3.6.
SPLITTING AND BOUNDARY BEHAVIOR IN CERTAIN H 2 SPACES*
Let # be a finite Borel measure with compact support in ~ ...
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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
0,
10(~
THEOREM T,
1
[3].
Let G satisfy
'l - t,)
(2)
O'(~)a4, 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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