Division and Multiplication by Inner Functions and Embedding Theorems for Star-Invariant Subspaces Konstantin M. Dyakonov American Journal of Mathematics, Vol. 115, No. 4. (Aug., 1993), pp. 881-902. Stable URL: http://links.jstor.org/sici?sici=0002-9327%28199308%29115%3A4%3C881%3ADAMBIF%3E2.0.CO%3B2-R American Journal of Mathematics is currently published by The Johns Hopkins University Press.
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DIVISION AND MULTIPLICATION BY INNER FUNCTIONS AND
EMBEDDING THEOREMS FOR STAR-INVARIANT SUBSPACES
1. Introduction. Let T denote the unit circle in the complex plane, and let X be a Banach space of complex valued functions on T possessing, in a sense, some smoothness properties. Denote by XA the 'analytic subspace' of X : XA X n H', where H1 is the classical Hardy space (see [GI or [K]) in the open unit disk, D. Suppose f E XA and 0 is an inner function, that is, 6 E H m and lo( 1, write a = k + ,B (k E N,,B E (0,1]) and set A" = Gf E ck(Q: f(k) E AP), where f(k) is the k-th derivative o f f , and differentiation is understood in the natural way. The intersection A" r lH' is denoted by A,: a E (0, +XI). Throughout this section, we denote by n and p the two numbers associated with the fixed a , a E (O,+w), as follows: n = [a]+ 1 (i.e. n is the smallest integer which is > a ) , p = ( a + I)-'. Note that p E (0, I), a =p-' - 1. A well-known theorem (e.g. see [F]),essentially due to Hardy and Littlewood, says that an analytic function f in D belongs to A,: 0 < a < +XI, iff
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KONSTANTIN M. DYAKONOV
Another well-known result needed below is the duality theorem of Duren, Romberg and Shields [DRS]. (See also [G, chapter ii] for 0 < a < 1). It says A: is the dual of HP under the pairing (cp, $) = J cpqdrn, where the integral makes sense at least for cp E H', $ E A;. In particular, for such pairs we have I ( ( P , $ )I~ cllcpIPI$I~AU, where c depends only on a and
We also need the following facts. THEOREM A. I f h E H m , then the Toeplitz operator Ti;is a continuous mapping of A: to itseg THEOREM B. I f f E A:, 0 is inner and f 8 E H1, then f 8 E A?. Both theorems are due to V.P. Havin [Ha]. In fact, the latter theorem is an immediate consequence of the former one, and the former is easily deduced from the HP - A; duality. Further progress in this area is due to N.A. Shirokov. He studies multiplication and division by inner factors in the spaces A:, whose definition involves an arbitrary continuity modulus w and even (as in [Sl, chapter I]) in classes with "varying boundary smoothness". However, at present we are interested in A" only, so we restate some of Shirokov's deep results for this special case. THEOREM C. Suppose f E A;, 0 m(Se) = 0 and
< a < 1, and 0 is inner. Then f 0 E A: iff
where So denotes the singular support of 0. THEOREM D. Let a > 1. There exist a function f , f E Am (i.e.f E H' and f ( k ) is continuous in ClosD for all k E N),and a Blaschke product B such that f / B E A", but@ 4 A:. THEOREM E. Let a E (0, +w),f E A;, 0 be inner, and f 10 E A;. Suppose each zero of 0 in D is of multiplicity 2 n. Then f 0 E A:. A generalized version of Theorem C is contained in [S2], those of Theorems D and E can be found in [S 1, chapter I, 521. Further references are also to be found in [S I]. Though the works of Shirokov seem to provide exhaustive information on the inner-outer factorization for A:, the theorems in the next section do not
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STAR-INVARIANT SUBSPACES
follow from his results, and they are obtained by a different method. Besides, a new feature is that we study divisibility without analyticity: condition (i) of Theorem 7 says fg E A" (not A;).
6. A": results. Let a E (0, +GO) and let n = [a]+ 1, p = (a+1)-' as before.
THEOREM 7. Suppose f E A: and 6 is an inner function. Consider the four statements. (i)
fB
E A".
(ii) f On E A;. (iii) For every E, 0 < E < 1, f(z) = O((1 - 1~1)")a s IzI (iv) For some E , 0 < E < 1, f (z) = O((1 - 1~1)")a s IzI
+
1, z E R(Q,E).
+
1, z E Q(O, E).
We have (ii) + (iii) + (iv) + (i). Iff on-' E A;, then (i) + (ii). Before proceeding with the proof, we make some comments. First of all, if 0 < a < 1, then n = 1, so the condition f Qn-' E A: reduces to f E A: and adds nothing to what has been already assumed. Thus, for a E (O,1) the four statements are equivalent. Recalling Theorem C one infers that condition (iii) (or (iv)) of Theorem 7 is equivalent, at least for a E (0, I), to Shirokov's condition (6). It would be nice to find a direct proof of this fact. Contrariwise, in case a > 1 the implication (i) + (ii) fails without assuming that f 6"-' E A;, as follows from Theorem D. Indeed, (i) does not even imply that f 6 E A;, which is weaker than (ii) by Theorem B. Neither does the assumption f On-' E A;, taken apart from (i), imply (ii). Proof of Theorem 7. (ii) + (iii). Let g E H m and h E K," = H m n O F . and the HP - A; duality gives Since hg E F , we have ~ ( h 8 E) ~
e,
where we write for typographical reasons point z E D and let
11 . llA
instead of
By Cauchy's formula, the left side of (7) equals
11
. 11~;. NOWfix a
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KONSTANTIN M. DYAKONOV
To estimate the right side of (7) we write
\ g b n / l p 5 2 n ( / I (dm(o , l ( n + l ) ~ ) ' I p < (1 - const ~z~)~+l-l/~'
where const depends only on a. The last inequality follows from the observation that (n + 1)p > 1 and the well-known estimate
in which the constant depends only on y. Taken together, (7), (8) and (9) yield
Hence (ii) implies (iii). (iii) + (iv). This is obvious. (iv) + (i). Consider the Carleson curves T, whose properties are listed in Section 2 above. We obviously have f$ = Tef + Haf, and Theorem A says Taf E A?. TO prove that H-$ E A" we use a duality argument Cjust as we did in the proof of Theorem 1). Namely, we fix a function g = zgl in H r with llgllp = 1 dm are bounded in and deduce from (iv) that the integrals J (H-$)g dm = J modulus by a constant, which is independent of g. A simple estimate for an HPfunction gives lgl(z)l 5 (1 - IzI)-'lp whence Igl(z)ll-P (1 - I Z ~ ) - ~ Now . we have
feg
1. We claim We know it is true if a E (0, I), so we consider the case a that (11) holds if 0 is a singular inner function (i.e. 8 has no zeros in D). This is because one can write 0 = 0?, where Q1 is another singular inner function, and apply Theorem 7. In fact, one replaces the pair ( f , 8) by (f01, 01) and uses the hypothesis (f 0 ~ ) 0 ~ -E' A: to deduce that f 0;" = (f O1)O? E A:. Then one applies the same theorem to the pair (fey, 81) and obtains f 0 ~ 'E~A z , etc. Finally, one concludes that fOyn = f O2 E A:. Moreover, i f f 0 E A2 and 0 is singular, then f Os E A2 for all s > 0. Now let 0 be a Blaschke product. Theorem D in Section 5 says (11) fails for a > 1, but Theorem E (which can be also derived from Theorem 7) asserts that (11) becomes true under the assumption that the zeroes of 0 are of multiplicity 2 n. In particular, for a E [I, 2) it is sufficient to assume that 0 has no simple zeros. In the next theorem we obtain a much weaker condition on 0 under which (11) is still true. Thus, Theorem 8 is a refinement of Theorem E for a E [I, 2). If B is a Blaschke product and {zk) is its zero sequence, we write bk(z) = (zk/lzkl)(zk- z)(l - zkz)-l (bk(z) = z if zk = o), so that ~ ( z =) nkbk(z) and set Bj(z) def = nk+jbk(z).A Blaschke product will be called sparse iff supj IBj(zj)l = 1. (Some authors use this term to mean that limj,, IBj(zj)l = 1). THEOREM 8. Suppose B is a Blaschke product, and B is not sparse. Let a E [I, 2), f E A2,@ E A2. hen@^ E A;. LEMMA2. Let a and f satisfy the same conditions. Let 8 be an inner function such that f 0 E A2 and let 0 be factored in the form 8 = 8182, where 81 and d2 are inner functions. Then for z E D we have
where c is a positive constant depending only on a. Proof ofLemma 2. Let g E H W , h E K r . Since ghe E
In particular, one can take
the duality yields
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STAR-INVARIANT SUBSPACES
where z is a fixed point in ID. One easily verifies that h E K r since kj E K c ( j= 1,2), For the chosen g and h the left side of (12) equals
by Cauchy's formula. On the other hand,
I
~
~5 I
(J 15 ) 'Ip 5 dm( 1. The desired estimate now follows from (12). Proof of Theorem 8. We are going to apply Theorem 7 with 6 = B, n = 2. More precisely, it will be shown below that under the stated conditions we have
for some choice of E. Once this is established, the implications (iv) + (i) and (i) + (ii) of Theorem 7 (the latter implication holds since jB E A;) yield jB2 E A;, and the proof is complete. Let {zk} be the zero sequence of B and let a df! supk Bk(zk)l.Since B is not sparse, we have a < 1 and we can find a positive number 6 such that a + 26 < 1. By p(., .) we denote the pseudo-hyperbolic distance in ID defined by
To establish (13) we fix a point z, z E R(B, E), where on, and consider the two cases. Case 1. infj p(z, zj)
E
will be chosen later
< 6.
Let zk be the zero for which p(z,zk) < 6. Since B = Bkbk,Lemma 2 yields
Clearly, Ibk(z)l = p(z,zk) < 6, and so 1 - lbk(z)I2 > 1 - 62. Next, we recall the well-known [G, chapter I] inequality
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KONSTANTIN M. DYAKONOV
which holds whenever F E HCO, IIFllm 5 1, F(C2)l 2p(C1, (2). Hence we have
_ 0
4),
-
and h E H m . THEOREM 11. Let p E (O,l], s E [2, +m), 2p[s/2] the following conditions are equivalent. (i) K f (ii)
> 1. Suppose w E (PA2). Then
c Lh.
Ow(z) = O((1 - IZ()'/P-')a s lzl
-t
1, z E Q(O,E).
Proof. As usual, one easily verifies that (i) =. (ii) by applying the inequality J I glSwdm 5 const( J I glpsdm)'lp to the family of reproducing kernels. This time we don't write out the details. Conversely, assume (ii) holds for some (every) E E (0,l). Note that for r E (1, + m ) K i is a complemented subspace of Lr and the operator Po = P+- OP+(~.) is a bounded projection from Lr to K i . Therefore it suffices to consider the C are established for case s = 2m, m E N. Once the embeddings K? P , and the missing embeddings are all m, 2mp > 1, we have Po E B ( L ~ ~L;~) immediately obtained by interpolation.
~t~
Now let N and h be such that N E 4 ,h E H m , Iu - 2?hll, d2fo < 1 (their existence follows from the above discussion, since w E (PA2)). Let g E K?. Without loss of generality we may assume g to be bounded, so that g E H m , g0 E H r We have then
STAR-INVARIANT SUBSPACES
Hence
where a def = P-1 - 1, A: dLfBMOA. Clearly, (ii) remains true if 0 is replaced by zNOm.It follows now by Theo~) rem 1 (in casep = 1) or by Theorem 7 (in case 0 < p < 1) that P - ( O , T ~ ~ E and so ( ( P - ( O , ~ ~ ~ ) ( ( - - < +me(Note that the needed implications (iv) +A;
of heo or ems 1 and 7 hold if f E HI, not necessarily f E A:). P-( f8) E c L;~. The theorem is proved. Thus, (18) shows that K?
8. Concluding remarks. 1) Suppose O1 and O2 are two inner functions such that
where E j E (0,l). (In particular, this happens when 1101- 0211co < 1). Let X equal BMO, QC, or Aa with a E (0,l); f E XA.The above results imply that the E X and fOl E XA are equivalent to similar ones with O1 replaced statements by 02. 2) Suppose B is an interpolating Blaschke product, i.e. infk (Bk(zk)l > 0, where {zk) is the zero sequence of B. Let X be one of the spaces listed above and f E XA.To have fB E X or@ E XA,it suffices to assume that the appropriate decrease condition on f(z) (see the table in Introduction) holds as z E {zk), k + m. In other words, in case 0 = B Theorems 1, 4, 6 and 7 remain true if the set Q(B, E) is replaced by the zero sequence {zk). The proofs become simpler than before, because now one can do without the Carleson contour construction, using Cauchy's residue theorem instead. 3) Given an inner function 01, one can apply Marshall's version of Carleson's method [G, chapter viii, $41 to construct an interpolating Blaschke product O2 satisfying (19) with some absolute constants Ej, E j E (0,l).
fel
Remarks added in proof. 1) One can drop the 'connected level set condition' imposed on 0 in Theorem 3: the inequalities IVO((, 5 llfllco constlVOll* hold whenever 0 is an inner function and f E K r . The proof relies on a recent result of W.S. Cohn. The improved version of Theorem 3 is contained in my paper 'Entire functions of exponential type and model subspaces in HJ", Zap. Nauchn. Semin. Leningrad. Otd. Mat. Inst. Steklova (LOMI), 190(1991), 81-100. 2) Conditions (ii), (iii) and (iv) in Theorem 7 are equivalent. They are also
