Splitting and boundary behavior in certain H2 spaces

3.6.

SPLITTING AND BOUNDARY BEHAVIOR IN CERTAIN H 2 SPACES*

Let # be a finite Borel measure with compact support in ~ Even for very special choice of # the structure of H2(#), the L2(p)-closure of the polynomials, can be mysterious. We consider measures # = v + Wdm, where v is carried by ~ and W is in L1(m). If l o g W is in Ll(m), H2(#) is well understood and behaves like the classical Hardy space H2(m) [I]. We assume that v is circularly symmetric, having the simple form dw = G(r)rdrdg, where G > 0 on [0, I]. Hastings [2] gave an example of such a measure with W > 0 m-a.e, and such that H2(#) = H2(v) ~ L2(Wdm); we say then that H2(#) splits. A modification of this example will show that given any W with [0, I] such that H2(#) 2, so that

l~W~=-m,

splits.

G can be chosen to be positive and nonincreasing

on

Suppose G is smooth and there exist C, C > 0, and d, 0 < d <

(I) for 0 ~ r < I.

Suppose further that for some c, ~ > 0,

10(~

THEOREM T,

1

[3].

Let G satisfy

'l - t,)

(2)

O'(~)a4, <+ Oo.

-~'~ _

(I) and (2).

and that W = 0 on a set of positive measure

,

Suppose that

~ < ~

for some arc F of

Then H2(#)

in ~

splits if and only if

4

<3) for small ~. This theorem settles the question of splitting only when W is well behaved. Conditions similar to (3) were introduced by Keldy~ and D~rba~jan, and have beer used by several authors in the study of other closure problems (ef. [4, 5]). Question I. Can W be found such that splitting occurs when the integral nite, or even when G z I?

in (3) is fi-

For h in ~ the point evaluation p § p(%) is a bounded linear functional on H2(~) (at least for those # that we are considering); let E#(h) denote its norm. If EV(h) is analogously defined, then E#(%) ~ EW(%) (an upper bound for E#). It is easy to show that H2(#) splits if and only if E#(%) = Ev(%) for all h in ~ At the other extreme, there is always an asymptotic

lower bound for E # [6]:

a.e. on an arc Y of

T

(I-~)E (~8) ~4/W(e)

m-a.e.

Sometimes

equality holds

:

b(4-~) E(~e) =w~) ~'-'~e. o,~ e.g., if l o g W is Ll(m), then (4) holds with F = T . One verifies a.e. on F and if (4) holds, then H2(~) cannot split. THEOREM 2 [7].

Suppose that IFb~W~ttp>-~

(4) that if W does not vanish

and

I

~-~

(5)

0

*THOMAS KRIETE. ginia 22903.

2214

Department

of Mathematics,

University

of Virginia,

Charlottesville,

Vir-

Then (4) holds, every f in H2(~) has boundary values f(e i0) m-a.e, on F, f = f m-a.e, on F, and I ~ 9 1 ~ I ~ , > - ~ i

whenever J is a closed arc interior to F and f ~ 0 in H2(M).

Every zero se~

--

for H2(~) with no limit points outside of J is a Blaschke sequence. The hypothesis on W is weaker than that in Theorem I, (5) is stronger than finiteness of the integral in (3) and the conclusion is stronger than the "only if" conclusion of Theorem I by an unknown amount. Equation (4) can fail if the hypothesis on W is removed. Fix ~, 0 < ~ < I, and let .

G satisfies

(5).

Define

G(~ = e/s.,p(- (4 _~),

~(0,$)=C~/~)?11,{~:~E[8-~,0 ~ ]

(6) , W(~)4e_x~p~-')}

THEOREM 3 [3]. ~ 7

and note t h a t 0 ~ ~ ~

If G is as in (6), there exist constants a, b > 0 with i0 for all re in ~ .

1.

E~C~e~e)>

If I. ~ W ~ > - ~ , then 9(0, 6) = 0(6 ~) as 6 § 0 m-a.e, on P and Theorem 3 yields no imPO formation near ~. On the other hand, for any d, d > I, one can construct W, W > O, a.e. with ~(O, 6) > (eonst)(--log 6)-d for 6 small and all 8 [3]. Thus (4) can fail even if (5) holds. Question 2. or that G ~ I.

Assume that the integral in (3) is finite, or even that G is given by (6), Is there a measurable

set E, E C ~

, with

where the first summand consists of "analytic" functions? Might such an E contain any are on which ~(0, 6) (or a suitable analogue) tends to zero sufficiently slowly as ~ + 0? If there is no such E with mE > 0, exactly how can the various conclusions of Theorem 2 fail, if indeed they can? Question 3. Let W(@) be smooth with a single zero at e = 0. Assuming the integral in (3) is finite, describe the invariant subspaces of the operator "multiplication by z" on H2(~) in terms of the rates of decrease of W(@) near 0 and G(r) near I. Perhaps more complete results can be obtained than in the similar situation discussed in [8]. Finally we mention that the study of other special classes may be fruitful. Recently A. Lo Volberg has communicated interesting related results for measures ~ + Wdm where v is supported on a radial line segment. LITERATURE CITED I. 2. 3. 4. 5. 6. 7.

S. Clary, "Quasi-similarity and subnormal operators," Doctoral Thesis, Univ. Michigan (1973). W. Hastings, "A construction of Hilbert spaces of analytic functions" (preprint). T. Kriete, "On the structure of certain H2(~) spaces" (to appear). J . E . Brennan, "Approximation in the mean by polynomials on non-Carath$odory domains," Ark. Mat., 15, 117-168 (1977). S . N . Mergelyan, "On the completeness of systems of analytic functions," Usp. Mat. Nauk, 8, No. 4, 3-63 (1953). T. Kriete and T. Trent, "Growth near the boundary in H2(M) spaces," Proc. Am. Math. Soc., 62, 83-88 (1977). T. Trent, "H2(~) spaces and bounded evaluations," Doctorial Thesis, Univ. Virginia

(1977). 8.

T. Kriete and D. Trutt, "On the Cesaro operator," (1974).

Indiana Univ. Math. Jo, 24, 197-214

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