UNIMODULAR FUNCTIONS AND UNIFORM BOUNDEDNESS

UNIMODULAR FUNCTIONS AND UNIFORM BOUNDEDNESS J . FERNÁNDEZ, S . HUI, H . SHAPIRO In this paper we study the role that unimodular functions play in deciding the uniform boundedness ofsets of continuous linear functionals on various function spaacs . For instance, inner functions are a UBD-set in H°° with the weak-star topology. 1 . Introduction Let M be a topological vector space and M* its dual space. We say that a set S C M is a uniform boundedness deciding (UBD) set if whenever ~D C M* satisfies sup fl W(u) 1 : cP E ~¿} < oo for each u E S, then -P is uniformly bounded on an open neighborhood of 0. If M is the dual space of a normed space and we endow M with the weak-star topology, then oP uniformly bounded on weak-star open neighborhood of 0 implies that it is uniformly bounded on norm open neighborhood of 0. In this case, we have sup UP 11(M,II .II)-< oo . (PES a a By the uniform boundedness principie, any set of second category is a UBD-set . Let H°° denote the Hardy space on the unit circle . We do not know if the set of inner functions is a UBD-set in H°° with the norm topology. Since the set of linear combinations of inner functions is of first category in (H°°, 11 11)[91, an affirmative answer cannot follow from the classical uniform boundedness principie. An affirmative answer would have as a simple consequence Marshall's theorem [4, p. 196] : Theorem (Marshall) . H°° is ¡he closed linear span of the Blaschke producís. The question of whether the set of inner functions is a UBD-set in (H°°, was raised in [3] . In the same paper it is proved that the set of inner functions is a UBD--set in H°° with the weak-star topology, which we denote by (H°°, w*) . 140 J . FERNÁNDEZ, S . HUI, H. SHAPIRO See [9] for a different proof. In [5], it is proved that the set of Blaschke products is a UBD-set in (H°°, w*). Suppose (X, p) is a positive a-finite measure space. Let LP(X,p) denote the usual Lebesgue spaces, and unless indicated otherwise, LP(X,h) can be the space of real-valued functions or the space of complex-valued functions . In this paper we prove Theorem 1 . The set of unimodular functions is a G6 set in ¡he closed unit ball of L°°(X, M) with the weak-star topology . Since every closed set in a metric space is a G6, the conclusion of Theorem 1 applies also to the unimodular functions in any weak-star closed subset of L°°(X, t.) . Combining this remark with Carathéodory's Theorem we have Corollary 2. The set of inner functions is a dense G6 in the closed zenit ball of (H°°,w*) . Using a technique similar to that used in the proof of Theorem 1, we prove the following generalization of Carathéodory's Theorem. Theorem 3. The set of Blaschke products is a dense G6 in the closed unit ball of (H',w*) . By the Banach-Alaoglu Theorem, the closed unit ball of the dual space of a Banach space is compact and Hausdorff with the weak-star topology, and thus a Baire space [10, p. 200] . The results of [3], [5] then follows from Corollary 2, Theorem 3, and the following [10, p. 2001 . Theorem. Suppose that X is a Baire space and that Y is a dense G6 subset ofX. Then a family of continuous functions that is poiniwise bounded on Y is uniformly bounded on an open subset ofX. The prooos of Theorems 1 and 3 are in section 2. Section 3 contains an application of Theorem 1 and Section 4 contains examples related to the hypotheses of the theorems . 2. Proof of Theorem 1 and Theorem 3 Proof of Theorem 1 : First suppose M(X) < oo . For a finite partition P of X, let ap(f) = ax(1 M(A) 1 1Afdl, l). It is clear that óp is weak-star continuous on the closed unit ball of L'(X, p) . Let P be the collection of finite partitions of X and let b(f) = inf 6p(f ) . PEP UNIMODULAR FUNCTIONS 141 Then 6 is a weak-star upper semicontinuous function on the closed unit ball of L'(X, ju) and clearly 0 < b < 1 . Since {ó = 0} = 00 1 n {6 < n }, n=1 we conclude that {ó = 0} is a G6 . We next show that {b = 0} is the set of unimodular functions. Suppose P is a finite partition of X. Then for each A E P, we have Summing over A E P, we obtain Therefore JA 1 f 1 dp ? ( 1 ap(f))p(A) . 6p(f) > 1 p(X) 1x 1 f 1 dp . 0<1_ 1 J f dp S(f) p(X) x Since Ilf jj, >,> _< 1, we easily see that {6 = 0} is a subset of the unimodular functions . Suppose f is unimodular . Divide the unit cirele into n equal parts Si , . . . , SI . Partition X with the sets of {f-1(S1), . . . , f-'(S I )} which have nonzero M measure. Call this partition P. Then for A E P we have Therefore p(A) I JA f I > cos . 0<b(f)<Sp(f)<1-cos 7'. n Hence 8(f) = 0 for f unimodular . Thus if p(X) < oo, the set of unimodular functions is a G6 in the closed unit ball of L°°(X, p) with the weak-star topology. For the general case, let X = U°°_1 SI with So = 0, SI C Sn+l, and 0 < h(Sn\Sn-1) < oo . Let _ XSn\S-I(x) J(x) n=l 2np(Sn\Sn_1) . 142 J. FERNÁNDEZ,' S . HUI, H. SHAPIRO Observe that 0 < J < oo . Define the measure v by dv = Jdp. Clearly v is a positive finite measure . It is also clear that L'(h) C L' (v), that L-(u) = = L'(v), and that F E L'(v) if and only if FJ E L1 (M). Therefore the weakstar topologies of L'(p) and Lw(v) are identical . By the finite measure case, we conclude that the unimodular functions is a Gá in the closed unit ball with the weak-star topology . Proof of Theorem 3 : We use the well known fact [4, p . 56] that if f E H°° with Ilf 11 < 1, then f is Blaschke product if and only if 21r lim log 1 f(re") 1 d9 = 0. r-.1 0 Recall that weak-star convergence implies uniform convergence on compact subsets of the open unit disk . By Jensen's formula, it is easy to see that for 0 < r < 1 the functions ;Pr(f) = exp(¡21r log I f(re'B ) I de) are weak-star continuous on the closed unit ball of H°° . Therefore the function CP(f) = sup cpr(f) 0<r<1 21r = exp(lim log 1 f(re' .0 dB) r.1 lo is weak-star lower semicontinuous . Hence {P=1}= I I{w> 11 } n=1 is a G6 . Using the fact stated in the beginning of the proof we see that {'P = 1 } is the set of Blaschke products .


a a
By the uniform boundedness principie, any set of second category is a UBD-set .
Let H°°denote the Hardy space on the unit circle .We do not know if the set of inner functions is a UBD-set in H°°with the norm topology.Since the set of linear combinations of inner functions is of first category in (H°°, 11 -11)[91, an affirmative answer cannot follow from the classical uniform boundedness principie.An affirmative answer would have as a simple consequence Marshall's theorem [4, p. 196] : Theorem (Marshall) .H°°is ¡he closed linear span of the Blaschke producís.
The question of whether the set of inner functions is a UBD-set in (H°°,  was raised in [3] .In the same paper it is proved that the set of inner functions is a UBD--set in H°°with the weak-star topology, which we denote by (H°°, w*) .
See [9] for a different proof.In [5], it is proved that the set of Blaschke products is a UBD-set in (H°°, w*).
Suppose (X, p) is a positive a-finite measure space.Let LP(X, p) denote the usual Lebesgue spaces, and unless indicated otherwise, LP(X, h) can be the space of real-valued functions or the space of complex-valued functions.In this paper we prove Theorem 1 .The set of unimodular functions is a G6 set in ¡he closed unit ball of L°°(X, M) with the weak-star topology .
Since every closed set in a metric space is a G6, the conclusion of Theorem 1 applies also to the unimodular functions in any weak-star closed subset of L°°(X, t.) .Combining this remark with Carathéodory's Theorem we have Corollary 2. The set of inner functions is a dense G6 in the closed zenit ball of (H°°,w*) .
Using a technique similar to that used in the proof of Theorem 1, we prove the following generalization of Carathéodory's Theorem.Theorem 3. The set of Blaschke products is a dense G6 in the closed unit ball of (H',w*) .
By the Banach-Alaoglu Theorem, the closed unit ball of the dual space of a Banach space is compact and Hausdorff with the weak-star topology, and thus a Baire space [10, p. 200] .The results of [3], [5] then follows from Corollary 2, Theorem 3, and the following [10, p. 2001.
Theorem.Suppose that X is a Baire space and that Y is a dense G6 subset of X .Then a family of continuous functions that is poiniwise bounded on Y is uniformly bounded on an open subset of X .
The prooos of Theorems 1 and 3 are in section 2. Section 3 contains an application of Theorem 1 and Section 4 contains examples related to the hypotheses of the theorems .

Proof of Theorem 1 and Theorem 3
Proof of Theorem 1 : First suppose M(X) < oo .For a finite partition P of X, let It is clear that óp is weak-star continuous on the closed unit ball of L'(X, p).Let P be the collection of finite partitions of X and let b(f) = inf 6p(f ) .PEP Then 6 is a weak-star upper semicontinuous function on the closed unit ball of L'(X, ju) and clearly 0 < b < 1 .Since {ó = 0} = 00  1 n {6 < n }, n=1 we conclude that {ó = 0} is a G6 .We next show that {b = 0} is the set of unimodular functions.Suppose P is a finite partition of X.Then for each A E P, we have Summing over A E P, we obtain Since Ilf jj,>,> _< 1, we easily see that {6 = 0} is a subset of the unimodular functions.
Suppose f is unimodular .Divide the unit cirele into n equal parts Si , . . ., SI .Partition X with the sets of {f-1 (S1), . . ., f-'(S I )} which have nonzero Mmeasure.Call this partition P. Then for A E P we have Therefore n Hence 8(f ) = 0 for f unimodular .
Thus if p(X) < oo, the set of unimodular functions is a G6 in the closed unit ball of L°°(X, p) with the weak-star topology.
Observe that 0 < J < oo .Define the measure v by dv = Jdp.Clearly v is a positive finite measure.It is also clear that L'(h) C L' (v), that L-(u) = = L'(v), and that F E L'(v) if and only if FJ E L1 (M).Therefore the weakstar topologies of L'(p) and Lw(v) are identical.By the finite measure case, we conclude that the unimodular functions is a Gá in the closed unit ball with the weak-star topology . is a G6.Using the fact stated in the beginning of the proof we see that {'P = 1 } is the set of Blaschke products.

Application
As an application of Theorem 1, we have the following generalization of the main result in [3] .Let (X, lí) be a a-finite positive measure space.Theorem 4. Suppose M is a weak-star closed subspace of L°°(X, p) .If the unimodular functions are weak-star dense in the closed unit ball of M, then ¡he set of unimodular functions is a UBD-set in M with ¡he weak-star topology.
Theorem 4 is trivial when M = L' (X, p) since every f in the unit ball of LO (X, ti) can be written as f = (u l + u2)/2, where ul , u2 are unimodular.When M = LO (X, p), Theorem 4 is a consequence of the following theorem of Nikodym [2, p. 309] .
Theorem (Nikodym) .Suppose -P ís a subset of the space of countably additive measures defined on a Q field E of subsets of X.If for each E E E we have sup I ;P(E) I< oo, WEID sup sup I cp(E) I < oo .

EEEsPE-P
An interesting question is whether Theorem 4 has an analogue when M is norm closed and <~C L'(X, Es)* .Since the unimodular functions of M are never norm dense in the unit ball, a natural hypothesis seems to be the density of the convex combinations of the unimodular functions .However, there is an example in [3] which shows that this is not sufficient even if -¿ C Ll (X, p) .When M = L°°(X, h), then the analogue of Theorem 4 is true with no density assumptions by the Nikodym-Grothendieck Theorem [1, p. 80] .
A necessary condition for the conclusion of Theorem 4 to hold is the weakstar density of the linear span of the unimodular functions in M. Example 1 below shows that it is not sufficient .We need the following .e Theorem 5. Let n l < n2 < n3 < . . .be a sequence of positive integers with the property that there is a sequence of positive iniegers ml < mz < . . .so that each mj divides all but a finite number of the nk's .Let M be the weak-star closure of the linear span of {1, zn,, zn2, . . .} in H°°.Then a function in M that is unimodular on an arc of the unit circle has the forro C,Znk .
Proof.Suppose f = ~~_ o aj znt E M has unit modulus on the open arc "y.
If aj = 0 for j > N, a positive integer, then I f la is a real analytic function and it follows that I f 12 = 1 on the whole unit circle .Thus f is a finite Blaschke product .Since the only polynomial Blaschke products have the forro czI, we are done if only a finite number of the aj's are nonzero .
Suppose an infinite number of the aj 's are nonzero .Let wj = e"/m ; and let Since mj divides all but a finite number of the nk's, Pj is a polynomial.It is easy to see that lim sup deg(Pj) = oo .
Choose wj so small that y (1 wjy :~0 and that deg(Pj) is greater than the order of the zero of f at the origin.Suppose pá(z) = bo + blz n, + . . .+ blz n, b1 9¿ 0 .
Since f(z) and f(wjz) are unimodular on y fl wjy, we have on that interval Equality then must persist throughout the open unit disk.We obtain a contradiction by multiplying both sides of the above by zn' -1 and letting z tend to zero.This completes the proof.Further examples .The following example shows that the set of all un¡modular functions in Theorem 4 cannot be replaced by the set of polynomials with unit norm.
Example 2 .Let P be the set of polynomials with unit norm on the unit circle .It is well knówn that P is weak-star dense in the unit ball of H°° [4, p. 6].Let cP n = 1 :~=0 e -'jo.It is clear that if p(z) = ~N0 alzj, then

Proof of Theorem 3 :
We use the well known fact[4, p. 56] that if f E H°°w ith Ilf11 < 1, then f is Blaschke product if and only if21r lim log 1 f(re") 1 d9 = 0. r-.1 0Recall that weak-star convergence implies uniform convergence on compact subsets of the open unit disk.By Jensen's formula, it is easy to see that for 0 < r < 1 the functions ;Pr(f) = exp(¡21r log I f(re'B ) I de)are weak-star continuous on the closed unit ball of H°°.Therefore the function CP(f) = sup cpr(f) Example 1 .Consider the sequenceIts structure is determined by taking arithmetic progressions of length 2,3,4, . ...Hence it is not a Sidon set[6, p. 51].Construct M as in Theorem 5. Then there is f = E' o alznj E M with Fji °0 1 aj 1 = oo.It is clear that the above sequence satisfies the hypothesis of Theorem 5, and therefore the only unimodular functions are of the form cznk .Let Aj 1= 1 and Alai =J aj 1. Clearly for each unimodular u E M, we have wherel 0 Therefore 11 ;PNJIM " is not bounded.