning arbitrary spaces E can be derived from theorem I. Here is an example. THEORE~ 6. Let E be an infinite dimensional Banach space and let T ~ o ( C A ~ E ) . Then T ( C A ) = ~ = ~ FROOF. Suppose to the contrary that T ( C A ) = E . Since ~o c ~ , the operator T factors through a Hilbert space and hence ~ theorem I.

is also a •

Hilbert space. It remains to apply

References. f

f

I. A.G r o t h e n d i e c k. Resume de la theorie metrique des produits tensorie]stopologiques. Bol. S o c . ~ t . SHo Paulo, 1956, 8, 1-79. ! 2. J.L i n d e n s t r a u s s, A.P e ~ c z y n s k i. Absolutely summing operators in ~ p -spaces and their applications. Studia Math., 1968, 29, 275-326. 3. A.P i e t s c h. Operator ideals. Berlin, 1978. 4. C.B.K H c x ~ E o B. 0 HpOCTpSHCTBaX C "t~J~Et~" aH~yJ~TOpOm.

363

8an. ~ay~H. c e ~ H .

~0~,

1976, 65, I92-I95.

5. G.P i s i e r. Une nouvelle fiant le theoreme

t de Banach veri-

classe d'espaces

de Grothendieck.

Ann. Inst. Fourier,

1978,

28,

N 1, 69-90. 6. H.K.H H E o ~ ~ c E H ~, B.H.X a B H H, C.B.X p y m e" B

(COCTaB~TeZa ~ p e ~ T 0 p ~ ) . 99 sepem~Hsax sa~a~ J m H e ~ o r o EoMn~eEcsoro a s a ~ s a . San. Hayes. c e ~ H . 20~g, I978, 81. 7. Y.G o r d o n, D.R.L e w i s, J.R.R e t h e r f o r d. Banach ideals of operators with applications. J.Funct.Anal., 1973, 14, 85-129. 8. B.N a u r e y. Theoremes lineaires 11.

a valeurs

de factorisation

pour les op~rateurs

dans les espaces L P . Asterisque,

/ / 9, S.K w a p i e n, A.P e ~ c z y n s k i° Remarks summing translation

inva~iant

ra and its dual. Michigan

operators

J.Math.,

on absolutely

from the disc algeb-

1978,

25,

173-181.

10.S.K w a p i e ~, A.P e & c z y ~ s k i. Absolutely operators compact wa,

and translation

abelian

groups.

invariant

spaces

Inst.Mat.PAN,

1974,

summing

of functions

preprint

on

N 162, Warsza-

1978.

/ 11.A.P e ~ c z y n s k i. Banach spaces and absolutely

summing operators.

of analytic

AMS Regional

ries in Nathematics, 30, Providence, 1977. / 12.A.P e ~ c z y n s k i. p - i n t e g r a l operators

functions

Conference

Se-

commuting with

group representations

and examples

of quasi

p -integral

rators which are not

p-integral.

Studia Y~th.,

1969,

ope-

33,

63-7O. 13.P.L.D u r e n. Theory of H P and London,

spaces.

Academic

Press,

N e w ~ork

1970.

I¢. C.B.E ~ c ~ ~ E o B. 0 p e ~ e ~ c m s ~ x ~o~HpooTpaHOTBaX npocTpa~. CTBa CA . ~yHE~. a~aJm~ ~ ero npH~o~., I979, I3, ~ I,

21-30. 15.A.Z y g m u n d .

Trigonimetric

series.

Vol.

1,2. Cambridge,

1959.

/ 16.B.S.M i t j a g i n, A.P e T c z y n s k i. On the nonexistence of linear isomorphisms functions 1975,

between Banach

of one and several

spaces

complex variables.

of analytic Studia }~ath.,

56, 85-96.

17.B.M u c k e n h o u p t. Weighted norm inequalities

for Hardy

364

maximal function.

T r a n s . A m e r . ~ t h . Soc.,

1972,

165, 207-226.

1 8 . R . A . H u n t, B . M u c k e n h o u p t, R . L . W h e e d e n. Weighted norm inequalities Hilbert

transform.

for the c o n j u g a t e

Trans.Amer.Nath.Soc.,

f u n c t i o n and

1973,

176,

227-251.

1 9 . R . R . C o i f m a n, C . F e f f e r m a n. W e i g h t e d n o r m i n e q u a l i t i e s for m a x i m a l f u n c t i o n s a n d s i n g u l a r i n t e g r a l s . ~ath.,

1974,

51, N 3, 241-250.

2 0 . S . S z a r e k. On K a s h i n ' s a l m o s t

euclidean orthogonal

p o s i t i o n of

. B u l l . A c a d . P o l o n . Sci.,

et phys.,

26 N 8, 691-694.

21.S.S

Studia

1978,

ser. sci.math.,

decoma~tr.

z a r e k, N . T o m c z a k - J a e g e r m a n n. On n e a r -

ly e u c l i d e a n d e c o m p o s i t i o n s Compos.r~th.,

1980,

for some c l a s s e s of B a n a c h spaces.

40, N 3, 367-385.

2 2 . L . D a n z e r, B.G r u n b a u m, V . K 1 e e. H e l l y ' s t h e o r e m and its r e l a t i v e s . Amer.N~th.Soc.,

"Convexity",

P r o c . s y m p o s . p u r e math.,

vol.7,

Providence,

1963. z f 2 3 . T . F i g i e l, S . K w a p i e n, A . P e ~ c z y n s k i. S h a r p e s t i m a t e s f o r the c o n s t a n t s of N i n k o w s k i astr.

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et phys.,

of l o c a l u n c o n d i t i o n a l

B u l l . A c a d . P o l o n . Sci.,

ser.

25, N 11, 1221-1226. r 2 4 . M . I . K a d e c, A . P e ~ c z y n s k i. Bases,

1861/62,

sci math.,

1977,

ces a n d c o m p l e m e n t e d s u b s p a c e s Math.,

structure

21,

161-176.

in the s p a c e s

lacunary sequenLv

. Studia

N,G.~akarov,

V.I,Vasjunin

A MODEL FOR NONCONTRACTIONS AND STABILITY OF THE CONTINUOUS SPECTRUM

O° Introduction~ I. The model of SzBkefalvi-Nagy and Foia~. 2. Model representation of an arbitrary bounded operator, Spectrum and resolvent. 3. Special choice of an auxiliary contraction. 4. Factorlzation of characteristic function. 5. The absolutely continuous subspace. 6, Scalar multiples. 7, The stability of the continuous spectrum°

366 O. Introduction.

B y t h e c o n t i n u o u s s p e ~ e t r u m *) of an operator ~ we mean the difference ~ C ( ~ ) ~ ~L) k~o, where ~o is the set of all isolated points of ~ ~ ) for which, the corresponding Riesz projections are of finite rank. Below, we consider only the operators with the continuous spectperturbation of ~ by some compact operator ~ , the continuous spectrum will either remain unchanged or fill the whole disc; the latter is possible, of course, only if ~ C IL) ~ - T (see

Lemma 5.2 [8] ; a t~pical example: i f

L#

d--~e#z#, ~ ~ ~Z(T~ ,

each ~ , I~1~ i , and gC( L+K)-{- S - S ~ , first case we say that ~ c ~ L ~ ) i s s t a b 1 d

ISI --~ ~} ).

In the under the p e r -

t U r b a t i o n ~ . When is ~ C ~ ) stable under any perturbation of rank one T under any perturbation of an arbitrary but fixed crossnormed Schatten's ideal F .7 (We denote below by T ~ the standard Schatten's ideals (cf. [8~ ); e.g. ~ i s t ~ e trace class ideal, ~ o o is the ideal of all compact operators). The answers to these questions are determined by the "thinness" of the spectrum ~ (~) , by the possibility to "draw" the resolvent's pole from the infinity inside the disc. The character of an appropriate "thinness" depends both on the *) We are not sure to have chosen an apt abjective for the introduced spectrum. "Essential" may fit better, the more so it would directly generalize the Weyl's definition [7] to arbitrary operators. However, some other set is meant by "essential spectrum" in many papers; we do not argue with the generally accepted terminology.

367

nature of the operator L

itself

and on the class of perturba-

tions. It should be noted that the stability of the continuous spectrum under finite rank perturbations is known in Weinstein-Aronszain theory L19~

(which gives explicit formulae of

, [201

the perturbation theory for the point spectrum) as "the regular case" (see L21~, Ch. I Y , ~ 6 ) .

It is important to distinguish the

regular case from the singular one and the main result of Section 7 can be considered just as a criterion of nonsingularity in terms of the corresponding characteristic function and of the spectral measure.

The principal tool to derive this criterion is

a functional model for noncontractions. The functional model by B.Sz.-Na~y and C.Foia~ plays an important role in Operator Theory.

The success of the model theory

makes it highly desirable to adjust the model not to contractions only - as is the case with its classical version - but to wider operator classes defined by some kind of closeness to unitary operators. Various approaches to the construction of such models are known.

Ch. Davis and C.Foia~ EI~ generalized the classical const-

ruction [21 , using the rator

L

~-unitary

dilation of an arbitrary ope-

. On account of geometrical

of spaces with indefinite metrics,

complications in the study

severe restrictions on the

characteristic function were to be imposed. Without such restrictions, even in the case of finite defects, usage of the

difficulties in the

J -unitary dilation become considerable.

There

proved to be more constructive

the S.N.Naboko's approach, used

in the study of nondissipative

operators

[3~ •

The method of S.N.Naboko consists in the construction, a given operator T T

L

for

, of an auxilliary dissipative operator

, and in the study of

~

in the Sz.-Nagy - Foia~ model of

• It should be mentioned that the idea to use the model of

an auxilliary operator in the study of an operator, which does not admit such a model, for example, in refs.[5]

can be found in other works as well.Thus, , [61

, [4]

the unitary operators,

which arise as perturbations of model constructions,

are studied.

The purpose of the present paper is to apply the S.N.Naboko's method for the generalization of the Sz.-Nagy - Foia~ model to the case of "nearly unitary"

operators.

The direct conformal

transplantation of the Naboko's model from the halfplane into the unit disc meets numerous technical difficulties. prefer not to use the results of ref.

That is why we

[3~ , but we frequently

368

explore its ideas in our parallel construction. We also display an application of the obtained model to an operator problem concerning the behaviourof the continuous spectrum under some class of perturbations. Let ~ be an arbitrsmy operator in a }Lilbert space H We represent the operator ~ as a special perturbation of some auxiliary completely nonunitary (c.n.) contraction T , "distorting" the action of T only on its defect subspace

L=T

* 2 Ao2

(o.I

Here ~ " E - ; ~ and ~ " ~ ~ ~ are the isometrical injections of Hilbert spaces ~ and ~ w onto the defect subspaces ~T and ~ T ~ of the operator T (for the definition of subspaces ~ T a n d ~ T ~ , see section I), and the bounded operator A 0 acts from ~ into [ , . For a given operator ~ there are many ways of choosing its representation in the from (0.1). In contrast to SoN.Naboko, we don't fix any definite way but reserve the right to choose those T smd A o which are the most convenient in this and that concrete situation. Next we replace the c.n. contraction T by its Sz.-Nsgy - Foia~ model. From now on, M is the model space, H is contained in ~ , the space of the minims~l unitary dilation ~ of the operator ~ , and

T=PHz H ( PHis the orthogonal projection of ~ onto ~ ). In the model representation of T , we study the operator ~ , thus obtaining its functional model. This model allows to reduce the questions concerning the spectral structure of L to the study of some operator-valued analytic functions. The latter turn out to be closely connected with ~ L - the characteristic function of L : they participate in a factorization of 0 L " In texans of these operator-valued functions we describe the spectrum of L and write out formulae for its resolvent. It is clear that the less the operator ~ s/~d the auxiliary contraction T differ one from another, the better the constructed model "works". In the process of our study we impose some restrictions on the connection between t and ~ . One of them requires the existence of a bounded operator A satisfying the condition

369 A o ~-A.O

A

where ~o=(~--0"(0) 0(0)) 1/% ~ = 0 T - the c h a r a c t e r i s t i c

A o ~

,

A#o:(~-O(O) f u n c t i o n of T

(0.2)

O*(O)) ~/2 ,

It turns out that any operator L can be represented in the form (0.1) with (0.2) satisfied, moreover, for t h e c o mplot ely nonunitary operators (i.e. for the operators unitary on no reducing subspace), we present two canonical ways to obtain such a representation. The restriction (0.2) not only allows to avoid some technical difficulties but also gives a possibility to study more profoundly the spectral structure of m . In particular under the condition (0.2), we define ~ - the absolutely continuous~subspace,_Hand N S - the singular subspace of the operator ~ ; ~ , ~ S C ~ ; we indicate the unitary operator Z ~ ( ~ is a subspace of invariant under Z ), whose quasi-affine transform is the restriction L I ~ • As usual, the application of the model is most effective in the case where analytic operator-functions admit scalar multiples. Certainly, it takes place now not for any operator L , but only for those sufficiently "near" to unitary operators. For example, this is the case, if the operator ! - ~ ~ ~ is of the trace class, and the spectrum ~ ( ~ ) does not fill the unit disc D

. In the "scalar-multiple" case we study the condition for the spectral measure of the unitary operator Z ~ to vanish on some set ~ of positive Lebesgue measure (in the following, we denote this measure by l'I ). It occurs, that this condition can be expressed in terms of boundary values of the characteristic function ~ ~ , and of the spectral measure of the unitary part of m . we apply the obtained criterion to the study of the stability of the continuous spectrum. The last part of the paper is devoted to this problem. Few facts about the stability of the continuous spectrum are known. The bilateral shift provides the simplest example of an operator with the continuous spectrum unstable under rank one perturbations ( ~9~ , problem 144). ~oreover, N. K. Nikol' skii showed [10] that, for~~the unitary operator ~ , ~C(~) is stable under perturbations of rank one (or of trace class - no difference) if and only if ~ does not contain the shift. The problem concerning ~c~-stability was completely solved in ref.

370

[I I] : in our situation ( L

is an arbitrary operator with

In the present paper we study the stability of the continuous spectrum of the operators, which admit the representation (0.1) with the aforementioned analytic operator-valued functions having scalar multiples. For such operators, the rank one stability is equivalent to the ~! -stability, and the corresponding criterion consists in the vanishing of the spectral measure of the unitary operator Z I N on some set ~ ,I~0 . This criterion can be formulated in terms of the invariant subspaces and in terms of operator algebras as well. Finally, we mention one very special case of the above assertion. Let ~ be the bilateral shift presented as the multiplication by Z on the space ---~I~~ . Let K ~ ( ' ~ ) ~ be an arbitrary rank one operator, ~ , ~ ~ ~V) . Then there exists an operator of rank one- ~ ,u ~----Io~ ~i~) ~ , such that ~ ( S + ~ + K~) ~ ~ . The latter is equivalent to the identit~y ~ + ( ~ - - _ ~ ) - ~ (~+~4)~ ~ 0 , ~ _~ ~ _, or

(the integration is performed with respect to the normalized Lebesgue measure on the circle T ). we would like to know how to prove the solvability (relative to ~I and ~ ) of the last equation for any ~ and I . Even in this simple case the application of the model preserves all its main difficulties and the outward complicacy, but it is only the model that makes it possible for the authors to solve the problem under consideration. We are grateful to N.K.Nikol'skii and S.N.Naboko for helpful discussions and reading the manuscript. I. The model of Sz~kefa!vi-Na~y an d Foia~ We begin by briefly recalling some of the well-known facts about the model for contractions (i.e., for linear operators T satisfying IIT II ~ ~) of a Hilbert space. Our approach is slightly different from the traditional one. We practically remove all references to the functional aspect of the model, using only the geometry of the unitary dilation space. Our point of view is close, in this sense, to the R.G.Douglas exposition in the survey ~12]. For the reader's convenience a "correspondence table "of the main initial objects and formulae of the model is provided: on the left-hand side of the pa~e we write them down in the symmetric form due to B.S.Pavlov ~13] , on the right-hand side - in

371 the original form due to B.Sz.-Nagy and C.Foia~ [2]. It should be noted at once that operators in the model notation are often determined only on some dense subset, but we consider them being boundedly extended to all of the Hilbert space, if possible. We now proceed to set forth some standard rules for notation. (X,Y) is_the space of all bounded linear operators acting

~ro,~ X

i~to /

space of X complex plane

C

, ;L ~,X) ~

$-(X,X> H ~ ( x )

~s th~ ~ d y

-valued functions on the unit disc (see [14] ; [ 2 ] , Y, §1) ~m(X)

0

of the

is

~bedded

in a natural way into the X -valued Lebesgue space L P ( X ) defined on the circle ~ hA: denotes the operator of evaluation at ~: # ~-~{ (~) __ where ~ is the harmonic continuation of f , ~ ~ ID • We use the same letters to denote functions and operators of multiplication by these functions in the corresponding functional spaces, and also to denote constants and functions identically equal to these constants. The letter Z denotes the identity mapping on the circle T (and thus denotes the operator of multiplication by E ). Everywhere I is the identical operator. Let F and F ~ be separable Hilbert spaces; and let 0 , ee H = ( a ( E E.)) be a contraction-valued function satisfying

LZfX)---*X

IIeo e II~ Uell for each nonzero vector

e

in E

,

=(I-@*@) y' ,

00 de{ ~ (0)

. We set

&eL~(¢(E));

A,.=(I - @@*)Y"

7

g , ~ L°'(£ (E,)) '

Ao =(I -~o~oj

,

A.oO( -eoe )

,

Z(E) ;

Since is purely contractive, the kernels of the operators and h-~ o A 0 and /k~0 are trivial. Hence the operators h o i are defined correctly on the dense subsets m 0 F and A ~ 0 F ~ respectively.

c[os Z~L~(E)

372 is used to denote the orthogonal sum of the indicated Hilbert spaces. An element of this sum is represented by the column of two functions. To introduce the Hilbert space denoted below by

I take the space of pairs

~I.

'

1 p :

endow it with the seminorm

[P (I,) 1~ = ~ * ¢~*, P~E~ + c e~ ,~,, ~o L~~E~' take i t s quotient space w i t h respect to the kernel of t h i s seminorm, and, at last, complete the obtained prehilbert space. Now we are able to describe the model space. To do this, set the following notation.

Sz.-Nagy -Foias' form

Pavlov' s form

f C- (E.~ = \ cB~a L~(E))

~*=

¢o) i,

~-,.~=(0,I)

The operators ~b and 9i,. ( E ~ ) respectively into

~~..=

('o) ," ,

imbed isometrically ~ :

,..=

¢~.:. ~ -= O

,

~-~,* ~-,.. = O ~.

1f (~',.-~% @)# tt1{ ' = it m~: llL~{EJor

,

L$ (E) and

Observe also that

Since

we

each Zunotion

373

, the equality

determines an isometry

~'.

cBs AL&(E)

-~

={_~) E', ¢~'=(a,o)ll ~:=(o},i s*=(o,:r_). Similarly,

the equality

cE:,~ = ~,

-~

L~(E.)

q:.:dosA.

de t ermine s

0"

-

,

It is clear that

'I:*~'= A, 'I;*~T. =0 ,

¢ . *~ T = O , q : .*~ T . = A . .

Next we set

P÷

If we denote by to

H zand

and

P_

onto its orthogonal

onto the subspace

m

the orthogonal projections complement

~

tion

on-

then the projection of

"Ji,P_Si',

acts the unitary operator T : m --~ m

La

can be w r i t t e n down as

PH = I - ~ P + ~ * On

of

Z

.

. We c ~ l l t h e

c.n. contrac-

, defined by

TdeS pHzJ H , t h e

m o d e 1

m o d e 1

c o n t r a c t i o n

s p a c e

o f

o p e r a t o r

c o n t r a c t i o n is u n i t a r i l y equivalent The proof of this fact a n d

a n d we eall

an

T

H

the

. E v e r y Con.

to some model contraction.

explicit c o n s t r u c t i o n of the model

374 can be found in [2], ~2], D5]" The operator ~ on ~ is the minimal unitary dilation for the operator

T

. The geometry of the space of dilation is

completely ~etermined by the isometries tion

~.~

=~

~

and

~.

; the rela-

provides the translation into the analytic lan-

guage. Now we return to the study of the model

operator. It can

be expressed in the form

Its adjoint is

T ' = P. ~ I H--{~-X £ . g : i% ~:}1 H. We call D T and DT~ (I- T T

t he

d e f e c t

o p e r a t o r s, and we call the closures of

their images

~T

and

~T*

t h e

d e f e c t

s u b s p a ee~

It is easy to check that

LO~

~

where

uJ=Z Since

@ ,

w.=

~L a)*OJ = &o

and

OJ. CO. = • . o

ZL&~=CO

,

, the relations

19_.~.o =CO.

define the isometrical embeddings

n:E

,H,

onto the defect subspaoes

£L..E. ~T

The images of the operators

and

~T*

,H "

Cd, CO. ~ ~ , ~

lie in

H

, so

375 their adj oints Since

00" , O0~*

,

.D_* , .('L.*

vanish on

~

oH

•

ko rlH =0, we have

and

(1.3) The operators

T

and T *

can be expressed in the form

T-- {~ ( l:-n 0.*)-_0_,,eon" } I H T ~={z (I-~,n,)-nOo~,} I H.

(1.4)

Indeed, for example,

l~rom

we in~er that

m =E-(Z ~ +_(~@Q)~* , as claimed. The established expressions for T and T ~ separate the strictly contractive parts of the operators, which map one defect subspaoe into another, and its isometrical parts which are the unitary transformations between the orthogonal complements of ~ T and ~ T ~ " i n the sequel the followln~ facts about the operator [ L - A | ) will be of great utility, We always assume that k 6 ~ PROPOSITION I. I.

-

(I_),T*)-~ w = ( z - x ) -4

~ ~:)IH

[~-~,

e (X)] g .o .

(1.5)

(1.7)

376

~Roo~ t l - X-F*) Therefore

Since

~0¢~

(~)

for each

% e ~ , we have

whence

x:C~ k~-~(~-x

k'k,

*

and (1.5) is proved. The formulae (1.3) and (1.1) together with (1.5) readily imply (1.6) and (1.7). @ 1.2. The formula (1.6) yields the inclusion

H In the same fashion one can obtain the formulae

and the inclusion

Consider next an analytic operator-valued function which maps the unit disc into ~ (m, E ~ ) :

We have

O Oo+~.o ~ao :

(1.8)

which follows by computations:

The formula (1.8) provides the following immediate corollary. COROLLARY 1.3. For every contraction-valued analytic function ~ , the operators ~C~) map ~ o ~ into ~ o E . @

377 The next equalities can be easily deduced from (1.8) (1.9)

(1.10)

~*~£ =

aoCl-O

Oo')go,

{r~* z .0_ = a.oCk*.

(1.11)

(1.12)

o

By (1.8) we conclude that the function ~,o@~ko belongs to Ha°(~(E,E~)) . Furthermore, (1.9), (Io12) provides

~" ao e :;r,,~,E, H~ c E,)), wher~s the function itself does not necessarily belong even to the class H ~ ° We are closing this Section by some more simple formulae which will prove useful below:

T.D. =-..q, eo . T ..Q.~=--..Q 0,,. D T = _Q. 5.o_Q*.

D..r, = £--1, A ,o

.

2. Model representation . of an arbitrar~ bounded operator. Spectrum an d r esolvent

The purpose of this Section is to begin the study of an arbitrary operator L on the model space of an auxiliary con. contraction T , which is connected with L by the relation (0.1)

L =T+_O_,/k ofL*,

/~o ~

( E , E ,,')

(o.1)

Any operator can be represented in such a form, and in many fashions. The only restriction on the choice of an auxiliary contraction imposed by (0.1) is that the defect subspaces of T should £ contain those of L , and, in addition, the equality ~ I ~ T =

378

=TI¢~

~hou~d be

va~id.

Representation (0.1) for a given operator ~ being fixed,we theSz.-Nagy--~oia~ model of the contraction ~ to study the action of ~ on the model space. We consider ~ , ~ ~ ~ having the same meaning as that in Section I. Hence

(2.1) where ~ ~ k o- e o . Thus, we obtain t h e m o d e 1 r e pr e s ent at i on of the o p e r a t o r ~. As is mentioned in Introduction, the less is the difference between the operators ~ and T , the more the model is adjusted for the analysis of the operator ~ . However, no additional restrictions on the relation between ~ and T are still imposed in this Section. We study the spectrum ~C~) of the operator ~ in the general setting. Let ~ be an analytic ( ~ ~,) -valued function, defined by

LE~

2.1.

(2.2)

~ C~gZ*IH= ~ CI-~T~-'C~,I-L) PROOF. Using (1.12) and (1.10) we verify the first equality:

The second equality follows from the first one, (1.6) and (2.1):

-'

IH

=Ac~O31H. 2 . 2 . ~he number

~,

•

i s i n t~e point s p e c t ~

~p (L~

of the operator L if and only if ~ ~4 CA) # {0}. If this is the case, then the associated eigenspace is equal to

:-

379

E~

C~-X~-~ C~A-LL.B) K~#~ CX).

PROOF. Let

be an eigenvector of

0 (L-x!)x OLnce ~ 0

~*

,

~

:

(~-X)x-C~

OC ~ 0

*

. Let us verify that

Consequently, ~eZt ~ (X) ~ { O}

,

~ = C~- ~)-~ C~~ - ~ , B ) Q * ~

and

~ E ~.

To complete the proof it remains to show that

E~ c.~£~(L- &I) - Let e e ~ ~A ~--~ ( ~ - ~ ) - ~ ( ~ - ~ ) e

~* O~

~ C

, and

~ (~), . Then

~* ~ = (~-x) -~ ~.o ~,e ~ H ~ (E,), m++mx = C~-~) -+ ~++ CmQ-~l+ B ) e =

-~[moCI-

@o)-~o

B]e L H'cE).

x~+H,

Hence

cL-&I)m~= (~- &) ~x- Cz~ - Q . B ) Q * ~ x = = CzQ - Q . B ) C+-~l+ m~) = 0, because

= ~-++oCI-~) -< ~ o [ c I - + * 8o)-~o+ * B]e =

since

0(,0)

=

L ~ A 2.3.

O.

•

~CL)

invertibleo In this case

i f and only i f the operator

~(~)

is

380

PROOF. We first prove the "only if" assertion. Let the operator ¢I [k) be invertible. Denote by ~ the operator standing in the right-hand side of the equality in the statement. We claim that ~ is both right and left inverse for the operator [h-k~). For any 0c~ ~ , by Le~m~a 2.1 and (1.9), (1.11) we have

(2.3)

~-¢M~= [ d-x]Tl[~-~oCI- ~0o- ~'B)k*~~ C~T~. •m~o~X~:~] i H~cE) hence

M ~ C H

. Since

we have

M Ch- xl~ = I- [ c~-XY~C~D_-D_,5)D_* + C~-XT ~C ~ f l - i l , B ) ~C )

~o

On the other hand, (2.2) shows that

~YM = ao koag-¢M=-~

*IH;

whence

cL-~I) M

= I-c~Q-Q,

+ ¢~-n.B~

B)~c~T ~~-' ~x~-~ + cxf ~ ~-.~ok x ~ IH = t.

For the converse, suppose that

k ~ ~ (L)

and set

W~ cx~ n ~cxl-Ly~cI-~J*)~I~. By Lemma 2.1

=f~.cl-xT'T~cxl-t?txl-LT~cI-x By Lemma 2.2

~

~4CX)={0}

)il. = I

, and we infer from the equality

381

that

~ (k) @~ Ck~ = I

, oom~leting the ~roof. @

We i ~ o d u c e n e x t a n o t h e r t h r e e o p e r a t o r - v a l u e d holomorphio f u n c t i o n s t o d e s c r i b e the spectrum of the o p e r a t o r L outside the u n i t d i s c and the speot~lm and the r e s o l v e n t of the a d j o i n t operator U :

BE a c o m p u t a t i o n s i m i l a r t o t h a t i n Lemma 2 . 1 , equa]ities can be established

the following

_~ k~ ~* C~,_ ~D.B~ )

.

._q.~~. ck~= cx i - L) t I - k T *

T~D_

(2.4)

q)~ (x~ = Eo~k~* C~O--~,B) D_ ~ (x] : (I-x[~)(I- xT*)-~il

~,¢x) n** IH :n*~ CI-xT'? ~ cI-xL~). If

X % ~ (L)

then

~ c ~ -~- ~* cI- ~T*) cxl- L~-~~ ~. If

k -~

¢ ~" (. ~*)

then

n ~ cx~-4= (I -xT*) (I- xU)-4 I/

(2.5)

~ cx~-~ IH=~cl-x~? ~d-~T*).

(2.~)

We state without proof some more formulae involving the functions~

382

(2.7)

We summarize the established results zn

I.

THEOREM 2.4 • I) ~ke ~'p (.L')

( > '~eYt 1.~(~,') # { 0 } ,~

3) ~-~

CI-XL) = Cl-Xa)-~C~fi-fi,B)N~ ~ (x).

it. i) ~ ~ ~cL) ~ ~%CX) is invertible.

@~C~)

is.in~ertible

<

",

I n this case,

(~-~ b-~= d-~z)-'[~-cn,-zn B*)~ c~)-~~ ~*~31 H. 2)

~-4 ~ ~CL) < ;" ~3 (~1 ~

(k)

In this case,

is invertible.

is

invertible

383

(I-AL*)-' =(z-A)-' [ z - X ( ~ . - z g B")4.(A)-'E~'o h,,~T2]1H 3. Special ChQl!oe of an auxiliary contraction The work with an arbitrary model representation of an operator ~ can meet some t e c h ~ c a l difficulties. We have indirectly come across one: in general, the functions ~ need not be bounded. As is already mentioned, some kind of difficulties can be avoided by choosing an adequate auxiliary contraction in the representation (O.1). ~or example, the following restriction imposed on the representation (0. I) enables us to obtain bounded operator-functions: There exists an operator A ~ ( E , ~ ) 7 such that /~a0

AA0 =A0

(0.2)

Any bounded operator admits a representation satisfying this restriction. Moreover, for completely nonunitary operators we describe in this section two standard choices of an auxiliary contraction. How to deal with unitary operators will be shown in Section 5Now we introduce some more notations. If

IHlde~(H*M) ~

~hen

o~ M

, M = V M IMI

, DM =II-M*MIs

we oo=.id~r sL~nO=t +

X ~ de~ i

( I -+ J M )

~ ~ ~ (~i~)

,

- polar deoompositlon

, JM=sB~(I-M*M) , i.e. J M I K e ~ ( I - H ~ H ) - - I

.

0 , ~ d X ~ ~enot~s th~ orthogo~

projection onto Ke~ D M Till now the same letter " L "

served both for the operator

and for its - ~ t a r i l y equivalent model.For a while this ement will be broken.

agre-

Let M be a completely nonumitary operator in a separable Hilbert space. We wish to construct its representations in the model spaces of two preferable and

Ti

(in some sense) contractions

-

~P~SE~ATIO~

I~ ~

MOD~

To

se~

H0

H ~0

T0

384

~0 is a c.n. partial isometry. Let ~0 be its Sz.-Nagy - Foia~ model and let ~ be a unitary operator which provides an equivalence between ~0 and ~ 0 : ~0 = ~ * ~ o ~ . Thus, the operator

L ~ To+~,~ kop._', O

y i e l d s a model f o r the o p e r a t o r M i n the model space o f % . For t h i s r e p r e s e n t a t i o n , we have

@o=o, ~=@, ao=l, a.o=l.

It is clear that (0.2) is valid ( ~ = ~0~ , and the functions ~% are bounded. Such a model is convenient for the study of operators with finite defect indices (i.e. with ~ t ~ 4
~c~)=~ ~ ,

M ~ ~I MI)

where

t ~ [0,~]

1 PROPOSITION 3.1. ~ ~ ( I~I~ is a completely nonunitary contraction. PROOF. The operator ~ ( I ~ I ) is a contraction because % I-IM ~ ~IMI)I ~= I-IMl~ff ~(IM I)-- D Mq OMlb O, c~. 1~ TO prove the complete nonunitarity of M q (I~I) , suppose that ~ q (I~I) has a reducing subspace on which it is unitary, that is, the defect operators of Mff(~MI) are ~ero o~ this subspace. Since the operators and q (IM*I) are invertible, we infer from (3.1) that and D~, are also zero on this subspace, and the operators IMI and rIIM*l are equal to the identity on it. Since ~(4)= ~ , the operators and M ~ (I ~ I ~ coincide on it, as well as their adjoints do. Thus, the chosen subspace reduces ~ , and the restriction of ~ onto this subspace is unitary. Since ~ is completely nonunitary, this can happen only if the subspace is trivial. •

~CIMI)DM

385 Let ~h denote the Sz.-Nag~y - Foia~ model of the operator ~I ~ . Let ~ be again an operator providing a unitary equivalence between T h and M ~ ( I ~ ) . Then

TL=II*MqOM~)V~ Define

L:T~+fl,Ao fl*,

Ao: fI~!%* M [I- R(IM~I~fl.

P~o~o~z~zo~~.2. ILLIL':M. PROOF. Since subspace of T~

~* is the projection onto the defect , we have

Similarly, ~.

-~

0

Hence

= M~cIMo +CI-)~M,) M [ I- ~ cIM~)](-~M) ° I ° = = M~(IMI)+M(I-)~M)[~I-~(I

O] (I-%M)=

= M~ (IMp) ~ M [I-R(IMI)]= M, because +

[ I-~(IMI)]%M=O , If, as usual,

~ ~

A o- @o

0

t

0

%~=%M%M

S

, then

This implies (since now we write %± , %+

instead of ~

~~ ,

@o=-B q CIBO: - ~ C%++IBI-~X - )

(3.2)

A0= BEI-W (IBI)]= BD~ IBF~% -.

(3.~)

By the~notation . I Bt -1%~-~ ~-~% (~,~) (~)

we mean the function

of the operator

I~I

)

~ b-*

, which is correctly

386 defined, though the operator of W(I~I)~ we shall write we obtain

I~l -z

may not exist. Instead ~ + + I ~ -I - ~ % . By (3.2)

Since

we have ~ =~is satisfied.

, and

t h e

c o n d i t i o n

(0.2)

REMARK 3.3. A given contraction T does not uniquely determine the operator h satisfying T =TL . Indeed, a model contraction T being fixed, we can choose an arbitrary orthogonal projection %of ~ , wit~ only two restrictions imposed:

~)

%-I eol=leol%-,

2) the operator ( t - % - ) + l e o t %we define in a standard way ~. = T + ~ ,

Ao=B+eo,

is

invertible. Then ~o ~ * , where

5=-0°[ (I-%-~+~eol~-] -~

and it is easily seen that T = T L • Now we return to the general case,

s u p p o s i n g

(0.2) i s v a 1 i d. Instead of functions duce bounded functions ~ defined by

~i

that we intro-

e,= <[ e,,-= ~I + eoA*)- e A* New functions are connected with the former ones by the relations

ff,oS~ =~'~ t,,o,

e~ &.o=t~.o~.

(3.4)

387

We can describe the spectra ~CL) and 5(L*) in ferns of 0~ in the same fashion as it has been done in Theorem 2.4. in terms

of ~

.

THEORE~ 3.4.

2) l ~ % (L*) 4==>~,,< O~ cx), {o},

3) &-~

* -'k

2[e~ C][-&LD= CI-I~)-4 C~k~- ~,A) t~e~ e s CA).

4) k-~ e 6"p (C) ~

~c!-xC) II. I) X~ ~(L)~ vertible ° in this case

~-~ 04 (x) # [0},

cz-x)-~(%k~ ~ e4(k)

)~0,,(~).

is invertible

~==)05(~]

is in-

CL-x1 ~-'-- C~-~# [ I- (~g~- ~. A~ O, (~)-~k x ~* ]1 H,

(t'- I) -- d-~?~ [~ - c~, ~o- ~ ~oW) e~ (x)-~k×~*~] 1H, e~(~4'= e~* +~* (xt-C~ ~ ~. (1 +A O*~), e~ (k~-' = eo"+([ ,0~* A) ~* (~,I- L)-~ ~,. invertible.

In this case

(I-~ L~-~= (~-~-~ [ I- C:~~*~-~.A) O~( ;,,~ k~, ~* ]1 H,

388

@~Ck~-~=l+k~

*

w..

CI-kL~f ~ 00 9

OUTLINE OF THE PROOF. The p r o o f of t h e t h e o r e m r e p e a t s

the

reasoning of Section 2. We have to use the equality (3.4) and the following identities

The only thing which differs from what we have had in Section 2 is the existence of the operators (mr

"~-~ ~{~ CL') .).

Qt~ (~')-~

if

~¢~'C~")

Let us verify the formula for

=l+[@(~)e:-l+O,C~)

t

@~C k y .

~£ c~,1- t)-~ ~o.] C! + A 8o)-- I,

because of

8~ (~,) w*Ckl-L) -~ w. = a,0 ~ Ck~II* C/,l- Ly ~ w. -= a.o 2". cl- kT*y ~(xl- L) Cxl- Lf~ ~. =

= k~q w. = I- e(~) e~. Here we have used the formulae

(3.4),

(2.2),

(1.6) and (1.2).

[e~, ~* c~,I L~' ~,(I, A d~)] e~ (;,,) =

= 1 + [ e~ ec~)-I +od (~,I-Lf' ~ . e~ ~,t,,,)]CI +e~ A)= 1 because of

~* c~t Lf' oo,,e~c~,~ = w* C;,,1 LZ~O, g~ (~,) a,,= = w*ckt-Lf 4CKI-L)ct-aT* y ~ flao=

=I-8~ Ocx). Here we have used the formulae (3.4),

(2.4), (1.7) ~'Jd (1.3). 0

389

In the end of this section we state the expressions for the functions

~

for the model with

~= ~L

i+eo

. We have

+,

hence

e =e%+-B% - ,

e-X;, B, 9~= f - X -

B"e,

e~ = ~;, - e B*XT.. 4. We define on

of

Factorization of characteristic function

t h e

the

c h a r a c t e r i s t i c op

e r at

or

L

functi-

by

eLeo -L~- Q . , [-LJL+ ~D~ (I-~,L* )I D~]Q @L(~,) is a whenever

~L (E, E . )

- ~a~ued function, ~olomorphic

"~-I ~{ ~ (L),

Our definition differs somewhat one, for the spaces fect subspaces of

~ L

and

~.

from the generally accepted can be "wider" than the de-

. Thus, the function

eh

depends on the

model representation of the operator L PROPOSITION 4. I. -contractive in the following ~L is sense :

JB*

J~- OL JB,~ e L~ 0 , PROOF. Since

i_Q~fi ~

o

,

0

%;,

DL~ =o,

%1-, o LJ~,Q -T _ ~T.O_ ° =0 we have

Q . e~(x~ = I - L ] L + xD ~(i- xL:)-~D~]£1,

390

from which follow~

J~-e L (~

e~c~= ]~- @~c~ O_~J~ _Q. @~ (~=

+ ~D~,(I-~L*) -~ D~]] A = : Q*D~ Cl-# L) -~ { cl-~ L)(I- ~L*) .S L ( I- ~U) + +~(I-~L)~- Ixl ~ (I-LL')I CI-X~)-~ D t £ = = (I-l~l~) Q'D~cI-X L)-~ (I- kt*) -~ D~_O_>~o, In the similar way we verify that

@

Js,-ek(x)]s0u(xf= 6- I@)~**D¢ d-x~) (I-XL)-~D~ ~ >~0 Now we exhibit

the factorization

of the characteristic

on. We consider at first an arbitrary +Ll,~011.

In this general

some noninvertible lity. Next,

case the factorization

constant

treating

representation

operators

the case

T=~L

L=T

will

functi+

contain

in both sides of the equa, we get rid of this con-

stants. THEOREN 4.2.

e L(~ D~JB=DB, ~¢~)~ c~), ]s, Ds. O~c~) = ~ c~ ~cxY 4D s .

PROOF. We s h a l l

use the

formulae

e u(x) D~ 7~= D~..0..'. (I- ~)-~ C~[- L)Q, ]~, Ds. OL(~) = fi~ (~I-L) (I-k L*)-~fiDa. For example,

the checking

of the first formula

proceeds

as fol-

1 ow s

QL~,~D~]~=D.**[-DL,L +xDL. cl-xL*f~d-Cb]O_=

=D~.fi~cl-~Uf'[-(l-

xC) L+ k(I- ~ L)]n =

=D~. D_*, ( l - k ~ f~(kl-h)fi. So f r o m

(2.5)

a n d Lemma 2 . 1

we h a v e

391

Ou(X)D B Je~ = DB*Q *~(I - )~ L*)- ~(I -)~T'~)(I - >,T*)-{( kI - L)@ =

= De ¢(a}-'Q: (i- aT*)-'(at -L) Q = D~, O~(a)-'< (X). Similarly

from ( 2 . 4 )

JB*DB*O,.(x~ = ~ :

and ( 2 . 3 )

follows

(xz- L)(Z - XT')-'(I - X T * ) ( I - X L*)-'

~D~ =

=Q:(),I-L)(I-XT'~)-~ &(),)-'DB =&(k)t~()~)"D~.

•

We can simplify the factorization presented in Theorem 4,2 in the case where T = T L , In order to do this we introduce additional operator-valued functions ~ L which differ from the functions ~ L by invertible operators,

e", de.~ e,(z÷_te, l-~:z.-)=e,++-v-x.-, e~ a~{ (x.,~-IB"I-'X.;.)e~--X.~e +'x.;.v.", e~ '~ (x.++l~,l-' x.-)e~ = x. +-x.-v*o, =)C.-OV.

X,,~ .

Here ~ f = ~ 5 , ~ / r ~ = V l ~ * a r e isometrical operators f r o m the polar

decompositions Since

~ = ~/5 1 ~I 7 ~

= ~ / ~ I~ i . -!

g3 =D;'~ Ao , Theorem 4 . 2 p r o v i d e s As follows from Theorem 2,4, the factors in the latter factorizations are responsible for the spectrum of I inside and outside the circle ~ respectively, In the sequel we shall need a relation between the operators J~-e~Js~Ob case

T=Tu.

and

I - ~ T OT

, We consider

t h e

392

PROPOSITION 4.4.

= O~ (Js

J6*O~) es,

ZPROOF,

e~ c,L-o"L, J~,,e,.)e3 =. o~ J~e3- o'~ Lo~ = ---(x:t- ov x-)(x++ ×-v*o)-(e*x; ,v,~x s)(x;, e -x,; v$) = = x+-O*xT~o - O * x ; o * ~ - ~ z - e * e

=

The second equality can be proved similarly. •

5. The absolutely continuous subspace. From now on we deal only with model representations of an arbitrary operator L for which the condition (0.2) is valid. In this setting we define the absolutely continuous ( N & ) and singular ( N s ) subspaces of L . For unitary operators and c.n. contractions our concept agrees with the standard ones (see, e,g., [I3] for a discussion on absolutely continuous subspace of c.n. contractions). In the present and in the next sections several different descriptions of the subspaces N a NS are given. Some of them essentially use the model lanand guage. The others appeal to the boundary behaviour of the resolvent not involving the model representation at all. Our exposition in this section often follows the approach of S.N.Naboko [3]. Let

us

denote

PROPOSTTTON

a~

U~

5. T. J~

Ke~[(I =

*

*

~OC.(L-#)p~(z-#)-'oc = pHo~

, ~eC,T] PROOF. We have successively

PH-(L-Ju)PH(z-p )-~ =(PHz -L PH) (z -ju)-~ ;

PHz-LPH =PHZPHl - ( L - T ) P~ ; the first ~ n d :

for

393

the second summand:

Consequently,

and

4re tt{, t P,-

P,

To prove the converse, suppose that

and check that the function

is identically zero. In fact, since ~e/t ~ , = 0 ~ the equality (5.I) implies that the Cauchy integral of the function vanishes on C \ T NOv 4 hence ~---- 0 • @ PH is calDEPINITION. The subspace c o n t inu o u s sub spaled the a b s o 1 u t e 1 y c e of the operator REMARK 5.2. Define N ~ as the absolutely continuous subspace of the operator ~ in its model with the auxiliary contraction T ~ . Then in the representation of ~ on the model space of T we have:

To make sure of it, let ~ , ~ ~ H etc. denote the objects having the same meaning in the model of ~ as ~ etc. in the model of < (in particular, t = ~ ). Note that there exists a unitary operatgr from ~ onto ~ which transforms E into ~ , ~ into ~ , , E , into ~ ,

394

into

{u~£ C

,

~

into

,Chic

into ~7" , into ~ , % into [t , ?, into ~" etc. ( C: ~(X)-~ --> tZtx) is defined by tO[) (~) : 7 (~) , ~T * is the characteristic function of T * , ~T~ (k) = O" cX) }. It is easy to see that the equality

corresponds to the equality

tI+k'eo)~'~x = A.<×. The a . f t r e m e n t i o n e d o p e r a t o r can "be e x p r e s s e d i n t h e B . S . l : ' a v l o v ' s form by

e_r

--" L~ 0

I

.

•

PROPOSITION 5.3.

X~

PROOf% Let us check, for example, the first formula. Let , i.e.

A~-0'x =CI+/k < ) < x~ and let

= PH x

(5.2)

. Then

~*~=e s P_ (~*-(9o ~.)x, < ~ : % P + (0; <. - ~rO)x. Indeed,

= P_~'x- Onp_ < x =

: Is-co< O;)A] P_~r~-E0<<0< <~(I+Ao[)] P_<x:

395

--e~ ?_ (~¢- eo s-~,**'x.) We get the expression for ~** ~ analogousl~ Conversely, if ~_ ~ H i_(~) , ~+E <E) is such that ~L,*~ = Os J~_, ~~ take the element

, and then we

(5.3) and verify that X 6 # and The latter is valid because ~ (~) J. m . To check that ~ * X and ~b; X :

~_Ht (E~.LH and X E~

, l e t us compute

(i+ oo

)(t-hb

~.'x - ,~* +Af_-e (I + o;A)L=

hence (5.2) holds. • S.N.Naboko in ref. [3] provides the description of the absolutely continuous subspace in terms of the "smoothness" of the resolvent. In the proposition to follow we present a similar result for the model with T = TL PROPOSITION 5.4. Let L be a completely nonunitary operator. For the model with T = T h , RH d =

{ ~' ~ ~ H ,

3 H - valued functions

X

and

X' such that ~=(~-~)X(~) =(~-~L) X~(~)~

DLx eH~(.H), Dux'~Ht{,H)}. PROOF. Since the operators D 6 and X/~o,, are slmil~r, the conditions DLX ~- H~(H HD E equivalent to the conditions *X 6 ~) , £O~X'~ (E). The formulae (2.7) imply the following identities:

H~'(X) H£are

Isx~: (L- x)=- O~(x~ co*t H, h.x~* (I-XL) = X 0 s (x)~* 1H.

(5.4)

396

Therefore, if X and k ~ are such as in the statement of the theorem, we set ~÷()x~=-CO*X()',') and ~ {,~,'}=~xCO~X'(~.'), @ Then the equalities (5.4) provide %**~=$~ ~÷ , ~*~ = ~ ~_ , i.e. ~ e PH ~ , according to the previous proposition. Conversely, if g~:~ = 0 ~ , ~+ ~ ~ i ( ~ ) and ~-~*~ = ~ ~, ~_ -C+ ~ t (~) , then it is natural (see the formulae for the resolvent from Theorem 3.4) to set

Direct computations provide that X(~') E ~ , X'(k) ~ ~ for any ~ ~ D and (~-k) X(k) =(~-~)X~(~) = ~ . In addition, CO~XC}x')=-~+(~) and CO~,X'(k)=~-~ ~_ C~,') , i.e. by the note at the beginning of the proof,

DL×'

HzCE),

DLX

~ ~%(~)

and

@

The next proposition is also adopted from

E3~

(cf. also

ch.=, PROPOSITION 5.5. If the spectrum the unit disc, the operator

6"(~)

does not cover

is a quasi-affinity intertwining ~ I~ and ~I~ / (cf. ~2] for terminology). PROOF. The equality 0 Z I~ = L Q follows from Proposition 5.I. Thus the only thing still to be checked is that Ke~ ~ = { 0} . it is not difficult to verify the following formulae:

e"4 P_ rC * = 0 and X ~ Ke% Q , then 01P+ ~* X = 0 . The operators 0q (k) are invertible in a neighbourhood of the origin, hence P_ Tv** X = 0 . Since d(~) does not cover the unit disc, @~(~) are also invertible for some open set of )k~ s , so P÷ ~. X = 0 . Therefore X~ H , and X = ~ x = 0 . • If

397

DEFINITION. The subspace is called the s i n g u I a r o p e r a t o r ~ .

S~larly,

N s , ~$ ~ ~ c9 N ~ s u b s p a c e o f

t h e

N~s : H ® It

Obviously, the subspaces N~ and ~s are invariant under rational functions (with poles off ~ (<) ) of the operator ~ . The study of these subspaces will be continued in the next section. 6. Scalar multiples. As usual, the efficiency of the model increases considerably if the corresponding analytic operator valued functions admit scalar multiples. In this section we study operators allowing a model representation in which the bounded functions ~ (~= ~ % , 5 , 4 ) h a v e s c a I a r m u I t i p I e s. The latter means the existence of the bounded analytic operator-valued functions ~ , such that

T~

and nonzero bounded scalar functions

This representation will be said to satisfy t h e c o n d i t i o n ( S ~) . we present here the results, having in mind the model's application in the next section. It is known (see [2] , ch. YIII, ~ I) that the characteristic function of a contraction 7 admits scalar multiple if i -T ~T ~ ~ and the spectrum ~ (T) does not cover the whole unit disc. The thing is that in this case the difference between ~ (~] and some constant unitary operator is of trace class for any ~ (so ~ ~ (A] exists; see [81 ), and ~ ( ~ is invertible for some ~ (thus ~ ~(~ is not identically zero). We use this observation to prove THEOREM 6.1. Let ~ be a bounded operator such that I-T*T , and D ¢ model representation satisfying

G(L) (8~)

PEOOF. Since ~ is the orthogonal completely nonunitary operators, it is theorem separately for unitary and for T h e c a s e o f c . n .

. Then

L

has a

• sum of the unitary and sufficient to prove the c.n. eperators. o p e r a t o r s. We

claim that (8 ~) is satisfied for the representation in the model space of the operator ~: II . Let us check, for example, the existence of a scalar~multiple for ~ • Note

398 that it suffices to find a scalar multiple for the function 7,,,+ 9 7~* " Ha(Tv+E) --* H a ( Z [ E . ) • In fact , if there exists a function T + ~ ~ ( ~ ( [ ~ ~)) such that +

T* = Z*T

+ - T * ?(,,

and +,

T.OK+=S7,

K .*~ 0 T + ={57..+

then the function

T~ dd- T+.iBI-~X, - B*0T +- ~515t-eZ-B * satisfies

T,O,=~I ,

OiT~=~I

But the operator ~Ig(~))Q*: • + E " ) ~ E . is invertible, provided ) X ~ ( L ) ((x~e(/k)~+)-i=~+e,(A)-'X;) , and the difference V s ~ 0(/~) belongs to the trace class, where ~ 5 is the isometrical factor in the polar decomposition S=~5]~l. Thus the function ~(+ 0 )~+ admits a scalar multiple, and the assertion follows. The case of unitary operators. Let U be a unitary operator in a separable Hilbert space H • Then there exists an orthogonal decomposition of H : k0

k=i such t h a t the subspaces

UI HK

m K reduce

[~

, and the operators

have s~ple spectra, i.e. H~ = spa,",(U'sc~ : n ~ Z )

for some Let

OCK , OCK~ HK • ~ be a trace class selfadjoint operator on a Hilbert

space E , dim ~ ~ k o , and 0 < R < I . Let ~ be an isometrical embedding of E into H , such that q O F = = s p o ~ lOCK] , a n d q D ~ K are the eigenvectors of R . Then the operator

T =U(I -@Re9*)

is a c.n. contraction. Indeed,

I - T ~ T = ~ ( a R - R 2) q ~ > O ,

399

and let H~ be a reducing subspace on which T is unitary. Then H 0 ?I ~ also reduces. Since the vector X~ belongs to the defect subspace of ~-I~K and T l~0 ~ is unitary, XK i H o ~ ~K • Hence TIHo = ~ I ~ o , and '~A, ~';~l

Z~o~=H¢0forany H~

~,

~eZ

. ~hus

H0n~={0}

}, proving the complete nonunitarity of So we can assume that H is the model space of X

, and . As

DT. = ~LcPc~R- R~)~/~q~*l~~,

DT = ¢ c%R- R~)Vz¢ * , we have

e¢~:O~ 1,LmJR-I-kC~R-R~)v~m*IZ~CI-~T*) -~¢ ~R- R~ ~I~]qb~, Since the characteristic function is determined up to constant unitary factors, we can choose

~cte that kn this case

Q{,h)+I ~_~

for all

k's. We

have also

@o=eo = R - I , 10 =

A .o : Cf.. R - R ~' )4I~,

Hence (0.2) is valid and

A=(~I-R) -~. Pinally, the analytic operator-functions

0~=-¢~

=

A (e-I)

admit scalar multiples. Indeed, the ftmction ~ ~ I - 0 ) is contractive. It is invertible at the origin, and~ I ~ -~ (I- e(h~= 4 = ~ ( I + Q (~)) E ~ for any k ; so the assertion follows. • Now we turn to the study of operators admitting model representation in which ~M) is valid. This assumption remains unchanged up to the end of the section. First~ give a new description of the absolutely continuous and singular subspaces.

400

PROPOSITION 6.2.

PROOF. We shall verify only the first formula. Let ~IZ!C~~7~~ = ~ ~ ~ ~ . Then

=d,e~T~

:c - ~ s e e , T , oT, :c =

hence

~,~:*o: = ~ ( I +eoA)T,~J o:.

Hence 3C is orthogonal to ~I ~ Note that C ~ 5 ~ i ~ = ~ and ~ ~ ~ i ~ . Define ~ Since Z ~ = ~ integer ~

Indeed, let

de~--~[][~*l~l[{+ /Iq~,~ ~ l[~.

, we have #, ~ K ~ i ~ , i.e. 0 = ( £ n ~ ~)~6=IK"~i~

Thus ~ f = 0 , and since ~ We ~ v e ~ o v e d that ~ J ~ Ke~ [~, To

•

m~

~

, f=0

C~

L~

for any V~

@

, and

=0

-

prove the converse inclusion it suffices to check that

-%[O ~-#eo)e'~-A*] T, $ --(d,e~Y~-~e, "r, )g -o.

•

@

401

COROLLARY 6.3.

-~G.< )L (E.)

j'¢, : oE)s(~,~ T~ -ga~.T~*)L~(E),

@

COROLLarY 6.4.

Ns ---Ke,(8,T2s*-SsT~sr2)rlH, N.s:Kex.(8~T~sr*-SzT~?)

nil.

,

In the sequel we shall use the boundary values of the resolvent (in the weak operator topology). In terms of such values one more description of the singular subspace is provided. Thus we ascertain that the definitions of the absolutely continuous and singular subspaces of the operator b coincide in all models of b satisfying ( S M ) • For arbitrary elements CC and ~ of the space m consider two functions:

~g,~(A) ~-~-(/k) ae{((L-X-I)

~,~), X-~¢6(L).

PROPOSITION 6.5.

~+(~)=((z-~)-'a,~)~-x-'(
-'

*

~t-(s)---X((~-xz)-~,~)x-X(A0[(x)-' hG*z, h~<~)E,. PROOF. To prove these equalities we make use of the formulae for~he resolvent from Theorem 3.4:

+(A) -(( z - ,x,)-~~:,~)Z -((z -A)-'(s~h;-u), A)O~(A)-' h#r2 ~:, ~)z =(( z- X)-~a:, ~ )X-( @~(A)-~hA072;c ,[(I +A*@o)ho~*-A*ho97.*](~-i)-~)E" Since

ho(2-X)-'flt** ~ = 0, ho(~-X)-~SF*~ ~ X-' h~F's~, we obtain the required formula. The computation of ~ - ( ~ )

can

402

be performed in the same manner. COROLLARY 6.6. Under the condition (.~') the functions ~+ and ~_ are meromorphic in ~ and of bounded characteristics. Indeed, the first summands are of Nevanlinna class being the Cauchy integrals, and the assertion is true for the second summands because ~?~ ~ ~ • This corollary implies the existence of nontangential boundary values of the functions ~+a.e. on ~ . It is the difference ~ + ( ~ 3 - ~ - ( ~ ) , ~ ~ , that is important. Applying the Privalov theorem about boundary values of Cauchy integrals, we obtain a.e. on T:

, b,~)E

+(m~x,m*~)~

-

Taking into account connections between ~ , ~, and we can rewrite this expression in the following forms:

~, ~,

or

-+-

p,J,

a,(l,Oo;<

(6.2)

PROPOSITION 6.7. +

~

N~s--{~:.~cH,~'x+,](~1:~(~)a.e.

~ ~T

, f o r ~,ny x , x ~ H }

PROOF. Let X6 N S , then (see the proof of Proposition 6.2) ~*X = & ( ~ + ~o*~)~T~*~× • Hence the formula (6.1) yields a.e. on ~+= ~T Conversely, suppose that ~+ = ?a.e. on ~ for

a~

~ ~ H

, and set

~ = ~ e,

e ~ E

and

Therefore we obtain from the formula (6.1):

. We have

•

403

+

*

+ #(i +Co•.~ A)]

~

,(7,~

C)

=

h e)a =

Analogously can be checked the expression for ~ ~ ~ - • In conclusion of this Section we study the spectral structure of the unitary operator Z t~ . Let ~z denote its spectral measure. We set the question when the Lebesgue measure is absolutely continuous with respect to the measure ~ Z ; i.e. when does the operator Z ~ contain the bilateral shift. Remind that we consider the condition (SM) being satisfied. PROPOSITION 6.8. Let ~ be a measurable subset, ~ C ~ • Then the following assertions are equivalent

2) The operators 0 ( ~ ) are unitary for a.e. ~ , ~ C ~ 3) The operators 0 ( ~ ) are isometrical for a.e. ~ , ~ - . PROOF. Note first that ~ 2 ( ~ ) =

a~

0

iff ~ Z

k = 0

. for

X e ..N I) ~ 2 ) .

zf x ~ H

,we have 5 g z T * x

= 0

and

~ X = 0 . Due to Corollary 6.3 we can take X r,.~.C~@~b_qf'~--~1"~.']rg*)1** , C ~ . G L~ ( . E . ) . For such X S t x = g , , ~ , T . ~ . , ~ x = - ~ o k . T ; a,~. uence % z k : 0 and ~ ~ , =- 0 , that iS the operators U ( b ~ ) are unitary for a . e . ~ ~__ ~

2) = ~ 3 ) i s o b v i o u s . tors

3)==:>I). By Corollary 6.3 it suffices to show that the vecg~. _ ~ ~ a - ~ 'b.'I~4 ) a . are zero for every ~ ,

=g4%~- ATZ C}.

; hence i f ~ ( 3 ~

= 0

for a.e. 3 , ~E_5-" ,

then [Xtl & ~ltcg~Xl~-k ~*Xl~=0, that is X = 0 , ending the proof. @ REMARK 6.9. Let the representation of a c.n. operator in the model of T = V h satisfy the condition ( $ M ) . Then by Corollary 4.3 the characteristic function O h has boundary values (in the strong operator topology) 0 h ( ~ ) , for a.e. on ~ . From Proposition 4.4 and the previous proposition,

404 it follows that in this case gz ( ~ =0 OL(~) are ~ -unitary (equivalently, ~ a.e.

~

,

~ 6 ~

iff the operators -isometrical) for

.

REN~ 6.10. For a unitary operator ~ , our concept of the absolutely continuous subspace coincides with the traditional one, which is expressed in terms of the spectral measure ~ U • Indeed, the subspace N a reduces ~ , thus ~ I ~ is unitary and, by Proposition 5.5, is unitarily equivalent to Z I~ (see ~ 3, ch. ll, [2] ). The latter is absolutely continuous as a part of the minimal unitary dilation of c.n. contraction. Hence, ~I ~ is absolutely continuous, and it remains to establish that the spectral measure of the restriction of ~ onto the subspace ~s =~ is singular. Proposition 6.7 shows that this is true. @ Two previous remarks provide the following corollary of Proposition 6.8 (see Theorem 6.1): C O R 0 1 ~ Y 6.11. If I L ~L ~ ~i and ~ ¢ ~ ( L ) , then the operator Z lJ~ does not contain the bilateral shift if and only if there exists a set ~ , I ~ I > 0 , such that the ope-

rator. 0L (~) are ~

-u~itary ( ~

-isometrical) for a.e. S

~e~ , and such t h a t ~(~)=0 ' where ~ is t h e s p e c t r a l measure of the umitrary part of the operator ~ . • 7. The,,,,,,stability

of the continuous spectrum.

The last section of the paper is devoted to the study of the continuous spectrum of the operators admitting the model in which the functions OL have scalar multiples. Our study is performed entirely on such a model. In Introduction we stated the problem of the stability of the continuous spectrum and discussed the definitions involved. Now we would like to note that the stability of ~C(L) is closely related with the properties of the lattices of invariant subspaces of the operator L and of its resolvent RXae~ ( L - ~ I ) - ~ ~ ¢ ~ ( Z ) . (we denote them LG~ ~ and Lat R~ respectively). The following general lemma is true for an arbitrary operator in a Banach space. *) L E ~ A 7.1. Let ~C([ ) C T . The following assertions are equivalent. I) ~c ( ~ ) is stable under rank-one perturbations, 2) L Q t L ~ L a L R ~ , f o r any ~ , X!~6(L) , 3) LQtL o Lot R~ for some h , k e D \ ~ ( L ) • PROOF. 3 ~ I. Suppose that L admits a perturbation • ) Suggested by N.K.Nikol'skii.

405

K= (" ,×)~ must show that Fix any

of rank one such t h a t

b~

Lc L~2~o

ko. The operator k ~

D

for any

L * K - ~I

G~(L+K)

. we

ko~D\~(h).

is not invert±~le for

. The identity

L + K-~I= CL-~I~(I+ ~K~, provides that ~ of ~ o . since

(I + ~A K) ~-~ 0

~t=O

k~CL) in some neighbourhood

ko %~'

we have

(R~o~, x) =-t, 1 (R ~k o l ~ x)-- 0 ? ~>~ ,

(7.1)

These equalities imply that the invariant subspace of the operator Rko does not - 4 ~ {R ~ko : ~ >t } belong to L&t b . I $ 2. Now assume that there exists a point ~o , k o ~ d (h) , and a subspace ~ such that G ~ h ~ ~ko , ~ ~ ~@~ b . Without loss of generality, we can consider the subspace ~ to have the form ~ = =~0~t{~o~ ~ Z } ~ L~ not belonging to G • By the Hahn-Banach theorem there is a vector X for which (7.1) holds. Converting the reasoning of the first half of the proof, we obtain that ~ ( L + (. ~ x ) ~ ) D ~ , i.e. ~c ~L) is unstable under a perturbation of rank one. Unfortunately, we are not able to present a similar geometrical interpretation of the stability of ~ccm) even for the case of perturbation of rank two. We do not know, in psrticular, whether the rank-~ne stability implies the rank-two stability or ~I - stability. On the other hand, the second assertion in the above Lemma ( b~t U p ~ t ~ k ) follows from the formally stronger one: L ~ ~ (~) ~(.) denotes the weak-closed algebra generated by the operators ~ and (') ). It is also unknown whether these conditions are equivalent. The affirmative answer to this question would follow from the well-known conjecture:

406

A , ~ are arbitrary bounded operators (see 96] ). We can give answers to the questions stated above f o r the operators studied in the previous Section. We assume since now that ~ admits a model representation i n which the functions

where

~$ have scalar multiples. We remind that the class of such operators contains all operators with ~ - ~ h 6 ~i , and

De

~cL~

.

THEOREM 7.2. If the operators 0 C~) are isometrical on some set ~ (~ q ~ ) of positive measure, ~ C~ ,

~c ( L ) is stable under the trace class perturbations. The proof is preceded by the following T,BI~tA 7.3. Let ~' and K ~ be operators of the Hilbert-Schmidt ideal ~i . Then for a.e. ~ , ~ ~ , the function

then

k ~ IIK' { L - ~ ) - ~ K' ~ ~ is bounded in a Stolz angle with vertex at ~ . The same is true for the function

k ~

II K
R~b -~

K" = ~, ~ ,' C', ~ )~ k~' be the Scbm~dt de~ .... K composition of K" , where { X~"} ' t ~"~ are orthonormal f a milies in ~ , and ~ I~ " }~ < ~ . Similarly, let PROOF. Let

=

2gC. , ~ e ) x '

II K"( L- kI)-~ K' I/~- =

. Then

II K"CL xlb ~ K ' ~

I ~ t ~ I Cc L

/xT-~

l~ x~, ~ ) ~iI ~

KII to all of ~ and by setting them zero on @~ . The extended operators are denoted by ~ and respectively.

We extend ~'~

11~.=

KI

i

II

11~"<~-~,)-' tK' II~ = 7', I~"~ I ~ I~,1 ~ I (~-~,~ -~ x~, ~ ~/~. Proposition 6.5 provides,

K ,¢¢I, The function II~{," (~-A)-~ ~ ' ll~..z is bounded in a Stolz angle with vertex at ~ for a.e. ~ , ~ ~ , as (Z-k~ -{

407 is the resolvent of the unitary operator E (cf. [I0] ). We wish to prove the same for

and ~', ~

K,~I, This sum is not greater than

¢.~CY

~I II ~,x~,

CZ

'~

To finish the proof, it suffices to invoke the following

S~_~. Let ll~ttH£cF_) ~ J .

J~ >~0 , Z ~ < ~ Then the function

, ~K~H~cE) ~,I-~J~KIi~KC~)I[ ~

bounded in a Stolz angle with vertex at ~

PRooF. If

~ ~H~cE) . then ~ ~

, is

for a.e. ~ , ~ ~

II~CXJ,~

.

is ~ub-

harmonic. Hence

p,

o

denotes the Poisson.~kernel,

~l~c~C~l/~_h .~!'Z~ tJ ~ ~ K I1~K ce~)ll~P~c~-b~ = ~I"~

0

o

is summable. Therefore the angular boundedness follows from the properties of the Poisson integral. ~ PROOF OF THEOREM 7.2 essentially repeats the reasoning of ~IO] . Assume that there exists an operator ~ , ~ ~ ~ , such that ~ the operators

(,~+K) ~ D I + R~K are

t~-K+K=ci~-~I)d:+RxK),

. Since not invertible for any

~

,

where ~% denotes the regularized determinant (see [8~ , ch. IY for the definition and properties). The function is analytic in ~ \ G ( ~ ) , and ~(~)=-- 0 in D . To prove

408

the theorem, it suffices to show that the angular boundary values

exist

and are

this le

is

the

equal

case,

to zero

for

a.e.

~

by a Lusin-Privalov

,

~ ~

theorem,

. l~deed, ~

0

if

on the who-

of

@ . This contradicts the obvious fact that ~(co)= { . Now let us prove that the limit in (7.2) is zero. Let = ~' ~ be any factcrization of K i n t o the product of two

Hilbert-Schmidt operators. Then

~cf~ = ~ C I + K"cL-~l~-~ K') Since ~ is continuous with respect to the ~Z -norm of its argument, the theorem will follow if we prove that

IIK(cL ~I)-~-CL-R-~I)-~K tlT=O for

a.e.

~ , ~ ~

. To do t h i s j

it

suffices

following conditions ( s e e Lemma 2.1 i n ( ~ ) the function d e d in an angle with the vertex at ~ (~) for a.e. ~ , ~ £~ , the ~,I C(b_~l)-4 _ C ~ _ ~ _ ~ ) - i ) m ~ has a

[17]

to verify

two

):

IIK"CcL ~I?~-(L ?~I)~ K' I1~,

is bounfor a.e. ~ , ~ ~ ; operator-function of k zero angular llm!t in

the weak operator topology, k tending to ~ . The first assertion was proved in the preceding lemma. The second assertion can be deduced from the formula (6. I). Indeed, given two arbitrary X,~ ~ ~ , there exists an exceptional set ~ , ~ C ~ , of measure zero, such that for every ~ , ~ ~- k ~ , the a ~ l i m i t

~,~ C[CL-~I?LcL-X-~ I)-~] K'x, K"'~) exists and is zero. Since the space ~ is separable, we can choose a dense countable set { X ~ O ~K } in H ~ m • To each pair X~ ~ ~K corresponds the exeptional set ~ , P~ C ~ . For all ~ , ~ \ U ~ , such that the function ~*

ll~'I(C~I) ~-(~-~-~)'~)~II

is bounded in an angle with

the vertex in ~ , the condition ( $~ ) is satisfied (by the Ban~ch-Steinhaus theorem). It is rest to note that such points form a subset of full measure in ~ . @ In the proof of the converse assertion we shall use the concept of simply-invariant subspaces. Let ~ be an i n v e r -

409 t i b I e operator, and let G be an invariant subspace of ~ . Then ~ is said to be s i m p 1 y - i n v a r i a n t, if G ~ ~d~t ~-~ . The following simple lemma shows that the preimage of a simply-invariant subspace by an intertwining transformation is simply-invariant. I ~ , ~ 7.4. Let ~ and ~I~ be invertible operators in the spaces ~ and H~ . Let ~ be a transformation of ~ into H~ such t ~ t ~ = ~Cp , and let X~ be an element~ of . Then if the subspace is simply invariant with respect to ~ i , then 5 ~ { ~ X ~ : ~ > ~ 0 } is also simply-invariant. PROOF. Observe that the subspace ~a~t { ~ X :¢~ >~0} is not simply-invariant ( ~ is an invertible operator on H , x e ~ ) iff there exists a sequence of polynomials { ~,lr such that ~(0)=0 , and ~ ~(~)X = X . Assume now that ~0M¢ { ~ X 1 % >~0 } is not simply-invariant, and { ~ ~} is the corresponding sequence. Then ~ ¢ ~ p ~ ( ~ ) X ~ = ~ X l and ~1, ~,~(,Ml)X£=X£ • Hence 5,120,/l¢{Mtt XI : '14,>0} is not s i m p ly-invariant. Now we are able to prove the following theorem, in which, as in the previous one, ~ is a model operator, and @~ admit scalar multiples. THEORE~ 7.5. If the operators @(~) are non-iscmetries for a.e. ~ , ~ ~ T , then ~ ~ ~ ~0~ ~k , for any ~

D \6"(;9

.

PROOF. The formula (5.3) provides an expression for the unbounded operator Q-~ (see Proposition 5.5):

Hence, the operator ¢ ded to all of ~ ,

£

~1 ¢ * 0 -{

can be boundedly exten-

and Proposition 5.5 yields

A function 9' ~ e ~ (E) , generates a simply-invariant subspace of the operator (~-~)-~ if

,

4t0

Z~

I

HI(t)IIEdt

0 (this is an easy consequence of the Szego theorem and of the

La~(L X),

fact that Lat E -i= Lat (E _~}-i we have Lat ~ ~ LG~ L subspace.

). sinoeLG~L = iff ~ has a simply-invariant

Note that ~ l q S ~ =~icC ~ Q - ~ ~ MG , thus, according to Lemma 7.4, it suffices to prove that there exists an element ~C , 0C ~ , such that

I

(7.3)

0

Hence, we can take a vector ~ ll~(~)II E =i

a.e. on V

= (~m + ~ T # A A) T

.

~:~ o~osA L~(E)

,

so that

. Let us verify that 0C = belongs to ~ and satisfies the

condition (7.3).

AsT'ac =A(~4A+ e"Y AA)f =

'<) T/AA

=

+Aeo)T4 AA~ : { I + A 6 2 ) ~ J z

i.e. 30 C ~ • Next, cg*OC = ~ T , _ (7.3) is valid. @ The obtained results can be summarized in

tt(£~$)(S)IIE =tde(B)l, _ _

and

THEOREM 7 . 6 . L e t an o p e r a t o r L aamt t a model r e p r e s e n t a t i o n in which the functions ~ L have scalar multiples. The following assertions are equivalent 1) ~C (L) is stable under rank one perturbations. 2) ~C(L) is stable under trace class perturbatlons. 3)LatL~LatR~for any (some) p o i n t ~ ,~EDxE(L). 4)LE~(Rx) f o r any (some) p o i n t ~ , / ~ D x ~ ' ( L ) . 5) T h e r e e x i s t s a s e t ~ , t ~ 1 > 0 , such that the operators (~) are unitary (isometrical) for every ~ , ~ ~ ~ . 6) The L e b e s g u e m e a s u r e i s n o t a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o t h e s p e c t r a l measure o f Z 1 ~ PROOF. We have already proved that I ~ 3 ~ 5 ~ 2 ~ 1 : the first implication follows from Lemma 7.1, the second - from Theorem 7.5, the third - from Theorem 7.2, the last is evident. We apply the obtained equivalence 3~=~5 to the operator

L(n) ~

Le .... • L

(

n

times).

Obviously,

the

ope-

411 rator h (~) satisfies the condition of the theorem, and the assertion 5) holds (or does not hold) for it simultaneously with the operator ~ . Hence to prove 3 ~ 4 it suffices to refer to Sarason's lemma [18] :

Finally, we note that the equivalence

5 <=> 6

was established

in the previous section. @ REMARK 7.7. Por an operator ~ with ~ - ~ * ~ ~ ~ and ~(h) ~ ~ , the established conditions are equivalent to the following: t h e r e e x i s t s a s e t [ , I~I > 0 , s u c h t h a t t h e o p e r a t o r s ~L(~) a r e -unitary for every ~ , ~ , and ~C[)=0 , wh e r e ~ i s t h e s p e c t r a 1 measure of the unitary part of t h e o p e r a t o r h . (See Corollary 6.11). In the case of unitary operators this provides the theorem of N.K.Nikol'-skii [I0] . In the conclusion, we mention that the spectrum ~c(h) is stable under ~ -perturbations, where ~ is an arbitrary cross-normed ideal, ~ ~ ~ , iff ~cCh) @ ~ . This is true not only for the operators considered in this section but in a

much more g e n e r a l s e t t i n g

(cf.

~111

).

References I

D a v i s

C., F o i a q C. Operators with bounded charac-

teristic functions and their

~

-unitary dilations. Acta

Sci. Math. (Szeged), I97I, 32, I27-I40. 2. S z. - N a g y B., F o i a ~ C. Analyse harmonique des operateurs de l'espace de Hilbert, Masson et Cie., Akademiai 3

Kiado, I967. H a 6 o E o C.H. AOCO~DTHO HenpepHBma~ cneETp He~DIOOWnmTKBHOrO onepaTopa x ~ W m ~ m o H a a ~ H a a Mo~ex~ I, II. S a n . ~ . ceM~H.~0MM, I976, 65, 90-i02; I977, 78, II8-185. 4. B a 1 1 J., L u b i n A. On a class of contraction pertur5

bations of restricted shifts. Pacific J.Math., I976, 63, N 2. C 1 a r k D. One dimensional perturbations of restricted

6

shifts. J.Analyse Math., I972, 25, I69-I9I. F u h r m a n n P. On a class of finite dimensional contrac-

412

tive perturbations of restricted shifts of finite multiplicity. Isr. J. Math., 7. W e y 1

I973, I6, I62-I75.

H. Uber beschrankte quadratische Formen deren Dif-

ferenz vollstetig ist. Rend.Circ.Math.Palermo,

I909, 27,

373-392. 8. r o x d e p r

I~.IL, K p e ~ H

M.r. BBe~erme B Teoptm Jn~-

H e ~ m ~ x Heca~oconp~meHH~X onepaTopOB B I ~ d e p T o B O M npOCTpaHCTBe. "HayF~", M., I965. 9. H a I m o s P. A Hilbert space problem book, Van Nostrand, Io

I967. H H K o ~ ~ c K H ~

cneKTpa ym~TapH~X

H.K. 0 B o 3 ~ e H ~ X

onepaTopoB. MaTeM. BaMeTEE, 1969, 5, 341--349. II. A p o s t o I

C., P e a r c y

C., S a 1 i n a s

Spectra of compact perturbations of operators.

N.

Indiana Univ.

Math.J., 1977, 26, 345-350. I2. D o u g 1 a s R. Canonical models, Topics in Operator Theory (ed. by C.Pearcy). Amer.Math. Soc. Surveys,

I974, I3,

161-218. 13. H a B x 0 B

B.C. 0d yC~IOBN~X OT~e~XMOCTE CHeETpa2BHHX EOMnOHeHT ~ccm~aT~BHoro onepaTopa. MSB. AH CCCP, cep.MaTeM., I975, 39, I23-148.

I4

H e 1 s o n

I964. 15. B a c • H ~ H

l<e~.;ll~B~-Ha~ I6-23. 16. R a d j a v i

H. Lectures on invariant subspaces. N.Y.-London, B.H. llocTpoeH~e ( ~ L ~ O H a X ~ H O ~

- ~i. ~Oi~m~.

3an. Hays.

H., R o s e n t h a 1

Springer-Verlag, I973. 17. IS ~I p ~ a H M.~., 9 H T ~ H a

ceM~H.

Mo~ex~ B.Ce-

~0J~4,

1977,

73,

P. Invariant subspaces,

C.B. C T a I ~ O H a p H ~ no~xoJ~ B

aOcTpaETHO~ Teop~g pacce2H~a. HSB. AH CCCP, cep. ~aTeM., 1967, 31, 401-430. 18. S a r a s o n

D. Invariant subspaces and unstarred opera-

tor algebras. Pacific J.Math., I966, I7, 511-517, 19. W e i n s t e i n A. Etudes des spectres des ~quations aux deriv~es p a r t i e l e s s Mem. Sci.Math. I937, 88. 20. A r o n s z a j n N., B r o w n R.D. Finite-aimensional perturbations of spectral problems and variational approximation methods for eigenvalue problems.

Studia Math. ,I970,

36, 1-76. 21. K a t o T. Perturbation theory for linear operators. Springer-Verlag,

I966.

Berlin,

N.A.Shirokov

DIVISION AND MULTIPLICATION BY INNER FUNCTIONS IN SPACES OP ANALYTIC FUNCTIONS SMOOTH UP TO THE BOUNDARY

O. Introduction. I. Two essential lemmas° 2. Some technical preparations. 3. The influence of the inner factor I on the rate of decrease of

4. Proof of theorem I. 5. Proof of theorem 3. 6. Proof of theorem 2. 7. Remarks about spaces

H P

8. Further generalizations.

4t4

O!ntroduqtion.

Let

N

be the Nevanlinna class of functions amaiytic

in the unit disc ~ [I] . Every function ~ , S E N ,admits the factorization in the form ~ = F, ~ , where F~ is an outer function and I f is a ratio of~two~ inner functions [I] , [2] , namely,

It: B~

being a Blaschke-product,

B (z)

N _[7_ -~

oL-z

-e(~-(o)nID)\{o}

,

tcl=~

,

and

where j~b is a real measure singular with respect to the Lebesgue measure on ~ . A function ~ N is said to belong to

415

the S m i r n o v c 1 a s s ~ , if t , ~ D in other terms if and only if the measure ~ is positive. Let X be a subset of ~ . We say (following V.P.Havin ~ J

or )

that the class X possesses the (F)-p r o p e r t y if for any function ~ , ~ X , and for any inner function I ~-I~c~ implies ~ ~ ~ E x . It is sometimes important to know whether a given class X has the (F)-property (for example in connection with uniqueness theorems, with the description of ideals in algebras of analytic functions and even in the theory of ~aussian processes). Classical Hardy spaces ~ P , 0 < ~ < O~ , [I] possess the it is the theorem of V.I.Smirnov [I~ . It is not very difficult to prove

(F)-property-

[2]

that the space

~A

of functions analytic in the disc

and continuous in its closure possesses the ( ~ p r o p e r t y . The first essential result applicable to a space of smooth functions (in a sense) was obtained by L.Carleson [3] • He has found a formula for the Dirichlet integral ~ l ~ q ~ from which the (~)D property of the class of analytic functions with bounded Dirichlet integral can be easily deduced. Further progress in the problem was connected with the notion of a Toeplitz operator. This method was proposed at first for some Hilbert spaces by B.I.Korenblum [19] and then in the case of "sufficiently regular " Banach spaces by V.P.Havin [6] . The clue of the method is as follows. Let X c ~4 be a space of functions analytic in and let a~E~ eO . Define the operator ~ (the T o e p 1 i t z o p e r a t o r

with the symbol

In some situations the space y

~

of functions analytic in ~ ' - ~

0J ):

is dual or predual of a space with the pairing

where the integral is somehow defined. Then the operat_oor T ~ is dual or predual of the operator of multiplication by ~ in the space y and provided the space y is invariant under such multiplication (this property is the base of the method)~ the operator

T~

is bounded, ,,m ~,H.I.~][X ~< C @ ""[[~I[X

. But if

I

is

an

416

imqer function and ~ / ] ~ ~ then ~ = ~/~ and the (~)property of the space X is established. The method of To~plitz operat be applied to the spa,

D~]

. A~,.

~4

' D~]

~,>~o ),.,

, 4<~p<~

,

[ J , [~], ,.~.a,.;<--?I ~ ( ~ ) -

~

~

. ,.~0. 0 < ~ < I

HP

4~(~l=o~O~(I~-~)[~]and to some " The method of To , for the spaces C A , b ~ spaces

~

, ~I,~

vertheless possess the

a,(B'; ~.__!(~-4/'e H ~ The operator T ~ , B(,

other spaces [~1, D~] ' D 4 ' ~'' ~tor is however useless ~(~8A or for the

• ~H (

~

F)-property *!

, which he-

So, for instance,

~=0 ~-kq-~ ')~

, is unbounded in

the space ~ , ~>11 The results of this paper complete the list of basic classes of."smooth analytic functions" with the (~)-property. We posshall prove, for example, that the spaces ~A and ~ ~ sess the (F~property. This fact was known earlier for ~ = 0 [2] and ~-~-~ ~ 5 ] • We shall prove also the validity of the ( ~ p r o p e r t y for spaces

where ~ 0;~... and ~0 is an arbitrary continuity modulus. These statements are contained in the theorem I (a more general assertion is announced in section 8 below) THEOREDi I. The spaces ~ A ' H~ ' ~ ' ~OO possess the (F~property for ~ = 0 ~ ° , . and for an arbitrary continuity modulus (~ The method of proof is, like in ~ 5 ] , founded on the analysis of restrictions imposed on a function ~ by the presence of an inner factor I in its Nevanlinna factorization *) Let us mention examples of spaces X , X C ~I , without (F~property. V.P.Gurarii has proved that the space ~i of all power series ~ 0 ~ ~ with ~ ]0~I ~ O0 does not possess this property, see also

[20]-

[22] .

417 (i.e. by the fact that ~ --I~I-IE H ~ ). The passage from ~_~A' to C A by ~ is connected with great difficulties and with some new effects. So, for X = A~ , A~ , H I or A the inclusion ~ I ' I E X implies the i n c l u s i o n ~ E X for every function ~ X and every inner function ~ . The next theorem shows that respective classes "of higher smoothness" behave differently. '

~

'

~+~

~A

'

'

then there exist a function ~ c X and a Blaschke product g such that '--~/]~ X , but ~--E~ X • For the validity of this theorem it is essential that we are dealing with a Blaschke product (not with a general inner function) and that the multiplicity of its zeros is less than .

The case of a singular function or that of a Blaschke product with zeros of higher multiplicity differs from the situation of theorem 2, as the next theorem shows.

THEOREE 3. Let X = A ~

,

~

'

~+t

, ~I

or

,

and l e t a)

then

~~

If

,

X

is

a

ingular function,--÷/

H '

;

b) if g is a Blaschke product, ~ / g ~ @ , and the multiplicity of the zero of function ~ at any point ~ - i ( 0 ) is ~ ~+~ , then ~ E ~ X R E ~ R K S . a) We note that theorems I and 2 prove the existence of a function ~ , ~ X ( X being one of spaces under consideration, ~ ), such that ~ T ~ ~ ~ (~ is "the inner part" of the Nevanlinna factorization of ~ ). b) It is sufficient to prove theorems I and 3 for spaces ~ only ( ~ ~ an arbitrary continuity modulus). c) The proof of theorem I will give more than is promised in its statement, namely the existence of a constant ~ depenwith ding only on ~ such that for any function ~

"'I~(~)(~)-~(~)(~)I~(~(I~-~I) tion ~

, ~/I

~H ]

~ ~

~

and for~any inner functhe following inequality holds:

Theorem 3 admits an analogous improvement.

418

I. Two essential

lemmas.

Here we shall state and prove two important facts necessary for the proof of theorem I. DEFINITION. a) Let us write - ~ - - - ~ E ~ ~ ) ( ~ 0 , ~ 0 an integer, 00 domain), if the function tinuous in ~-~ and

a continuity modulus, ~ C ~ a convex ~ is analytic in ~ , f(~) is con-

We shall write B A t ( ~ ) instead of B w A ~ ( ~ ) b) From now on C ~ K will denote various constants ding only on ~ . LE~ I*). Let~-~ let__ ~ C B ~ ( ~ ) contains ~+ ~ multiplicities).

be a convex domain with the diameter ~ ; , ~0 , and suppose the domain zeros of ~ (with the account of their

Then

PROOF. We shall use the induction. ~B

Ao60 (~''~)

' ~(@)-'- 0

,

If

~-----0

,

then

Let us suppose the statement of lemma is true for all 0 ~

~-J ---~ ~4 has

, and take a function ' ~(~) ~ ~

depen-

0

, ~(~)_

zeros in ~

~BA~.,(~) ~

~(~)~-.

~

,

. Put The function

and

~=4

t

@

*) i am indebted to V.P.Havin for this version of Lemma I.

419

~=o .-~

('zr ~,)"~÷~

(.i)-~C,., ~ ~, I

We shall now estimate every term inthe right hand side of the formula (2) of the form

de~ We assume further that l~i-@l~
~.~-,.~

i ~<

I~g-~l.~

T A

,,~

~,~

...... A

Then I ~ - ~ 1 1 > / ~

• We will A

and

.

(4)

*) In the inequality (4) (and further) we use the well-known property of continuity moduli: if 0 ~ ~C~ ~ , then

420

Case

2:+A
I¢~,-'~ I

.

~,e~

, ,#~- ~,A.

Then

4 c~

...

+

*[ii(~,-t~-(~ct

(~ca~+,

- -

~'I +

)

(#%-}(%)#

+

%% ~ %3

We have now

I-,k qO,~)d"

(5)

(6)

(7) The inequalities (4)-(7) give

(8) if we utilize the we obtain also

estimates

like (4)-(7) in the formula (3),

la-~j

(9)

The inmquality (8) implies c -4- ~ - - ~ e B A @~-~~ ) , and therefore the induction conjecture gives

421

(~o) At last,

and then we get for

and for

~ ~-0

~-~--< ~- {

l#~(~I< $. C~ and for

(10) gives

~ ~--- ~b

(8) and (9) give

and our induction goes.

2. a)

T,~,~

~e D

.

C~ f'~-~#'~,

$~-~-~ox,#~ :

•

Let S c ~ A ~

,

(I]))

~

,÷(@)=0

Then

#o(~ ) & # b)

Let ~-~

the domain

t ~ e O :I~-Zol < l*

O<~-
# (~,) = o the number Then

be

i~,i,I, depending only on

},

where

#eBA~v(~)' I a l ~>~1,ko, 4 'ana ~b

.

PROOF. The arguments in the proof of assertions a) and b) are similar and are like the proof of lemma I, therefore we shall describe briefly the main steps of the proof ofthe part b) only. It is sufficient to check the statement of lemma for any points

.) The sign

0

here and below denotes the end of the proof.

422

~4 , ~ E ~ - - ~ and then make use of theorem of P.M.Tamrazov D T] about connection between the continuity moduli on the curve and in the area. The Taylor formula gives

%

Then, like in the proof of lemma 1~ we apply to (11~ the formula of Newton-Leibnitz and consider the same situations as in (4) (7) taking into account the inequality

4-g I which holds because of the condition[65-~01 ~ ~ at last we obtain the necessary estimate for ~I)~bES~-~

, and

.

2. Some technical preparations. Here we shall establish facts, which will be used in the proof of lemmas 6-8, essential for the proof of theorems I and 3. DEFINITION. Let I ~

r]

~-Z d. ~+%~(~))

LE~{h~ 3 - Let % ~

J

~00

be an inner function, I = B S a Blaschke product, a singular function. We define

~~

,

I ~

H

be an inner f u n c t i o n ,

,

423

¢,%(1~F4)¢~ IT(~)I ~< 6c.~(fgl-~)~ ,

where the constants

0 < C$< ~1 < O0

PROOF. See .[15] LE~ 4. Let ~ D tion,

~= ~t

Then for J ~ - ~ l ~

@

,

are absolute,

H 03

,

[I ~

(%~p~c,I) :- o ~

I~I~

d

be an inner func-

, s'=v ~ ( ~ , ~/IIQ~) ! ) .

and for ~ = ~ Z ~

,..

(12)

(13) gives

e~Cf

.

(14)

The Cauchy formula for the circle ~ : I ~ - Z I = ~ (14) implies (12) and (13), @ L E ~ A 5. Let ~ ' 0 , ~0 , ~
together with , ~>~

PROOF. It is sufficient to consider the case ~ - - ~ Inequalities

,-U

. Then

only.

,

imply

We utilize now the estimate

and the harmonicity in ~ We get then

of the function e ~ I ~ - ~ - ~

'/d 12~ •

3. ~he influence of the inner factor I on the rate of decrease of ~ t • In this section we find the rate of decrease of Ill near

424 speo I caused by the smoothness of ~ and to the presence of the inner factor I in the Nevanlinna factorization of ~ . The idea to use in this connection the Taylor expansion of ~ at a certain point of ~ has been proposed for the first time by .l. ore bl

LE~'~ 6. Suppose ~ ~ ( ~ )

, ~#

ra++{~:l~-Zol=r},c-BeO~:l~(this

notation

will

be u s e d a l s o

in

I z~} the

next

, 0<~<~,

.

Let

us

denote

lemmas):

de~ Then

(15)

A]

"+' j~

(16)

~

PROOF, The Cauchy inequalities

+.r and the Taylor formula

go

+='1 @!

(~-1):~o

imply the following estimate (we suppose I ~(~0)I---~ ;36

)

-, because A ~-"400 Applying the Taylgr formula to the points ~ and T o and to the function~ ~(V), ~ = ~ . . . ~ ~ , we obtain the inequalities

~+)

'

A"

-v

Z

w(j.~,

~ = o,'/,..., ~, •

Again the Taylor formula at the points ~ imply (16). @

and

~

(17)

and (17)

425 LE~,I~ 7. Let the number ~ ~ from lemma 5, and

ber

be taken from lemma 3, the num-

A=

Let the number ~ ~ (it is the most important number for us) be defined from the equation

C t¢~-u~,¢~A

•

A~ +~

_~_

~

'

, -r~- H (~.

, IT ~hen for

~-~

, I~-~1~

~---~

~

be an inner

~'> T

"

the following ine-

qualities hold:

I ~oo~. We write J' = A ~'

"~

:

Dp4,...;

~

•

(18)

5~ and with the (we remark that A 6"< ~/~ 6 ) and with the function ~ we associate the arc , as in , the circle F ' the numbers ~ = O O ~ ~ ~-- ~ lemma 6. Let the point ~ a e F be such that ] $ ( Z ° ) J = % . We denote by F the outer factor in the Nevanlinna factoriza~ion of ~ Using lemma 3, we get

points

, ~0=(~-~)

'~.,~.e, with the numbers ~, f~ A

We write further Z 0 / ] ~ o l = ~ 1 ( C ~ ] Lemmas 5 and 6 imply

, 4-1Z°I=yl

"

(19)

~D

~U)

~ e ~quality C~,9.1~,~!. ~,~"*~----- O,a~-lows us to give (19) the form

426

(20)

5

and using the definition of the number

4

A

, we obtain

+-~I~ + C~f~ ~(3)

and

(2~) The number

A ~4

depends

only on

. Lemma 6 and

(21) g i v e t h e n

~,..< 2A ~

.<~

The e s t i m a t e (22) i s ~---0 • Applying

t h e r e q u i r e d i n e q u a l i t y (18) i n t h e c a s e the Taylor formula with the center at

~o

together with the Cauchy inequalities,

to the point

%

lem-

ma 6 and (21) we obtain remaining estimates in (18)(~=~...,~). • DEFINITION. Let ~ o ~ be an inner function, ~ = B ~ , where

i s a BlascM<e p r o d u c t ,

is a singular function. a) We denote for

o~E ~pec T

g

the m u l t i p l i c i t y

of the zero ~

,

if l~l< 1

b) We define also ~ ~ v ~ ( . ) = ~ , - r (.,~¢~t~/___. ' i f Z i s a Blasohke product w i t h l at most ~ zeros;

c) Let

E c~

be a closed s e t ; we w r i t e

427

B'E ~pecJ-~BnE where ~l~_j~[~ if ~

~~

.We write ~#.~Cll~,

~urtherZl de~lE ~[E

, we denote

RE~ARKS. a) The expression under the infimum in the definition of ~ attains its lower bound, so instead of "inf" it is possible to write there "rain". b) I f ~ 5 ~ 5 ~ C I N ~ and ~ A ~ ( ~ ) , ~ EH ~ , then the point ~ is a zero of ~ with the multiplicity at least ~+ ~ . Suppose not. Then the Taylor formula gives

p:

,

,

!

and }(2~(~)=bO , so for a oertain S > O ~ e ~) we ~et

I

I

ol -z I P

for 0 ~ I ~ - ~ I < ~

,

)

}(~)

(23)

The inner factors of the functions ~ ~oO and./~(~). ~ H being equal, the point % belongs to ~ p e 6 ~ ~-%)~ , which is impossible because of (23) - see [2] . Therefore , we have the inequality

c) If I is not a Blaschke product with not more than zeros, the domain i ~ " [~-~5[~(%)} contains at least

~t J

points belonging to speo I

.

428

be a~ inner function, ~ / ± ~ H

~

, ~

, ,~=~"~(~)~

0

Then the following inequalities hold:

~ (~,)

~(~)

where "1)=02#)..., 1~ . PROOF. We begin with the estimate (25). We can suppose without loss of generality that the function ~ is not the Blaschke product with at most ~ zeros (if it is we use lemma 2), so lemma I holds and gives

I , I But if ~ ( ~ ) ~ ~ , where the number A~ is taken from the lemma 7, the ~nequalities (24) follow from (26), and if 6~(~)~ ~ established.

, we can use (18) from lemma 7 and (25) is

We pass now to the proof of (24). The number A = A ~ is again the number from lemma 7. We shall consider two cases.

~.

~,(~) 7

The domain

~-A

~()-Itt"~. -{

• We write 6"-- ~t1,(~ ) ~-~: I ~ - ~ 1 < . ~ }

finition of the number ~(~)_. contains at most points of ~ p ~ G ~ . Therefore , there exists ~ , ~ such that the domain ~ k : { ~ ( 5 ~ . - - ~ has no p o i n t

of

~p~,I

,4_ZI <

. ~or t h i s

k~ -P ~(li+~) ~ ~=[~: I~-~l'
k

~y the de~

~+~

we denote

} '

Let ~(~)-----I I~/)e~j~.)) .

where 4(I;) is a Blaschke product with not more than ros. Let S0 = ~ / ~ ° Using lemmas 4 and 7, the equality

~ - ~ o- c~, ~

,

tl<+"~v

(7-)(~-e,

ze-

429 and taking into account the case under consideration, we obtain

We shall show also that ~ ~ B 6

points ~

~

~(_,C~.)

• Let us take any

. ~he~ ~

i (~)<;'}-e;(~,)

_<

b.-4

"Jr

/U ,1

4

The mean-value theorem~lezmma 4 and (27) give

leT}

tTj (q) ] ~
I ;,-;~1

and therefore (28) gives (we remind that I . ~ i - # ~ I ~

~-~

l~-Z~l ~ ( ~

@=O If 3 ) ~

6"

"

,

I÷{~(~)_~(~>(~;~)I.<~(I ~( q t), t c~ and we get at last

i~J(~-~0 (u]-< c~, ~(1~-~1), The choice of the domain ~ and (29) make it possible to apply le~ma 2 to the function ~oand to the finite Blaschke product @ .

430

~o/~eB~iA~(2)

• However ~o/~We apply now ~he Cauchy formula to the function F

So

~o= (~

~,~

)~

.

at the point

We get

-

~ (~;-~o)~ The domain ~

~/~ T~ ~

-

~ ¢ ~ ~(~;) (z;-~0)~÷

i s chosen p u r p o s e l y so t h a t i f

~

~-q-P- then

and so (27) and (..30) i m p l y

Now we find from (29),(31) and from the Taylor formula that

'Zo

(33)

-~= 0~..., ~-1.

0rr~5 ¢~"~(~),

The inequalities (32) and (33) accomplish our reasoning in the ease 1" 0 - ~ ~A ~._~, 2. ~(~)~ ~ ) . Put 6~ = (~) and be gin the proof as in the case I, but after the choice of the domain use lemmas 2 and 4 instead of lemmas 4 and 6. After that we can finish the proof exactly as in the case I

4. Proof of theore m I. i Let ~AS(~) , let I c H ~ be an inner function, F ~e~ ~ E HI . Using the theorem of P.~.Tamrazov DT] we shall restrict ourselves only to the proof of the inequality

I for ~I ~ ~ E sibilities.

-

FC~(z~)l~< C~o ~(I~-%1)

~

. We have to distinguish between two pos-

I.{~I-~{~/~(~(~i)

~) The numbers ~ ( ~ )

~ ~(~))

and ~ ( ~ ) w e r e

~). Then owing to lem-

defined in section 3.

431 ma 8 we have

! F(*)(~a - F<~)(%}! ~< q,0 c ~ ( l ~ - ~ l ) 2.,l~r~,l<~(~ example

'1

•

'f ~,,,,(z4~, ;d,~(~)).

~1.1.,(~.1)~/i~(~,~,.).

We c o n s i d e r

2.1. I~,-,',&i~ ~ (

'f

Supposefor

two su.bcases.

-,

'/

) . *gain o,,,ing to

lemma 8

I

-/).<

L

We knew that

,

, ,~.

'

l~r&l<~&(~a,

'.

W~,~,(~)

%(z~)~ ~(~,)

cal considerations imply

'fO0~,~(~)"

~

,

~(~)~(~)

too. The

geometri. The domains

y ~ < ~ (?_ ~ contain at most ~ points belonging to the GD~C I each and the d o m a i n ~ = ~ ~E~:/~-~.~<~(~)~ doe's not intersect the d o m a i n ( ~ ~ ) ~ ( ~ - ~ ; ) '. Therefore reminding of our case we have

(~4)

C~4~ We denote

Then(,,~)~.<~~

i ) v

The do.in ~ = [ ~ : I ~ - ~ , I ~

<~1-~J+(k,~)d contains at most fore at least one of the

~ points of ~ p ~ C ~

• There-

domains~k=[~:l~f-~l~
~i~'~I+(~+~)~}~ ~=0j...~ ~ this ~ we put ~ - ~ = { ~ .

does not intersect ~ p 6 ~

. For

l~-~I~l~4_~It(~+{)~}.

The definition of the number ~ 2.2 show that

and the condition imposed in

432

....~ . <

d~

~q ~< C~ o ¢ .

(35)

We a r e a b l e now to complete t h e p r o o f o f theorem 1. We denote

The function

most

~

~

zeros.

is a finite Blaschke product with

Let

~0~--- # / }

. From t h e lemma 4,

at

(34) and

(35) we obtain

Expanding the f u n c t i o n # i n the T a y l o r s e r i e s w i t h t h e c e n t e r at ~1 and u s i n g t h e i n e q u a l i t y (24) from lemma 8 we g e t f u r ther

Utilizing

(36) and (37) exactly as (28) and (29) we find that

But again as in lemma 8

F=÷/T= The function #o and every zero of the Blaschke product satisfy the conditions of lemma 2. Its application gives

F = :tlo/~ ( ~ -t~~BO~%6A~ , ) - "~ ~c~

and in particular inclusions

imply

Proof of theorem I is finished.

433

5" Plroof Ofl theorem ~. The statements a) and b) of theorem 3 f o r E ~ S ( ~ ) can be embraced by the following assertion: let I ~ ~ be an inner function, and suppose # ~ B ~ ( ~ ) ~ ~/Z ~I and suppose every point ~ 5~6~ is a zero of the function ~ of the multiplicity not less than ~ + ~ ;then # l ~ ] ~ p A a, The main steps of the proof of theorem 3 are similar to ~he corresponding steps of the proof of theorem I, so we shall only describe it briefly. Let ~ be an inner factor in the factorination of the function ~ , ~= FJ then theorem 1 implies F~ B~ ~ and the inclusion ~l/T ,~_..H1 implies . Then, if ~ e ~ p ~ l n , the multiplicity of the zero @ for I does not exceed the multiplicity of the zero ~ f o r ~ and if ~ and ~ are nonnegative singular measures corresponding respectively to ~ and ~ , then ~ and taking into account the conditions imposed on

[D)

we deduce

the inequality:

(38)

iZr(l.< Estimates (38), (28) and the Newton-Leibnitz formula give

(39)

Taking into account that ~(~) d~.~ ¢,T(,~)-~I~I,,~ (~) we can transform (39):

(40)

From now on t h e p r o o f

of

theorem 3 follows

literally

the proof

of theorem I with anon~cbsnge : the inequality (25) which was made use of in cases I a~d 2.1 has to be replaced by the inequality (40)• The domain ~ does not contain points of ~ e O ~ , V and therefore we put ~ - ~ and repeat reasonings exactly as in (28)-(29), and afterwards the proof ends because in the sent situation ~ ~ • @ 6. Proof of theorem 2. We begin with two technical lemmas.

434 Then

I-

] ~.(~,11 o,, ~ ~(~k .

¢, (n)[ c~ .'l]

nl 1,

..

~'~. (41)

)>4

PROOF. A straightforward verification. LEiG~ IO. Let us define the moduli I~I of functions ~ , ~--- ~; ~ . . . . in the following way:

(

-IOI)

,

~,,~

Then

where

0. =

~

~e

C $1~

~-2., o~

m~~+ 1

and moreover

is independent of ~ "'"

outer

. If ~ - ¢ ~ - - - ~ ~--0

U~['/J)~-[e

then

(44) PROOF. The assertions (43) and (44) can be checked with the help of a traditional method originating from ~ 8 ] • O We begin the construction of the counterexample needed in theorem 2. Let ~ ( ~ ) be the inverse function of OJ , ~(~) ~ ~ . We denote

435

~=

@

(~-% )~-~-

~k

B~(~)=fl ~ ~-~ k=~ I~kl ~-~k ~

c~k - ~

~-~

Now we can exhibit the desired function: e<)

#(~)=~

I-~(~) B ~

~(~)(c~/~--~') ~

•

(45)

Let us check that ~ . A ~ (~)) , #/'/~ ~ H t (and, therefore, owing to theorem I, ~ / B ~ A~(~) ) but ~ At first, the estimates (43), (44) and the construction of ~I) give

At

O0

~M,,t

SO ~ ~ A ~ ( ~ ) ) for every continuity modulus ~ • Further, every point e~@ is a zero of the function ~ of the multiplicity ~ , what one can see from the formula (45). Let us estimate the expression

v-~=(~]~ )(~(.L~)- ( ~E )c~(~$ Taking into account the definition Of ] ~ we get

0, k ""k~fJ : hence

O,

0 ~< ~,<~ FI,

). and ~.~ f o r k=ik'l~

436

For

~<~

~

the estimates (41),

(43),

(44) give

(47) but for ~ - 0

(41) shows that

.(48)

Z The assertions (46)-(48) imply the inequalities

"~=~3, However i f the function ~ then for a sertain constant

wou

0

(49)

....

. on= to

the inequalities

= C co ( ~ ( ~ - ~ ( ~ * ~ )

))= Cz- ~C~+~)

would be true contradicting (49). Theorem 2 is proved.

•

7. Remarks about spaces H P The method used in the present paper permits to check the , ~ and (F)-property for the spaces I ~P , ~ < p < ~ to prove the analogs of theorems I-3. Namely, the following resuits are true. THEORE~I 4. There exist a function ~ , S e H ~ , ~z~ . a) if ~ e H °° is a singular ~ function, ~ / S e H i , then

437

b) If ~ is a Blascke product, ~ / B ~ ~ 4 and the multiplicity of the zero of the function .~ for every I~e~-4(O) is not less than ~ , then { ~ E H ~ ~ ~-~--~3~ .... It is not clear whether the assertion of theorem 5 holds 1 4 for the spaces ~ , ~2/~ (for the space M s these assertions 4 hold what one can deduce from the (F~property of H ~ ).

8. Further ~eneralizations. In this section we announce results concerning (~property in domains with angles. These results cover (even in the case of the disc) the statement of theorem I but their proof seems to be too long to be published here. DEFI/~ITION. Let ~ be a bounded Jordan domain, a function ~ be analytic in ~ and bounded, I ~(~)I--< C . Let ~ be a conformal mapping of the unit disc ~ onto ~ . Then M ( ~ ) ~~. S(~(~))E H ~ . We write the Nevanlinna factorization for ~ :

H= where

~

is an outer function,

We define the o u t e r p a r t of the S by the equalities:

where

~-~

I

is an inner function.

and the

is the inverse mapping of

~

i n n e r

p a r t

.

THEORE~/~ 6. Let ~ be a Jordan domain whose boundary consists of infinitely smooth arcs ~ " " ~h. and the arcs ~ and ~I (we set ~ I = ~ ) form t'he angle qJIT/~I,- , ~--_~;,0,~;~are positive integers.Let an outer(in ~ ) function F satisfles on ~ the A i -Muckenhoupt condition:

Irl z

z

'

Let X be either the space of functions such that

or the space of functions

~

analytic in ~

~ analytic in

such that

438

where

v

t0

is an arbitrary continuity modulus, the point

[~:

= and

8~)~ ~ sesses t h e literally

,^~ ~$~ (F)-proper~y

like

fS

(~) ~ )

in

=D

watch zs defined

t h e case o f

,

. Then the space

F=4

the unit

, we g e t

for

X

pos-

the domain

disc,

theorem1.

RE,LaRKS. a) The assumption concerning the angles is essential: if their sizes are q C / ~ , ~ c ~ < cO being not all positive integers,then the conclusion of the theorem fails even for ~ ~ . b) An example of the function ~ with A-Muckenhoupt condition: let be a nonnegative measure on ,~(~)<~0, V ~({~})<~ for any point ~ . Then the function

is an outer function satisfying the Az~uckenhoupt condition (the domain ~ is as in theorem 6). I am highly grateful to V.P.Havin for careful reading the present paper and its correction in mathematical and linguistic aspects. References. I. H,E.I~ p H B S J~ O B. F p a a n q ~ e

U2~. ~ Z ,

2. K.H o f f m a n. Banach ce-Hall,

CBO~OTBa aHaaHTnqecEI~X ~yHE--

YHTT~, I950.

Engl. Cliffs,

spaces

N.J.,

of analytic

Math. Z, 1960,

Prenti-

1962.

5. L.C a r 1 e s o n. A representation integral,

functions,

formula

for the Dirichlet

73, N 2, 190-196.

4. B.M.K o p e H 6 a D M, B.C.K o p o a e B ~ 9. 05 a s a z a T ~ e c zi~_x @yHE~Z~X, perya~psHx B zpyre z rza~Z~x Ha ero rpaHm/e, MaT.BaMeT~z, 1970, 7, } 2, I65-172. 5. B.M.K o p e H 6 a D M. 3 E c T p e M a z ~ e COOTHOmeHAH ZJ5~ BHeE~MX ~yHEuH2, MaT.SaMeTEH, 1971, I0, ~ I, 53--56. 6. B.H.X a B ~ S, 0 ~aETopHsaH~Z aHSaHTm~ecEEx ~yHEu/~, rJ~AEHX BHJ~OTB ~0 PpeHHII~, 3anzcEE HayqH~x ceMHH.ZOH4, 1971, 22, 202-

-205. 7. ~ . A . ~ a M 0 S H. ~eaesHe

Ha BHyTpeHHDD ~ y H E ~ H D B seEoT0pHx

439

npocTpaHcTBaX ~ym~/m~, a H a 2 H T ~ e C E H X B Epyre, 3anacEH Hay~HNX ce~H.Z@&[, I97!, 22, 206-208. 8. C.A.B H ~ o r p a ~ o B, H.A.~ ~ p o E o B. 0 ~aETopHsasHE a ~ a ~ T z ~ e c z z x ~ y a ~ m ~ c npOHBBO~HO~ HS H P , 8an~cxH Hay~m~x ce~mH.AOMM, I97I, 22, 8-27. 9. B.M.E o p e ~ 6 a I0 M, B . M . ~ a ~ B ~ m e B C E ~ ~. 06 O~HOM ~aacce c~m~a~m~x onepaTopoB, c ~ s a m ~ H x c ~eam~ocT~o aHazaTz~ e c ~ x ~yszum~, Y~paHsc~m~ ~aTem.~ypH., 1972, I4, ~ 5, 692-695. I0.B.3.K a U H e a s c o s. 3a~e~aa~e o E a H o s ~ e c E o ~ ~aETOp~a-L~E B HeEoTopNx npocTpaHCTBaX a H a x ~ T ~ e c E H x ~yHFJ/~, 8anHcEz HayqH~X ce~mH.~0~, I972, 30, 163-164. II.M.R a b i n d r a n a t h a n . Toeplitz operators and division by inner functions, Indiana Univ.Math. J.,1973, 22,NIQ523-5~% 12.E.~.~I ~ H ~ ~ ~ H. i~Jm~ze ~yHEIIZH ~ta n ~ o c ~ x MHo~eOTBaX, AoE~ AH CCCP, 1973, 208, ~ I, 25-27. 13.~.A.~ a M o ~ H. 06 0rpaHH~eHHOCTH O~HOrO EJ~aooa onepaTopoB, CBHBaHHNX C ;~eJH4MOOTBD SHSJIZT~-qecI
a h a n e.

R~th. Soc.,

1974,

est approx

atio

in

C(T)

,

80, N 5, 788-804.

15.H.A.~ H p o ~ o B. Zu~eaJm ~ ~azTopnsazzm~ B aare6pax a~aJmT~qecEl~X { y H E L ~ , rJ~a~EHx BIIaOTS ~0 rpaHl41~, T p y ~ ~S4AH, I978, I30, I96-222. 16.~.A.[~ a M o ~ H. 06 o~Hom Enacce T~II/mideBNX oHepaTOpOB, OB~-saHHNX C ~eal#4OOT~D aHaZaTi4-~eoEzx (~yHEE~, ~yaEL~.aHaJms ;[ eFo HpM21., I979, IS, ~ I, 83-84. I7.H.M.T a ~ p a s o B. KOSTypm~e ~ TeaecsNe c T p y ~ T y p ~ e CB0~CT-Ba PO~IOMOp~HNX ~ y H E H ~ EOMIL~eKCHOPO llepeMeHHOFO, Yoneym fC~TeM.HayE, I973, 28, ~ I, ISI-I6I. 18.B.M.K o p e H 6 a D M. 8aMEHyTNe m ~ e a ~ ~ o a ~ a "A " , ~ysz~. aHaa~s n e r o npHa., 1972, 6, ~ 3, 3 8 - 5 3 . I9.B.H.]? y p a p z ~. 0 c ~ T o p ~ s a u a ~ a6COJ/DTHO OXO//~L~HXOS p ~ o B TeAaopa H m{~erpaaoB {ypBe, 3anac~H H a y ~ x ce ~#LR.JIOMM, I972, 30, 15-33. 20.H.A.I]] H p 0 E 0 B. He~{oTopue CBO~OTBa npzMapH~x m~eaaoB a6coamTHO C X O Z / ~ C ~ p ~ O B Te~aopa ~ ~/~Te~paaOB ~yp~e, 3 a n H c ~ Hay~H~x ce~H.Z0~M, 1974, 39, I49-I62. 2I.H.A.~ z p o ~ o B. AeaeH~e Ha BHyTpeam0D {ys~mao He Mea~e~ Enacca P.~a~ZOCTH, A o E n a ~ AH CCCP, I98!. 22. J.M.A n d e r s o n. Algebras contained within , 8a[H4C-~ Hay~s~x ce~1~.~O~M, 1978, 81, 235-236.

H

A.L.Volberg

THIN AND THICK FAMILIES OF RATIONAL FRACTIONS

Preface I. Introduction. Part I. Thick families. 2. The first step of the construction. 3. The second step of the construction. 4. The proof of the main theorem in the particular case. Part II.Thin families. 5. The proof of theorem 2. 6. Compactness of the inclusion map and related questions. Proof of theorem 3.

441

Preface The problems investigated in this paper can be formulated as follows: given a finite positive Borel measure ~L on the real line ~ (or on the unit circle T ) and a family of rational

(or A C D oLe~ ~ ~ C ~ ~I <]~), which properties of this measure and family will imply: I) the completeness of the family ~ in the space ~2 (~) 2) or the equivalence of the norms l],IIh~(#)andII.IIL~(~) on the linear span ~ of the family ~ A ( here D'~ denotes Lebesgue measure on the line ~ or on the circle T ) 3) or the compactness of the inclusion map ~ ~ - - ~ L~(~), i~ = ~ , ~ c ~ , ~ being endowed with the - ~ ( ~ ) -nOrmo If the first possibility holds we call the family l~Athick with respect to the measure ~ ° If the second or third possibilities occur we call the family ~ A thin with respect to the measure ~ , (we are interested mainly in the case of unbounded dens~tYthi~s~pa ;erthhl~fofLwl USseathed~sh~r~ ~th~n r). p i b the measures with one "singular" point and some properties of regularity. Thin families are described for the measures ~ with the bounded ~ensity with respect to Lebesgue measure ~ . The paper is subdivided into two parts and six sections. I contains main definitions and the statements of three

442

main theorems. We point out the connections with some other problems. We s_lso discuss basic concepts of the proofs of these theorems. The proof of theorem I is based on the construction of a special domain in the complex plane

~

. The construction of

this domain consists of two steps. In

§ 2 the first step of the construction is made. This

constz~action follows the scheme of the articles

~]

and

[2]. In

It

this paragraph a half" of theorem I is proved. §3 contains the second step of the construction. Here we also finish the proof of theorem I. In

~4, we prove theorem I in a very particular case by an

essentis~lly different method (using a tecb~aique of the quasi=analytic classes). In

~5, we consider thin families of rational fractions and

prove theorem 2. In

~6, we prove theorem 3. Here we present some propositions

concerning compactness of the inclusion map

~

and related

questions The results of this paper were partially announced in

ACE~OWLEDGENLENT. The author wishes to express his deep gratitude to N.K.Nikol'skii for formulating the problem and for valuvv

able advices. Thanks are also due to S . V . H ~ s c e v

for advices

which shortened some proofs and to P.Koosis who kindly supplied me with the prepublication version of the paper [1]. Finally, the author is sincerely grateful to V.P.Havin for readir~g the manuscript.

I.

Introduction

A. Thick families. Let

~

be a positive finite Borel measure on the real line

. As we have mentioned before we consider or~ly a special class of measures with one " s i n ~ l a r " DEFINITION. Let ~ = ~ D ~ , moreover,

point. ~E~

A (~)N

in a neighbourhood of the point

t: ° =

where

t 0*

d: -Oand

~ ~

~ 0

( ~ )

~

443

~-*0+

~-*0+

~ ~>

"

) is Then we s h a l l s a y t h a t t h e m e a s u r e j ~ ( a n d t h e f u n c t i o n regular. Let __~v A be a family of rational fractions ~--~-~_~ whose poles lie in the S t o 1 z d o m a i n K '~e~' ~= X~ ~1~ ~ : l~j~'l"~] } . Recall that ~ - ~ n o t e s the linear span of Now we are in a position to formulate our main theorems THEORE~I 1. Let

(1.1) If

#>o ~ (1.2)

J

then

~,~(Itl) d,t, < oo t~0 Itl "---- O0

(~.3)

On the contrary if

I

then

t
,(Itl) t =

(1.4)

oo

I1;I

oo

.

(1.5)

The fact that the measure ~ has the support on the real line is unimportant. The analogous theorem is valid for measures ,~ ~ , ~= ~ v . Certainly, we suppose that the function ~ ~ ~(~$) is regular and the family ~ has the poles in some Stolz domain ~(~) in the unit disc 0=[~E~" I~I ~ ~ } (let us recall that the S t o 1 z d o m a i n ~(~) i ~ D is the interior of the convex hull of ~ } and the circle ~ ; I~I= ~ v } . To obtain the statement of the theorem for the circle ~ we must only substitute (1.3) by

: -Ixl)

CO .

(I .3a)

444

Note that from (I .1) of the theorem we have

In the opposite case the following theorem(see[4]) gives a complete description of thick families ~ ° In this theorem there no restrictions on the set of poles A , except A C ~ \ ~ THEORE~ (see [4] ) • If

are

•

From a different point of view the rational approximation for the case (1.7) was investigated by G.Tz.Ttunarkin L5], L6]. The basic method of proof of our theorem is the construction of a region in ~ with some special properties. This method is similar to that of P.Koosis ~I]. Now we shall enlist some useful remark s. REMARK 1. It is clear that if ~_/k is thick with respect to ~and ~i~ a.e., then ~ A is thick with respect t o g 1 ~ . R ~ 2. If the function ~ is such that --

and the remark

condition

~

1~I,t~ - - - - - ~

(1.1)

1 we c o n c l u d e

is

ful£illed,

V ----- 0

(1.8)

then from theorem 1 and

that

B ~ it turns out that in this case we may deduce this result with the help of more direct methods. (See §4, which treats this subject). We shall need some notation below., i_lf~ is a measure and ~ is a ~--measurable set then ~ ~ will denote the measure de£ined b ~ ( ~ l F)(X) d ¢ ~ ( E NJX) for every measurable set . REMARK 3. We may consider more general systems of rational fractions.f_ Namely ~.A,~ ~ I~) ~ : ~eA, 4~<~<~(~)) . Here I~ is an integer-valued function. Theorem 1 is still true for these f~nilieso REMARK 4. One can suspect the restriction on the poles in theorem I, n a m e l y ~ c ~ , is unimportant. That this is not the case im mhewn by the following

~

445

proposition. PROPOSITION. Let ~ be a positive finite Borel measure on the circle T . If for every sequence ( A ~ ) ~ , ~E D~ ~4j ~ V~~ ~ we have

then there is a positive number

8

such that

~[4} being the one-point measure. ~PROOP. Suppose (1.9) does not hold. We construct a sequence p~-i.eo

It

is

Blear

that

~C{4})-- 0

withont

.

:et

loss.

of

~cA) ~

We choose the points

A~

generality

we m a y s u p p o s e

that

a({-A}-'ll,~{~). such that the vJec'tors

form a Riesz basis in their linear span in ~ ( ~ ) . ~irstly we show that we may choose ~ such that ~ - - - ~ " in the weak topology of ~(~) . To show this we note that

t~ ~upp ~

(~. lO)

since (1.9) does not hold and note also that

for

Rt,-

a,e,

point

~

. Indeed i t

is sufficient

to prove

e$~6£{P~E

?or an a r b i t r a r y

p o s i t i v e number g and every p o i n t there exists a neig~hbourhood ~ of this point such that . The family of intervals ~ ) ~ q ~ covers the set F and it is well-known that there is a sequence of disjoint intervals C ~ ) ~ 4 such that

j~(I~}~ II~l E~Ua

,

,~:

IT,ll ~ ~'~

,~ for a n arbit-

(1.11) is prove~. Prom (1.10) and (1.11) rar7 6, 6:'0 we deduce that there is a sequence ( ~ ) ~ I ~IT~ ~ = I such that

446

~---~ eO

(1,,,12)

I f we prove t h a t f o r every intez~¢al ~ C

~---* oo

(1.13)

I

then we shall show that ~ - - ~ Pix an arbitrary positive number ~& , ~ ~ such that

~'(~&)

in the ~ea~ topolog~ of

~ <~).

and choose an open interval

"~< S ~

(1.14)

(remember that we suppose that ~ ( [ ~ } ) = 0 ). Now we have the following chain of the inequalities

g~A,~} I I

"g- A.,

d~c~>

T

'I .0(~),

4

.&(~,D'y~Cas) ~/z

~<

o(~)

Taking (1.12) into account we see that (1.13) holds. It is clear that we may choose a subsequence /~ ~ such that

So (6~K)k~

form~a Riesz basis in its linear span. @ B. Thin families Let HP denote the standard Hardy space in ~ a~d let ° If @ is an inner function, then K @ , where the bar stands for the complex conjugation and U P ~ i { #~ ~P : ~(0)~- 0 ) • Let ~ be a positive finite Borel measure on the unit circle ~ and let ~ be Lebesgme measure on ~ . The following questions are interesting from many points of view (see the papers cited below). We may ask: what measures have the property

HPN~-$

q- C qr

I

T

447

where ~ ~e and the constants C~;65) 0<~4 < C~ ~ O0 independent of ~ ? On the contrary: f o r what measures ~

are the

operator : , = ÷ i s compact ( i . e . the norm II'U L~(~) is essentially weaker than II'll~ ( ~ ) )? Note some previous well-known results. When p__~ ~ , ~¢3cp -~-~i , 65,0, the first of the above questions has been considered by B.P.Panejah [7], M ' V.Ja.Lin ~ ,

V.N.Logvinenko and Ju.F.Sereda [10], V.~oKacnelson [1 1]. Here the first question is equivalent with the following: for which measures ~ on the real line ~ the norms ( ~ I ~ I ~ ) 4 / ~ and ( ~ I ~ I ~ ~%)4/Z are equivalent in the space of entire functions ~. of exponential type not exceeding ~ / ~ and such that ~ I ~ 1 % 1 ~ < CO ? A complete a ~ w e r has been given (in ~-dimensiona~ case and for every ~ ~ 0 ) for measures shall answer the above questions f o r ~ - - W ~ j and for an arbitrary inner function ~ . The answer will be gi^ ven in terms of the harmonic continuation W of the function W into the unit disc. Some results concerning- this problem with a measure ~ of more general form may be found in [9]. It is interesting to note that in [12] D.N.Clark showed that for every inner function ~ there are many ~-singular measures ~ with the property (1.15). If ~ is a Blaschke product whose zeroes are ( ~ ) ~ then ~ 0 = 6p6L~C~)~ll (here ,pal% ~A~ denot'es the c 1 o In this case the property (1.15) and the fact that the family ~ is thin with respect to the measure ~ are equivalent. Now we state our theorem about thin families (more generally, about the equivalence of norms in the space ~@ ). THEORE]~ 2. Let p e ('~,003 , w~l,°°(~,},~) ,W~/0, iijfl,=Wi~yH,. The following statements are equivalent: a) the norms II'II~(~) and II'll~.~(~) are equivalent in the space K @ b) if ~, ~ ~ , ~gF~ W ( ~ ) ~ 0 then~[~(~)l~-~ • ~^ --~ ~ ~'~ A similar theorem is true if we everywhere replace the disc by the upper half-plane ~ + . Then the statement c) for %q~ , 0=~%~, ~ > 0 , transformS to the property of the relative density of the set E and this is the criterion found in [8]-[11]. The problem of compactness of the inclusion m a p ~ : K @ - ~ ( W ~ )

448 is considered in ~6. First af ~Ii we remind a definition. Let ~ + be the orthogonal projection of ~ ( ~ ) onto ~ and let ~ _ : I I being the identity operator. Fora bounded measurable ~ , the T o e p 1 i t z o p e r a t o with symbol ~ is the operator T ~ on ~ Z defined e~ ~+~k and the H a n k e I o p e r a t o r with the same symbol is defined by ~ '~'~.t~. ~k~ k ~ N $. By Fatou's theorem the algebra--~ °O of all bounded analytic functions on ~ may be considered as a closed subalgebra of L°°(T) , the algebra of all essentially bounded Lebesgue measurable functions on ~ . For a given $ in ~ let ~ [~] denote the uniform subalgebra of ~ generated by H ~ and'@. -Go o An interesting example is the algebra M [ ~ ] = H + C(T)~ where C denotes the space of all continous functions on V . THEORE~ 3. The following statements are equivalent: a) the operator ~: Ke ---¢- I x,(w~D@) is compact ; b) the operator ~ [ ~a is compact;

byT~k

e) the operator Hw~ H ~ is compact. In conclusion we introduce some additional notation . If ~ is a Hilbert space, T is an operator in ~ , T: ~--~ and E is a subspace of ~ , then the symbol T I ~ denotes the restriction if T onto E~T IE : E--~ • We use the symbols C~ C~ C~ ~3~ 0 4 for constants, moreover the letter C may denote different constants even in the same inequality. PAINT I 2. The fi,rs,t ,s,,tep of,,,,,,,,,,,,,,t,,,h,e,,, construction We begin to prove the sufficieneg in theorem I. Remember that the function k is regular and the condition (1.1) is fulfilled. Let the family ~/i be not complete in L~ ( ~ ) ° Then there is a function e , pELI(~) , p=~0 , such that

Let

449

. The functions ¢ + and ~ _ bein ~ + . It is well-known that ~q~(t)~(t)=¢~(~)-¢_(~) a.e. in ~ . Now it is almost obvious that ¢ ~ 0 Indeed, if it is not the case T t h ~ n ~ h a ~ , ~ ] ~-(t)I ---~ ~ I ~ ( ~ ) l " l~(~Jl and so in the upper ha!f-plane

~e

long to the Hardy space ~

~,

~'itq "r- ~, ~',~I A.

~ + t 9,

~_.Y~ = since @ _ ~ H 9"

co

. But this implies that ¢__~ 0.. . Thus p ( t ) ~ ( ~ )

i~ a tcontradiction h e r e f oand r e .

=

O

a°e. This

~.~is a non- zero function in

From (2.1) we see that the zero-set of @ + . Let

Let us now introduce two auxiliary functions

contains the set

'~/1~ U~ :

]

It is clear that

(2.2)

,j,,~/e

and that the functions "4 > ~ tiable and

oled:

are twice continuously differen-

We consider the curve T ~ ~ = ~ + ~ : ~--~-~(~)} and estimate ~ } Q ~ ) : ~ ~I¢$~Q~) ~ ~=4~ for ~=XJ~(0~)E~ and for 06 sufficiently small:

450

0

-00

Now we estimate the gradients of functions

(~= ~,~ 1

on the c~r~e

~

:

0 ~<~.

The same e s t i m a t e i s t r u e f o r ,

~ ~*

C~)

' ~ T

. There-

f o r e on T (2.4)

LEN~A I. Let Then

W

is

W

be the conformal homeomorphism of the up-

smooth up t o t h e boundary and i s

distortion-free,

i.e.

for

every i n t e r v a l -'7, ~ C ~ . PROOF. The function ~[(~) belongs to the class -C ~ It is easy to see that the angle formed by the tangent of the boundary ~ of ~ and by the line ~ considered as a function of the arc-length parameter on ~ is continuously dlfferentiable.

The lemma now follows from the Kellog's theorem

~ 3 Let , p.411] From the pre~ } ~ ) ~. ( WId¢ ( ~ )~) ) ~ E ~Wvious lemma and the inequality (2.4) one can easily deduce the following inequalities [ C

(2.5)

451 for 06

small enough, j = 4, $, :Prom the definition of the functions ~: it is clear that they are bounded outside of some neighbourhood of the point O, We denote the set ~4/'~(~I) by ~ again. Without loss of generality we may suppose that ~ A . Now we introduce two functions :

"¢j ~'* 4

~

;

"

The inequalities (2.5) imply that the functions and analytic in ~+ and I~4~(~6)I~<~(~06)

for 06

From the definition of S~

~j

are bounded

sufficiently small,~-~_~.

it is clear that

The aim of the above reduction was the construction of the function F ~a~ C~,1+,~9,,,)(.~4_,~3a )=,/4%~t which has the following properties : i) F is a bounded analytic function in ~+ 2 ~ 0 % ii) F C A ) = 0 for A ~ , where ~ is a set in Stolz domain

ii~)~l;Ff(~)l~ A t

F coc) l

for 0 C E ( - ~ o g )

I

i. l

l, lJ &l, l

Here ~ ( ~ ) ~ ¢ ~ k ~ C C ~ )

b) that C ~ ~Tt ~~ v w

)

has the following properties:

f

The functions ~ ~, ~4 are from the Without loss of generality we replace

R O ~[~ii I

and for some constantA9

. ~

statement of theorem 1.

~(~ot)

, %Cgo~)

by

. To prove the sufficiency

of the first part of theorem I it remains to prove lemma 2 below. The proof of this lemma is similar to the reasoning in [I], [2]. So we shall prove it omitting some tedious technique details.

452

LE~S~A 2. If the conditions i)-iv) fo~ the functions F are fulfilled and

and

@ then the z e r o - s e t A o f

the f u n c t i o n F

has the property

PROOF. Consider the set

Then and

~o=~¢

since otherwise I

F<~>I-

I~l F~t~l~i ~-I@

0

o

and this- contra~_Icts ~). For every point ~ o ~ ~ 0

~

=-~

for ~e(O,~> ~

we construct the interval ~From ~de~

iii) and ira) it is clear that

IF~>I~<~, L"e~ ~ % #l ~~ , , %J

~e U 6~o •

<~.8>

I~. ,

be the union of finitely many disjoint intervals, I ~ . - - 2¢.11,#( )u ~ , the intervals bein~ enumerated from right ~o left. "~

Let ~ = [ 0 ) ~ obvious :

\ (~,'

IF(~>)~-~(~>,

. The followin~ inequalities are

(2.9)

Here ~ stands for the principal value of the argument which varies from ~ to ~U . Let

.......

~) We shall say that s imul t ane ously.

if these integrals diverge

453 Thus from (2.9) we have

I Let

~

denote the harmonic conjugate function ,v cO

K

I

~

'

Without l o s s of g e n e r a l i t y we may suppose t h a t t h e . f u n c t i o n is SO small that ~ l ~ [ ~ c ( ~ ) ~ ~-~~(~+~) . We introduce the auxiliary function

The properties of ~

~

are the following :

are real. It means that there is an analytic continuation of the function ~ across ~ from ~+ into the lower half plane ~_ . This new function will be called ¢ ~ also. We have

~ is analytic in the domain ~ .~] ~ - U ~ . Let ~ ~enote the boundary of ~ , )~ , ~= ~ ~ ~ . By the reasoning similar to that o: e can choose a point in the first (from the right) interval so that I¢~(~)I>~ ~ 0~ where the number ~ is independent of ~ o We should like to apply Jensen's inequality to the function ¢ ~ . That it can be done was shown in []] and [2]. We write this inequality in the following form:

~

I

But the first integral in the right hand part does not exceed the constant c (since F is bounded) and ~ I ~ ( ~ )~i ~<~ 4 So we have

454

~ere

gw~(.,g)

denotes

the

harmonic

measure

of

the

aoma:Lm~

corresponding to the point g ~ andG~(.,g ) denotes the ~b Green function of this domain Now our aim is to prove that J ~ d ~ ; ~.v oj@(°) {) is bounded independently of ~b : ~

r~

e~

r. ~_t

jd,t=

is the first (if counted from Here v ~ ~ the right) interval of the set ~ . Without loss of generality we may suppose that the first two intervals ~ = ( G W ~ 0 ) and ~ = Q0~, ~I) of the set ~ become already stable, independent of ~b ° We may suppose also that I ~b[ ~ ~ . Then

do

do c~

iI'V,¢,l(f ~to

q

m

dt ~
£)+cf~~J~l

4

,~

ao

1 ~o'-,:x:, I

~ e...,< ~ .

Here d 0 O ( ~ )

denotes the harmonic measure of the

Thus it remains to estimate Now introduce the notation

It is clear that

domain

(~) ~

. ~

~C,R,(,,~ ) ~ C~((~L), CT'G,I~(k~)=-j

::10-1; -)

455 t~'~-+)$

< 0

(2.12)

for

Taking it into account we see that

Now let us recall that (~'J= Ue~

, ~O~:~'o and { j 6 o = ~o-~~--~t/~(~o)) ° . We denote by L l o the interval with the centre OG O and the length 5 z t is a w e l l -

...~_(9~0-~1/$(~0),

t¢5ol-

known fact that there is a sequence ( ~ ~ ; such that ( ~ . ) are disjoint and ~.fCU ~ L~} ~ 0 ~ ~ then n ~I

,

~E~o f . If~}

~. But the function ~ that

is increasing and

I@ 11- A

(2.13)

. So we obtain

(2.14) $

f Now we take into accotmt (2.t2) and see that the f u n c t i o n ~ ~.(',#) decreases on every interval -~f~ . Therefore it has at most one zero on this interval and so

From (2,13)-(2.15)

we deduce that

f But f o r

t ~

Here ~ ( ~ )

j ~ ' we havre~

denotes the Green function of the d~mai~

So from (2.16), (2.17) we deduce

456

f

(2.18)

Now we rewrite the inequality (2.11) in the following form:

~.

r~

(2.19)

o. We note that the domein£

where

~('

=

~{)

U<~

~

increase and

As the harmonic measure of the domain

~

gr~ i~ its ~reen ~unction. ~here~ore f~o~

~f

(2.19) and ~atou s lemma we may deduce that

(2.20) !t,_

,iteA

r

where [-- de:~ ~Z)f] [O,o&] It is clear from the construction that

~(~>) for ~ F

(see (2.9) and ivb)). Thus (2.20) implies c~

(2.21) o

LEN~A 3. Let the function

be increasing,

0

~(OJ ~-- O~

(A)

C

J

eCm) ~ m = , o o

(B)

o

and let an increasing on [0) ~ ] be such that

¢

x~

continuous function ~)6J(0)= 0~

457 .

for some positive number ~

Then

o ~-*0

(/))

o

~-'

~'4' ~'~

for every pair of positive numbers

satisfying ~ ÷ ~ :

= F~'dPROOF. Without loss of generality we may suppose that

- -

T'

~

(0, ¢-) •

So the function decreases on C0, ~) . Now from (C) we see that (D(~) ----~"0( ). Applying (C) and the integration by parts combined with the facts mentioned above we see that

£

(cO

oo o

It is clear that (A) implies: S ( - ~ - - 0 ( ' ~ - ~ ) and therefore it is clear that the integralU~ ~

W (~) ~¢~' (~0(5) 064

converges. Let

for e v e r y ~ ~(~O) ~

O~

o ~ Then

t o

and so

o

W(~) ~

0k £--~

"

Another application of the integration by parts gives the chain of inequalities : £ ¢

• ~

I;

f e.C~)wc<)-

Thus ~.f 8~W~')'~,~
o T

~z~+Y,

(~D)is

false.

, and we have

458

This contradicts (B). @ Let ~ be a small positive number. Taking into account the inequality (2o21) and lemma 3 we may choose a number ~ so small that the following inequality holds:

0

and t h e r e f o r e ~ ~ .

(2.22)

0

If we prove that

~m(X, ~,)m c(J~A} ~/~ for the points A from the Stolz domain ~ inequality (2.20) we shall obtain that

(2.2~} , then applying the

Xej~ and so l e n a 2 and the s u f f i c i e n c y of the f i r s t p a r t of theorem 1 w i l l be proved, NOW we prove (2.23).~Let4#~¢--~/ ~ U ( O ; O v ) , ~ be i t s Green function and let ~,~i-x-. ~ be the domain homothetic to ~ o It is clear that the following inequalities hold: #,

4

O

o i 6

....... !

not depending on ~ appears where Xl°= ~x , the constant from the application of Harnack's inequality and the symbols ~ 4 and ~,-~^ denote the Green fnnctions of the regions ~# and ~, ~ C ~ ~ respectively. Now if ~ ( ~ , ~ ) P ~ I~+* i ~-~-i~l--~. ~ is the Green function of the right half-plane, we may rewrlte the latter integral as

j=oJ

0

459 Integrating by parts we obtain ( ~ d6~A/~,-- ) I

(2.24) 0

It is clea~ that

l{e,~ ~<

o~.<-~-T

%,

<

o
(2.25)

,~

since ~ = d,, and ~ K r . LE~MA 4. The condition (2.25) being satisfied) the following inequalities hold

k-£~- J ~

G-c~,'l) = o(1~ I ) ,

for "t;'e ( %'1 ).

~oo~.

6C~,~;)[ = ~1¢1 i ~_tF. i~.+1;1~- ~< - - F r -

~rom lemma 4 and (2.22), (2.24) we deduce that

o

Here the constant

C

is independent of

•

. Thus we have

0-< It is easy to see that ~ A ( ~ , ~ ) , - c,l l w constant C 4 . Indeed if~ ~o d e ~ ~ m ( ~ _ U [ 6 5 ; + o o ) ) Now (2.26) implies (if we choose 0"< 6~/~¢

th

an

aboo~ute then

)

04 and therefore (2.23) is proved. Thus the proof of lemma 2 is finished. • To prove the first part of theorem I it remains to show that if ~ A C ~ I ~ ~)~/~ < oo then the family ~ j ~ is not complete in ~'~( k ~ ) -

460

Recall

that for some positive number ~

=~6p~---~

for ~

(-o~> 0)

. Let

we have ~ ( $ ) =

~

be the confor-

mal homeomorphism of the unit disc ~ onto the domain ~ : C~ [-o~ 0 ] , ~ ~a_~~-~ . Let~be the outer function in ~ with the modulus

This function is correctly defined since

~-oj-~--@~ > - ~

.

It is clear that the set ( ~ ( ~ ) ) ~ / k has the Biaschke property in ~ ,A~A(~.I~(~)I )< co , and so there is a Blaschke product B whose zero-set i s ( ~ ( ~ ) ) ~ ] % . Define the function ~ : ~ - - ~ . ~ - - I t is bounded and analytic in ~ and

~,CIJI/(.A))~__0.

A~/~_

o Therefore

. , , . , .

'P

- J - ~-A ~G-

3G-

CWc~z)- V'cA)) c ~-'~)

~G-

It is clear that the last integral is equal to zero. Now if ?(t) d£~ e ~

~C~(t+~))~0~

- ~(~(~+ ~))

then

o

But from the definition of the functions viou~ that J ~ ( ~ ) I ~ ( t ) and that finish the proof. Let &

m

pc~)= Thenp~ b~ ct,,.~)

~

~

and

~

~0

~ it is ob. Now we may

t e (-~, 0) ~ct> ' 0 , t ~ R,',("~,O) . 0

for

461

every

~

.

@ 3. The second step of the c ~ n s t r u c t i o n

Lemma 5 below contains the proof of the second part of theorem I. LE~ F and

5. Let the conditions i)-iv) ~ be fulfilled and let

(see @2) for the functions

(3.3) (3.2) 0

Then the zero-set ~ of the function F is finite. PROOF. The proof of this len~na repeats very closely the scheme of the proof of lemma 2. We begin with the function

where the functions

~

,~

,~

were constructed in len~na 2o

The function .~s. is analytic in the region ~ + obtained form the right half-plane by cutting out a finite number of closed intervals (one of them is infinite) of axis ~ T " On the imaginary axis we have

~ (~) =

the positive half-

F(-~b e ~ ~

and so it is clear that

~1 F(-V")I-<1 '~ (~,7)1 -~ c,, I I:'(-,#h I , where CI and C~ do not depend on of lemma 2 we consider the set

At first we note that ~ . = ~ ~

P~

(3.3)

. Following the scheme

~/~,(

. If it is not the case, then

Fc~,)~,.~/~c~)~ ~e~,p(.4/~, ~~,,£~!~ ) Applying the inequality (2°20) we obtain

for ~E{-~O)

462

where ~ O

is the harmonic measure of the d o . i n

~

in lemma 2. But from the properties of this d o . i n

constructed one can deduce

(see [2], p.7 ) t at

for ~

sufficiently small. So we have

o and it contradicts (3.2). Now we proceed exactly as in the first step of the construction. Namely, we introduce the open subset --~[ of the imaginary axis ~ ,

~'~=

U , I~ °

,

d,÷.. Let the'function I ~ the boundary values

be harmonic ~ n the right half-plane with

Then I ~ $ ~ ( ~ ) i ~ ~ @ # / ~ ( - 7 ~ ) . Let ~ be the harmonic conjugate function. We introduce the auxiliary function

It is clear that there is an analytic continuation of this function~ across ~ from the d~maim~ ~,~ to the symmetric domain ~M_. We denote this new function by ~ , ~/ again. We note that

is analytic in the domain

The function •

~

~----- ~ .

be the boundary of ~.

U

, then

The application Jensen inequality (see [I], L2]) gives the following inequality which is similar to (2.1 I) :

463

(3.4)

Here ~OJMiC', ~) denotes the harmonic measure of the domain ~ , corresponding to the point ~ , @ ~ , ~ is independent of~. Our aim is to estimate the integral ~ L ~ : 4 ; ~ , @ . The inequality (2.10) shows that eO

0

0

since the function

@

decreases so fast that ~ ~ ~ < O O

.

In a similar way we can obtain that It~(l~61)l ~ C . Therefore ~Z,I ~< O , ~,~ 0 , where C does not depend on }$. To estimate ~ , ~ we note first that CO

co

O

-0~

where

It is

clear

that

~,pp ' ~ c c6e~

V~

_]

where

Thus

t

I

"

We see t h a t t h i s i n t e g r a l i s s i m i l a r t o t h e i n t e g r a l ~ , estimated in § 2. So we may reduce the estimate of this integral to

the estimate of the expression (see (2.16)):

r

I"

464

Here ( I ~ ) , T ~~ " ~ ~ ~I,

is a , ~__~ ~ ~ ~ ~"'

.

sequence of disjoint intervals, ~, is the interval with the same ~" #

c-,-0,,,,(-I;,~)'~ 1 ~l~-ti

d,~(~,~,,,.

It is necessary to note that I T 9 ~ . ~ I : ~ ( [ . Now we have the obvious estimate from above for ~ C, ~ ~) :

r~,~

,t

for t ~ [ - ~ ,

qg].

Now we h a v e

X'E

,].r

,l

I

'

..¢-g

The estimate of

J~$

C,~: oo .

is quite similar to that above and so . Therefore from (3.4) we deduce

The application of Fatou's lemma gives

q

r~

the Green function nic

measure,

of

From t h e

the domain ~ construction

(3-5)

and ~(o~6) of this

region

is it

is

its

harmo-

easily

seen t h a t

~(~)

i R~l~zd~c~'~~'(

- ~

~ ), ~f-~',~]~

Thus the inequality (3-5) is equivalent to the system of the

465

inequalities ( ~

~g~

~--~)"

E (~)

(3.6a)

(3.6b)

(3.6C)

~*(~) ~__~~(~)

(~>--- ~ C~~)

Here . It is clear , &~ that the functions ~ ~ have the properties (A) and (B) of lemma 3. We want to prove that the set ~ is finite. It is clear that the set is symmetric with respect to the real and the imaginary axes. So it is enough to prove that the intersection ~ ~ ~ •0 , ~<0} is finite. To do this it is enough to establish that if A C / ~ N [ ~ 0 ~ ~<0} then ~ ( ~ ) ~ for a fixed number ~ C ~ and some positive number C not depending on ~ . Let ~ be a very small positive number. The application of lem~na 3 gives a number ~ , ~0 , so that

Therefore

~,et

~ a~--~d~ ~ U ( - ~ , 0 ~

. It ~s olea~ that the following chain

of inequalities holds:

O

To obtain the last inequality one must apply Harnaek's inequaluty

to the f~notion

---~(X~ ._

(remind that ~ e A ~ 0

Thuswe have

> ~ositive ~

)

~ i ~t,~

harmonic i n F ~ < 0 k ~ - ~ - m ]

466

~

.....~ .. - j

;

~C~.C~(~,~)

t~

The last inequality follows from (3.7). Now the choice of 4 ~'< ~6' ' gives

The s i m i l a r reasoning shows t h a t .

I~1

(3.9)

I~1

I'

Another a p p l i c a t i o n of lemma 3 shows that

~-*0

where._.. _ . ~ ( ~ , g)

%g.l~l
(3.10)

o

.

"

denotes the harmonic measure of

~a~in~

(3.9),

(3.~0)into

an arbitrary small positive number ber ~. such that

~

~ocou~t

there

~n

we see

that

for

is a 9ositive num-

v

Now we estimate

~6(~,-~)

from below (we may suppose that

0¢

where

It

is clear that

We may suppose that

T 3 ~ O. I Xl i s small enough and so

The integration by parts gives

(3.12)

467

Cr[n

_ e~ ¢'1;,- ~ )

The last integral does not exceed

O jo

t

~

since the point ~ lies in some Stolz domain. Thus, after another application of Harnack's inequality we have

(5.14) 0

Now from (3,11)-(3.14) we deduce that (3.15) Now we c h o o s e

~'

small enough ~

so s m a l l

that

~"<

~ ~@~

@8

and note that for . So from (3.15) we

have .

Applying Harnack's inequality once more and taking (3.8) into cocount we obtain that

for some constant C independent of second part of theorem I are proved.

, Thus lemma 5 and the

@

4. The proof of the£rem I in a particular case the weights In this paragraph we consider o n l y with the following properties: a) keL (l ) n ; , where the function ~ is logariShmically convex and rapidly increasing ~) ; c) the function % is of the form "$(~)= eOC,p(n---~) , where k

5) The properties of logarithmically convex, rapidly increasing functions may be found in [14] .

468

THEOREM"If. Let the function ~

satisfy condition a)-c)

and let

0

1

~.

Then the family ~ = } ~_/k of rational fractions with the set of poles ~ In some Stolz domain is dense in -~(----~) provided that the set ~ is infinite. A sequence of moments corresponds to the function ~ :

We introduce two functions constructed with the help of the sequence (

~I,)N~, ~ ~.>

of

the

reformulated

Tn t e r m s

in

o lorry,

~.:. o ~

"moment s e q u e n c e " the following_

the

(4.1)

may be

form

In what follows we shall need the fact

~,,~1~(~5)

condition

"

{/{}5 =

that (4.1) implies

o0

(4.2)

If it is not the case, then we have

This implies that -'A--'---~0 C ~ ) since 4ecreasing sequence. ~herefore, we have

~{

is

a

469

and this contradicts (4.1). Thus (4.2) is proved. PROOF OF THEOREM Ir Let the set ~ be infinite, lie in some Stolz domain K ~ and let the family ~ A be not dense in ~(~t~). Then there is a~function P~ P ~ 0 ,-pE~(~d~) such that

r We have the analytic in ~+ function ~ ~J~ ~_ which is not identically zero (see the beginning of @2) and whose zero-set contains the set ~ . Moreover, F is in the Hardy class H~ in @ + . It is clear that this function is infiniSely differentiable in the closure of every Stolz domain ~ 0 <~
F(~)(~;) =

(4.3)

0

~e ~tr ~ ~;-+0 since the infinitely differentiable function many zeroes in ~ . Let

If ~ r ~ , estimate:

o~(0,~)~J(~)

~E)

F

, we have the following

ct'l, [

3 We i n t r o d u c e a f u n c t i o n t~)g a n a l y t l c i n d e p e n d e n t of o~ i n the h a l f - p l a n e 2 ~

has infinitely

~." (4.4)

and bounded by a c o n s t a n t

q =om

we

we~P~

t.i..r {w I ~" '1

~ e.,

~! R.,.

(~'~'~)~lwl~" =

(4.~)

470 It is clear that (~)~4(W) so we have a function

analytic and bounded in the domain . It is easily seen that for W ~- t~+ $1r~ ~ where ~ 0 and I~'l~ Z we have

Now let K be a positive continuous increasing function defined for 0 O ~ - ~ , K(-~) ~ 0 , so that the curve ~= C~-06+~ :~ = ~(06)~ lies in the domain ~ . Suppose also that co

The function ~ constructed above is analytic and bounded in the domain ~ c J ~ \ { ~ _ _ - - ~ + { ~ : I ~ I ~ ( ~ ) } . Moreover, from (4.5),(4.6) we deduce that CI

l ~,

for

(4.7)

t~l~t~ot-

In order to prove that ~ 0 we consider the conform! homeomorphism ~QCO) of the domain ~ Ne~ { (D~---~'t"~ : I t l <

onto the domain ~

_d¢£ [ ~__~+{~: tUI<'~.-~ ÷-" -

2Kce.~) ~, ,

p s

J

and we prove that the function^ ~(~0~__a¢~ ~(e~(~) ).P-" is identically zero. Let ~(~) I~¢~ ~q~_ @ ~ . By the well-known inequalities due to S.~Vars~awski ~5] we have

~ - ~ ~<~ ~ ~ ~ The function ~ ~ and that

~ ~< ~ ~( ~ )

6~C~} Then for

K ( C ~)

~I

increases and ~--~-~(~)

decreases

. Taking it into account we see

.<

large enough we have

i2,erk~!~< c,. ~'h° ~'(D.I K'C~~

C

471 since (4.9) and ( ~ * ) (4.8) we deduce that

hold. ~rom the last inequality and from

~-<~I ~g +c Thus we obtain taking into account (4.7) and ( 4 . 1 0 ) that for F £~r ~ e I ~ ± - ~ = .- , ~+I ± - T I we have the following estimate for the function ~(~0):

BI I Thus we c o n c l u d e

that o<1 (4.11)

"1

Suppose now that we know how to construct the function ~ ving so that: I) ~ is a smooth and increasing function; decreases, ~ ~ (~)--^ ~

beha2)

. Granted this our theorem

4)

d~÷ ~clr~ ')

follows easily. Indeed if we choose K ( 0 0 ) - K(~) satisfies (*•) and co co

C~

, then

O0

1 So from (4.11) and Jensen's inequality we deduce that ~ ~ 0 . Then the function ~ is identically zero also and therefore F~0. ~(30)

This is a contradiction.

de~

~,

.~Tz~r~

d~

But the function

obviously satisfies I) and 2). Indeed,

~ Now set ~ ( ~ ) ~ [ ~ Now it is clear tha~t

l~ % ~

into account t h a t ~ ( O ~ ) ~

~

"

. Then (4.1) shows that ~.~.~:.T,~--~O W ~ ~ < CO -- and, taking

~

we obtain

IX)

4

472

~(~) t

4

~(~

-~-00

.

@

PART ii 5- The,,,,Rro,R,fq,f,theorem 2 The proof of theorem 2 is based on two lemmas due to L.Carleson [16] and S.-Y.Chang, J.Garnett ~ LE]@~ I (L.Carleson ~6]). Let ~ be a function of norm 1 in ~ . For $ positive and sufficiently small there is a number ~(8) ~ ~(8) ~ 0 , and a system ~ of closed ' ~--*0 rectifiable curves (~$)$~I in ~60~ D , with disjoint interiorS~ with the following properties:~

ii)

for ~

Fn ~ ,

~l~(~w.< ~(~)

iii) arc length measure on rN ~ is a Carleson measure with a Carleson constant C($) which depends only on ~ ~). LENNA II (S.-Y.Chang, J.Garnett ~7]). Let ~ be a contour constructed in the previous lemma for an inner function ~ and a positive number 8 Then for every function ~; ~ ~I

T

r

a ) = > b). We suppose that the normslI, ll~(~) andll.I I ~ ( W ~ ) are equivalent on th~space ~@ Theref~e, taking into account we have

v

~.~

e

I

for every ~,I@l<

(5.1)

Now we estimate both parts of this inequality. ~) A Carleson measure ~ is a measure on ~ such that ~II~(~GII@H N' for every function ~ in the Hardy class --H 1 . T~b Carleson constant of this measure is the best constan~in this inequality.

473

II~e(~d)lP~¢~)-,_-Z~ !l]-e(~e¢~)lPI4-~gl ~-P~-I~l~t~.m~l,@~ ,1_1¢1 ~ j I --~_1~1 ~ mo e s t i m a t e

(4_ g~)~(~_

~,~ 1~-~i~_~i ~.

C]-I ml~)P-s

(4-1~I~}P "~

t~e ri~t

h~d

part

~

=

ohoose a number ~O<~.~m~m(~,#gD;

Thus from (5.1) we deduce that

cw{~)>~(~-IO(~)I ~ -

and

so b) h o Z d s

b ) ~ c ) . It is obvious. c)=>a). Let W ( ~ ! ~ + I ~ C ~ ) I ~ > 0 , for e v e r y ~ D . Let ~ d~--~---e{ ~ : W ( ~ ) ~ / ~ } . If ~0~ denotes the harmonic m~asure of the set E then it is clear that

We s h a l l p r o v e , t h a t f o r

every ~ E ~

I1~ liec.)-~ o11~IlacE> ~,~ o(i l ~ i ~ )

~/~

~.~>

It is obvious that this implies a).E Suppose first that we know how to construct the function I~ with the following properties: I ) ~ - - ~ E ~ ~ ; 2) ll~IIi~(Er)~ where E f ~ e ~.....T \ E and ~ is a small positive number. Granted this it is easy to prove (5.3). Indeed I) implies thatT~IK$=T_iK~

474 where

I

stands

ry ~

Ko

for

the

identity operator. Thus for eve-

we have

E Now if we choose

G-

so small that

and (5.3) holds.

El

~Pcp.~4

then

E

Our aim is to prove that

Granted this we may find for an arbitrary

1~, bell ~

~

so that IIg+sllu~¢~')< ¢

the properties

,~

(

~

F

0~

so small that ~ ( 8 ) ~

is the function from lemma I and

~

as in the inequality (5.2~.Am ~pplication of lemma I contour

, a function

I) and 2) are obviously fulfilled.

It remains to prove (5.4). ~ix a ntur~ber ~ ~/~

O

. Then f o r ~ = 4 f f .

is the same gives the

on which

~1 ~1-< ~(~)< ~/~ Now from ( 5 . 2 ) we have O O E ( g ) ~ E / ~ be an outer function with the modulus

Taking into account t h a t l F 1=4

for

~'I~F

. Let

F

on E z and that F-+C N ~

we have (for an arbitrary positive integer ~

)

~%¢E,)(kH~) -- ++~ II~+I+IIL,+CE,)-- +~ IIF*#+F+~IIL,cE5 k~H ~

~1.1 ~

Now by the standard duality argument

++4 ~H°~

llF+~,kll~ = ~ ~P

(IfF+~4~l).

~ ~Ir,ll~ll-~1 T

An application of lemma Ii gives

475

But the last expression does not exceed

IFI ~ I~1

e-T F

By the choice of ~ the last expression can be made arbitrarily small, and (5.4) is proved. @

6. Compactness of the inqlusio n map ~ and relate 4 questions, l~Rof of theorem ~.

In this section we shall be interested in the following question: for what symbols ~ the operator T%01 K@ is compact? This question is closely related to the problem concerning the criterion of compactness for the operators ~ H~ ~see D8])" Indeed, the compactness of the operator H ~-~ ~ is equivalent to the compactness of the operator T ~ I K @ . ¢~e know the answer only for the "alr~ost" real symbols ~ (see proposition 7 below). But for the proof of theorem 3 we need only positive symbols ~ . Lemma 6 treats this subject. Let N stand for the set of positive integers. Let a ~ n ~ d ~ ~ ~ ( ~ ) ~ S , LE~9~A 6. If the function

~ ~ ~

denote the Poisson kernel

satisfies the inequality

then the following statements are equivalent: a) ~ ~ is compact; b)T~ I K6 iAS compact; I%1--*I

--

n N~[~] c N ~ , C . PROOF. a)(-~-->b).It is obvious. b)~--->c). Let ~ ÷ C and suppose t h a t l l ~ - ~ I I ( ~ < l . We denote ~ 4 : ~ + ~ , ~'-~ ~-- ~ . Now take a sequence (~)~,~ D ~ sucht~l~(~;pN~,fitl~l~'l. Then it is clear that the functions '~=('~-I'~l,[ ) ~@(~,~) d)

H'[~e]

converge weakly to zero and that ff,~ ~

II P,~$,*~)~II~o, and so I IIz Ik - II

. Therefore

I-*- o.

476 H

__~H

Taking into account that ~ , - ~ ~- - - ~ " @ ~~ that II ~- ( ~ i - ~ ) ~ l l ~ ~ 0 and so

is compact, we obtain

I II P_ ~,~14-11 P- ~11~ I --~

o.

Thus, we have

after computing this expression we see that

I

I

I

T

~ow

A

Thus I ~ I C ~ ; ~ )

- - - ' - O.

c)--->d). Here we shall apply the reasoning of D8]. Using the reasoning absolutely similar to that of lemma 5 [18] we may prove that for every pair of numbers ~ and ~ ,~ ~ the operator ~ ~ H~m is compact. Thus for every ~a --

and ~ ,

O0"r

v

v

.,~

''

--

~I"

"

in ~ the operator I'~WI,~I~'~. I"l~l~l~~ T ~ ~M4~H~I~T1. is compact and therefore for every function 9 ~ l ~ [ ~ . ] r l ~[~i ~ the operator is compact. It means that ~ + C. d):>a). It was proved in [18]. • To prove theorem 3 it remains to show, that the conditions a)-c) of the theorem imply tha~ the inclusion map ~ : ~$-~b~(W~) is compact. But the compactness of ~ is equivalent to the compactness of two operators:

H~H~

:E w 'i~OP_ ~ 1 ~ ~ ,

~_ w~P_ ~1 H~

The compactness of the first operator is equivalent to the compactness of H t II ~ H and the compactness of the second to the compactness of -w~l~ ~ ~ . But if we take into account the condition c) of the theorem and use the reasoning of lemma 5 ~ 8 ] we shall see that the operators H W d ~ ~ ~ and HW~I~H ~ are compact. Lemma 6 does not give any criterion of compactness of the

477 operators T ~ I K $even for a real function ~ PROPOSITION 7. If there are numbers, ~ sat isfying

.

g>

!

0

, and ~)

k~

- < then

the

statements

a)-d)

o f lemma _6 a r e

equivalent,

PROOF. It is clear that the only implication that we need to prove is b)-~c).

~et the sequenoeC ~ I ~ ~ , ~ For the functions

D

~atisfyl@~}I~ti~,k£

%1¢,d~g'"(~-l~l~)4/2" ~(~)we

have

. Now we compute T 9 % ~

Let % ~ ¢ J ~ ( ~ ) ( ~ - ~ Q ~ ) O ( ~ ) )

:

4-~ (6.2)

where ~

i~,~h

Schwartz kernel,

~X(~)-- ~" A ~ T , A~P

T Taking into

~ou~t

(6.1)

and (6.2)

we s e e

that

T

By the Holder inequality it is obvious that

Thus we have

I~-+eo ~ and therefore

""~

B'+ m

~p

'~"

~"* I qP~ ~ = Now introGuce the following notation

"

~--,.m

= 0

o. :

~li(:(~)= PI,.£.~VI,,(~),

~)

478

then ~ denotes its harmonic conjugate, ~ ( 0 ) ~--- 0 notation we have

• Using this

'II'

Taking into account C6.4) and the fact that ~ have

"

~---~

0

we

~p

q?

Now we introduce an auxiliary function,

P~cX)~ E ~ ) ,

~C~¢A)-~))]

O-~(%)O(A)).

This function p~ belongs to every Hardy class ~ P , p < 013 in D . Noreover, it is clear that for every ~ , k > 0

where 0{~) does not depend on ~ , Now the following chain of inequalities is obvious:

-

9

~

But ~ I ~ C ~ ) ( ~ - ~ ( ~ t ~ ) O ( ~ ) ) = number

X)l 2~cA)g~¢1)<.

K

~<X)lW-g{~)O(A)I ~

~

. Now l e t the

be the same as in the statement of proposition 7.

, since

~FI~, (~l~C~) =

0

(see ( 6 - 4 ) ) ,

~.

479

1P

T }=o Taking into account --} (6 .5) and (6.6) we see thatl~,=0(~). But I#v%~Q~KI~ g and therefore ~ 0 } ( ~ ~or some constant C~ C > 0, Thus we conclude that

EI I

4

T gi so

) - - o.

Te

COROLLARY 1. If the function ~ has a finite range, then the conditions a)-d) of lemma 6 are equivalent. COROLLARY 2. If the function ~ takes the values ± ~ only, then the operator is not compact unless @ is a finite B!aschke product.

T IK o

References I. P.K o o s i s. Harmonic estimation in certain slit regions and a theorem of Beurling and Iv~alliavin. Acta ~ath. 1979, 142, 275-304. 2. A.Z.B o ~ ~ 5 e p r .

0~oBpeMeHHa~ annpo~c~Man~s n o ~ o M a ~

3af~CI~ H a ~ H . C e M . ~ 0 ~ 1999, 92, 60--84. 3. A . £ . B o a ~ 6 e p r . IIOAHOTa paasoEaa~H~x ApoOe~ B B e c o ~ x L~-npocTpaHCTBaX ~a oEp~HOCTZ. { y H ~ . a H a a a 3 ~ e r o n p s a o ~ e ~ S , 1980. 4. H.H.A x ~ e s e p. Z e u s no Teop~s a n n p o E c s M a ~ , orz~, rocTexHa

OElOYrd~ocT~ ~ BH~TpM Epyrs.

~S~aT, 1947.

5.

r . H . T ~ M a p E s H. H e o 0 x o ~ M ~ e ~ ~OCTBTOqHRe ~CAOBZ~ ~ S BOBMOEHOCTM np~Oa~zeH~s ~yH~n~s Ha OEpy~t~OCT~ pSZI~OHSABH~Z ~ O 0 S ~ , BRpa~eHHRe B Tepm~HaX, Henocpe~CTBeHHo CBSSaHH~X C pacnpeAeaeH~e~ no~ocoB a n n p o E c ~ p 2 # e ~ x ~poOe~. MSB.AH CCCP. Cep~s ~aTe~. 1966, 30, ~ 5, 969-980. 6. ~.H.T y ~4 a p E ~ H. H p ~ d ~ m e H s e ~yHI{E,Z~ p a n ~ o H a ~ H ~ ~pod~m~ c 3apaHee 3aAaHHm~z no~0ca~s. A o ~ n a ~ AH CCCP, I954, 98, ~ 6, 909-912.

480 7. B.H.II a H e s x . 0 He~oTop~x 3aAa~ax r a p M o ~ e c ~ o r o aHam~sa. ~Io~aAu AH CCCP, I962, I 4 2 , ~ 5, I 0 2 6 - I 0 2 9 . 8. B.II.l~ a ~{ e ~ x . H e ~ o T o p ~ e HepaBe~cTBa JD'i~ ~ H I 4 n ~ a~cnoHe~uaa~Horo

T a n s ~ a n ! o a o p s a e o ~ e H I ~ AAS O O ~ X

onepaTopo~,

Ycnexa MaTeM.Ha~K,

Aa~epem/zaasHax

I966, 2I, ~ 3, 75-I14.

9. B.g.~ z H. 0O ~KB~BaAeHTHRX HOI~aX B npOOTpaHOTBe

C~M~zI~eMRx

C KBa/LpaTOM &eARx ~HEIIZ~ SECnOHeHIlaaJLBHOFO Tana, MSTeM.cO. I965, 67(I09), ~ 4, 586-608. I0.B.H.

Z o r B a H e s E o,

HOpNR B HpOCTpaHCTBe ueaux

Teopas

SyH~,

D.(#. C e p e ~ a .

SyHEa~A

3EBaBaAeHTH~e

sEcnoHeau~aasHoro

TZHa.

~HEIL~OHa/LBHRM 8HSJIM3 n ~X npaaomesas. B~n.i9.

PeonyOJL~EaHcF~ HS~qHNM C00pHZE,

~IDBEOBo

I973.

!I:B.3. K a U H e a s c o H. 3KB~BaAeHTHRe HOpeR B HpOCTpaHCTBe ~yHEAZ~ sEcnoHeHn~aaBHOrO T~na. MaTe~.c0. 1973, 92(134), ~ I. I2. D.N.C i a r k. One gimensional perturbations of restricted shift.

J.anal.math.,

1972, 25, 169-191.

I3.~.M.F o a y S ~ H. YeoMeTpsqecEas

TeOp~S # ~ H E A ~

EOMDJIeECHOPO

nepe~eHHoro. MOOKBa, HayEa, I966. 14.S.M a n d e i b r o j t. S4ries ad/q4rentes. R4aularisation des suites. Applications.Paris.

1952.

15.S.E.W a r s h a w s k i. On con_formal mappin 6 of infinite strips. Trans.Amer.Math.Soc.

1942, 51.

16.L.C a r 1 e s o n. The corona problem. Leer.Notes in Math. t18, Sprin~er-Verlag,

1972.

17.S.-Y. C h a n g, J. G a r n e t t..Anaiyticity of functions and subalgebras of L~

containing H e~ . Proc.Amer.Math. Soc.

1978, 72, N I, 41-46. 18.S. A x 1 e r, S.-Y. C h a n g, D. S a r a s o n. Product of Toeplitz operators.

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1978, I, N 3, 285-309.