Integr Equat Oper Th Vol. 20 (1994)
AN
INVARIANT
0378-620X/941020243-0651.50 § 0.20/0 (c) 1994 Birkhiuser Verlag, Basel
SUBSPACE
PROBLEM
FOR
p = 1 BERGMAN
SPACES
ON SLIT DOMAINS WILLIAM T. ROSS ABSTRACT. In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces At(G) and AI(G\K), where G is a bounded region in the complex plane and K is a compact subset of a simple arc of class C 1. 1. INTRODUCTION For a bounded region U C C, we define the Bergman space AI(U) as the space of analytic functions f on U with fv Ifl dA < cc (Here dA is Lebesgue measure on C) and the operator S on An(U) by (Sf)(z) = zf(z). Characterizing the subspaces A4 of An(U) for which SAd C A4 is a difficult and unsolved problem which has received considerable attention over the past 40 years. In this paper, we give a complete characterization of the S-invariant subspaces A4 with (1.1)
AI(G) C .M C AI(G\K).
Here G is a bounded region in C and K is a compact subset of a simple arc of class C 1. This paper will be a continuation of an L p version of this problem [5] to the largest of the Bergman spaces p = 1. Different techniques will be used here since the papers mentioned above use duality and the reflexivity of L p, a luxury not afforded us in the non-reflexive setting of L 1. Our main theorem is: T h e o r e m 1.1. For .A4 of type (1.1) and S-invariant, there is a closed F C K with M =
AI(GkF). The author would like to thank Prof. Peter Jones and Prof. Dmitry Khavinson for a helpful conversation. 2. PRELIMINARIES Before proceeding, we point out that some of the techniques used here are somewhat standard and fall under the general name of 'Havin's lemma'. We refer the reader to [3] and [8], Ch. 4, section 2, for further details. For the sake of completeness, we will outline these results here. For a bounded region U in the complex plane C, we identify the dual of La(U) = Lt(U, dA) with L~(U) = L~(U, dA) by the bi linear pairing
< f,g >= ~ fgdA,
f E LI(U), g e L~(U).
244
Ross
For a linear manifold X in LI(U) we let X • be the annihilator of X and note that X • is weak-star closed in L~176 For a linear manifold Y in L~176 we let • be the preannihilator of Y and note that • is norm closed in L1(U). We also note that by the Hahn-Sanach theorem • • is the norm closure of X in LI(U) and (•177 is the weak-star closure of Y in L~ L e m m a 2.1. At(U) • is the weak-star closure of OCt(U), where -0 = l(O/Ox + iO/Oy).
Proof. By Weyl's lemma [2], p. 172, •176
= AI(U), hence
(• (-OCt(U)))•
(2.1)
AI(U) •
By Hahn-Banach, the left-hand side of (2.1) is the weak-star closure of-0C~~
[]
R e m a r k : We will show, in Proposition 3.1, that in fact -OCt(U) is weak-star sequentially dense in AI(U) • a technicality that will be important later in the paper. We now relate AI(U) j- with a certain type of Sobolev space on U via 0. Let }/Y = D2(C) be the Banach space of f E L ~176 = L~176 dA) such that Of (in the sense of distributions) belongs to L ~176We norm/41 by IIfl[w = []f[]oo + I]Of[[oo. R e m a r k : We pause here for a moment to mention that W contains, but is not equal to W I ' ~ ( C ) , the Sobolev space of f E L ~ whose first partial derivatives (in the sense of distributions) also belong to L~ In fact, if f E Fr then the first partial derivatives belong to BMO but are not always bounded (see [91 and [101, p. 164, and [41). For g E L ~176 with compact support, define the Cauchy transform Tg by
(Tg)(w) = _ 1 [
(2.2)
7r .l
g(z) dd(z) Z
--
W
and note that Tg is continuous on C ([11], p. 40), analytic off of the support o f t , (Tg)(oo) = 0, and r = T(Or for all r E C ~ ([2], p. 170). L e m m a 2.2. Every f E 14] has a continuous representative.
Proof. Let f E 1,V and r E C ~ . Note that e f E 14] and, by distribution theory [2], p. 174 175, e f = T(O(r a.e. (dA). Since T(O(r is continuous, we can conclude that f has a continuous representative.
[]
Assuming now, and for the rest of the paper that all functions in t'Y are continuous, we let 1/Y0(U) be the subspace of functions in }/Y which vanish off of U. (Wo(U) is not the same as the closure of C~(U) in the 142 norm.) For f E }'V0(U), one sees from Lemma 2.2 that f = T(-Of), and thus for w E U (2.3)
If(w)l
< l fu I-Of(z)ldA(z) < CgHSfll~. -
Iz -
~l
-
Here Cu is a positive constant depending only on the region U. Hence an equivalent norm on D20(U) is Ilfllw0 = Ilaftl~.
Ross
245
If f E )d0(U) and w r U, then
But since AI(U) is the closed linear span of {(z - w) -1 : w • U} [1], then -Of e At(U) • and moreover -O: )d0(U) --, AI(U) • is an isometry.
Proposition 2.3. -O: )d0(U) --~ AI(U) • is invertible.
Proof. Since -O is an isometry, it suffices to show that -O is onto. To this end, let g E AI(U) • By distribution theory [2], p. 174, -O(Tg) = g E L ~, so Tg E )d. Since g e A~(U) • then by (2.2), (Tg)(w) = 0 for all w r V. Hence Tg 9 Wo(U). [] Proposition 2.4. I'V0(U) can be equivalently re-normed to make it a Banach algebra.
Proof. If f = T(-Of) and g = T(-Og) both belong to )d0(U), then one has ([2], p. 178, Lemma 3.11) f g = T(f-og + g-of), hence -o(fg) = f o g + g-of,
(2.4)
from which we obtain H-O(fg)H~ -< []f-ogllor + l[9-ofl]~. Using (2.3) will yield II-o(fg)]loo _< 2Cul]-of][oo]l-ogllo~. We conclude from this that 14;0(U) can be equivalently re-normed to make it a Banach algebra. []
3. INVARIANTSUBSPACES Define the operator R on AI(U) • by (Rg)(z) = zg(z) and M on the Sobolev space I'Y0(U) by (Mh)(z) = zh(z) and notice that R and M are well defined and continuous. For f 9 )d0(U), observe that
-o(zf) = z-Of, and thus -OM = RO. If 3/[ is invariant with AI(G) C .Ad C A I ( G \ K ) ,
(3.1) we can take annihilators to get (3.2) with R3,t • C 3,4 •
A I ( G \ K ) • C .M • C AI(G) • Taking T = ~-1 (Proposition 2.3) of both sides of (3.2) will yield
W o ( a \ I ( ) c T M • c Wo(G) and using -OM = R-O, we get that T M • is z-invariant. We will eventually show that TAd • is an ideal of the Banach algebra I'Y0(G) and that T.M • -- I'Yo(G\ZM), and hence
.Ad = A I ( G \ Z ~ ) , where (3.3)
Z ~ = {z e K : (Tg)(z) = 0 Vg e Adz},
but first we need a few preliminary lemmas. In Lemma 2.1, we saw that -OC~~ is weak-star dense in Aa(G) • us slightly more.
This next result gives
246
Ross
Proposition 3.1. -OC~~
is weak-star sequentially dense in
AI(G) •
The proof of Proposition 3.1 will depend on the following l e m m a which uses a certain "mollifier" of Ahlfors [1]. L e m m a 3.2. Let h G Wo(G). Then there is a sequence hn E 1Wo(G) with supp(h~) C G and -Ohm~ O h weak-star.
Proof.
Since h - 0 off of G, then one can show [11], p. 40, that for all z and w in C Ih(z) - h(w)l ___Clz
Thus if
d(z)
-
w[I log Iz - wll-
equals the m i n i m u m of e -2 and dist(z, OG), t h e n
(3.4)
]h(z)] _< Cd(z)l log d(z)l.
We now construct the "Ahlfors mollifier" wn as follows [1]: Let differentiable function on lie with 0 < j < 1, j(t) = 0 for all t < 1, and FornENandzE Glet / \ n (3.5) w~(z) = j ~ log log d(z) )
j(t) j(t)
he an infinitely = 1 for all t > 2.
and notice that w~ - 0 near cOG. Thus define wn on C be defining w~ ~ 0 off G. Since d(z) is Lipschitz continuous with constant 1 and j'(t) = 0 outside 1 < t < 2, one can check [1] that C
1
I"Own(z)l < n d(z)llogd(z)l Vz c G.
(3.6)
-
Hence w~ E ]W0(G) and so, by Proposition 2.4, h,~ =_ w,~h also belongs to }42o(G) with supp(h~) C G. We now show that "Oh~ ~ "Oh weak-star. If f C L 1(G), then by (2.4) (3.7)
I L f("Oh,~-Oh)dAl 0 be giveiz. Then there is a polynomial p(z) and a 9 CI(c) with (i) ~ = r on 7
(ii) l i P - ~ll~ < c (iii) IIO(p- ~)11~ < ~.
248
Ross
As a consequence of this lemma, we have t h a t TAd • is not only z - i n w r i a n t but invariant under multiplication by any C ~ ( G ) function. C o r o l l a r y 3.5. / r e E C~~
then ~b(T.Ad • C T.hd "L.
Proof. Let r E Cg~ and ~ > 0 be given and kO and p be as in L e m m a 3.4. If f E T M • then q t f C Wo(G) and t g f - C f = 0 on K , so tgf - C f E W o ( G \ K ) C TAd • Hence dist(r
= d i s t ( ~ f , TAd •
TAd •
< I[O(pf - ~f)[[oo < Ce]l-Of][~
[]
This i m m e d i a t e l y yields the following: Proposition
3.6. TAd • is an ideal of Wo(G).
Proof. Let f E TAd • and g C Wo(G) and notice that f g E W0(G). E m p l o y i n g the weakstar sequential density of O C t ( G ) in AI(G) • Proposition 3.1, we can find a sequence r C Cg~ with ~4,~ ~ 0g weak-star. By Corollary 3.5, r C Tfl4 • and by L e m m a 3.3, 0(Ca f ) ---* O(fg) weak-star. Since .A4• is weak-star closed, then O(fg) E Ad• hence f g C TAd • [] Proof of Theorem
1.1
Let Z ~ be as in (3.3). Since W o ( G \ K ) C TAd • , then TAd • C W o ( G k Z ~ ) and so To prove A ~ ( G k Z ~ ) • C A d • we apply L e m m a 3.2 to see that it suffices to show that ~ r E Ad• for all r E W o ( G k Z ~ ) with support in GkZM. For this, we use an a r g u m e n t of Sarason [7], p. 41, L e m m a 1, along with the fact that TAd • is an ideal to find a g E TAd • with g - 1 on the support of r Thus, since T.Ad • is an ideal, r = g r E TAd • and hence ~ r C .Ad• []
Ad• C A I ( G \ Z ~ ) •
REFERENCES 1. L. Bets, 'An approximation theorem', J. Analyse Math., 14 (1965)~ 1 - 4. 2. J.B. Conway, The Theory of Subnormal Operators, Math. Sur., 36, Amer. Math. Soc., Providence, Rhode Island, 1991. 3. V.P. Havin, 'Approximation in the mean by analytic functions', Soviet Math. Dokl., 9 (1968), 245 - 248. 4. T. Iwaniec, 'The best constant in a BMO-inequality for the Beurling-Ahlfors transform', Mich. Math. J., 33 (1986), 387 - 394. 5. W.T. Ross, 'Invariant subspaces of Bergman spaces on slit domains', to appear, Bull. London Math Soc. 6. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. 7. D. Sarason, Invariant subspaces, Studies in Operator Theory, Amer. Math. Soc. Surv., 13 (1974), 1 - 47. 8. H.S. Shapiro, The Schwartz fnnclion and its generalizations to higher dimensions, Wiley, New York, 1992. 9. E.M. Stein, 'Singular integrals, harmonic functions, and different• properties of functions of several variables', Proe. Syrup. in Pure Math., 10 (1967), 316 - 335. 1O. E.M. Stein, Singular Integrals and Differentiabilily Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970. 11. I.N. Vekua, Generalized Analytic Fnuctions, Addison-Wesley, Reading, M~ss., 1962. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF RICtIMOND, RICHMOND: VA 23173, USA
E-mail address:
[email protected] MSC 1 9 9 1 :
Primary
Submitted: July Revised: January
47B38,
5, 1 9 9 5 30, 1994
Secondary
32A57,30B40
