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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 N.A.Shirokov has proved (unpublished) that if ~ has co~ner points then one cannot replace ~-M in (9.3) by ~ - ~ , were ~ , 0 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 < ~ 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
-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)
~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 , $=t~,,...).
These functions are analytic in the upper half-plane
~,~ k'~.{~÷~.~)( ~-*÷0 ~ ~ --~-,-~''C:~ ))= l,t~ ~'("~,)whenever
~
, and
~ is an interior ~oint of I-~, b. , J~, ~ e 2 " ~ ~ "~\p:~ ¢/~ (~6 being the characteristic function of the set ~ ). Consider now a Jordan region ~ ~ C + with the smooth
169 ( C~ where
, say) boundary ~ such that ~ n ~ = ( - ~ ~ 3 , ~ such that Up~ = ( - ~ ) It is not hard to see that ~ belongs to the Smirnov class EI(~) (see E3] , p.203), ands~ I~4(~)]I~ I ~ , ~> being a constant depending on ~b , ~ ~ an~d ~ only. If 06EPI" we have K ~ ( ~ , ) ~ - - ' ~ ( ~ , } ~ - ~ " P ~ , ) _ I'~,I~,~IP~ ( ~ ) . But "a
-PI ~
-
~
and (I6) ~ p l i e s
~ 11~(~)1a~ ~-~ I~_ I E~ .~,~
-~
. This estimate, the inequality
and (I7) imply
t(~(z)= 0
~ut .~'~, ~-(~)=f
--.--~
(~). ~
(~
~ ~'0)
, where
It is not hard to deduce from the last identity that it holds with o~--0 too (see EI~ , p.144) and therefore the measure , ~ being ~ is ~-absolutely continuous, ~ = ~ ~ afunction from the Hardy class ~ " But l,r~ T-~
~ II~l~
,=n~
~1~ I ~ ~
0 and let E be a closed subset of the set of density points for K , ~ = 0 • Let _/i be a subset of ~ with Hadamard's gaps. Let, finally, the operator ~ be defined by
a÷
s v(K).
C V ( K)~-- C(E) x///q'(.~) . There is a definition of the space V(K) equivalent to the previous one. If # E V ( ~ ) then the Cauchy transform ~ ("t3-~) -'l #(~) ~ % ( ~ ) is holomorphic in ~ k ~'. So the space of Cauchy transforms ~ ($-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< ~/ ~ > ~ 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 ~,
+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
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
.c 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 { ). 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.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 { ). 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.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 '~eYt 1.~(~,') # { 0 } ,~
3) ~-~
CI-XL) = Cl-Xa)-~C~fi-fi,B)N~ ~ (x).
it. i) ~ ~ ~cL) ~ ~%CX) is invertible.
@~C~)
is.in~ertible
,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.~ ,
(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 ~ 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 =
' 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 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 ~
. 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
(~;~)I.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: ez~ . 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 ,~(~)
"
) 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 ' 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-
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 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) 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.
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