INNER FUNCTIONS IN THE HYPERBOLIC LITTLE BLOCH CLASS. Wayne Smith

Transcription

1 INNER FUNCTIONS IN THE HYPERBOLIC LITTLE BLOCH CLASS Wayne Smit Abstract. An analytic function ϕ mapping te unit disk into itself is said to belong to te yperbolic little Bloc class if te ratio (1 z 2 ) ϕ (z) /(1 ϕ(z) 2 ) converges to as z 1, wile ϕ is in te little Bloc space if just te numerator of tis expression converges to zero. Several constructions of inner functions in te little Bloc space ave recently appeared. In tis paper we construct a singular measure on te unit circle suc tat te associated singular inner function is in te yperbolic little Bloc class. 1. Introduction Te yperbolic derivative of an analytic self-map ϕ : D D of te unit disk is given by ϕ /(1 ϕ 2 ). To explain te terminology, we note tat integrating ϕ /(1 ϕ 2 ) over a rectifiable curve γ in D gives te yperbolic arclengt of ϕ(γ). Tis notion of derivative as been used by Yamasita to study yperbolic versions of te classical Hardy and Diriclet spaces; see [Y1] and [Y2]. More recently, in [MM] and [SZ], yperbolic derivatives ave been sown to be pertinent to te study of composition operators on certain subspaces of H(D), te space of analytic functions on D. An analytic self-map ϕ of D induces a linear operator C ϕ : H(D) H(D), defined by C ϕ f = f ϕ. Tis operator is called te composition operator induced by ϕ. Recall tat an analytic function f on D is said to belong to te Bloc space B provided (1 z 2 ) f (z) is uniformly bounded for z D. Similarly, f B, te little Bloc space, if (1 z 2 ) f (z) uniformly as z 1. Te yperbolic Bloc class B is defined by using te yperbolic derivative in place of te ordinary derivative in te definition of te Bloc space. Tat is, ϕ B if ϕ : D D is analytic and (1 z 2 ) ϕ (z) sup z D 1 ϕ(z) 2 <. Similarly, we say ϕ B, te yperbolic little Bloc class, if ϕ B and (1 z 2 ) ϕ (z) lim z 1 1 ϕ(z) 2 =. Note tat tese are not linear spaces, since ϕ is required to be a self-map of D. It is an easy consequence of te Scwarz-Pick lemma tat every analytic self-map of D belongs to B, and in fact te supremum above is at most 1; see [G, p.2]. Membersip in te yperbolic little Bloc class, on te oter and, is nontrivial Matematics Subject Classification. Primary 3D5 Secondary 3D45, 26A3, 47B38. Key words and prases. inner function, yperbolic Bloc class, smoot singular function. Typeset by AMS-TEX 1

2 2 WAYNE SMITH It is easy to see tat C ϕ : B B is bounded for every analytic self-map ϕ of D, wile C ϕ : B B is bounded if and only if ϕ B. It is a recent result of K. Madigan and A. Mateson tat C ϕ : B B is compact if and only if ϕ B ; see Teorem 1 in [MM]. Membersip of ϕ in B as a simple geometric interpretation wen ϕ is univalent, since (1 z 2 ) ϕ (z) is comparable to te distance from ϕ(z) to ϕ(d). Tis results in C ϕ aving very strong properties. In particular, te autor sowed tat if ϕ is univalent and in B, ten C ϕ : L p a H q is compact for all < p < q < ; see Teorem 6.4 in [Sm]. Here L p a and H q are te classical Bergman and Hardy spaces. Tis paper resulted from an effort to understand B wen te univalence assumption is not made. Our main result, Teorem 1.2 below, sows tat B contains inner functions. Tus ϕ B does not even imply tat C ϕ is compact on H 2, since an inner function can not induce suc an operator. We introduce te notation τ ϕ (z) = (1 z 2 ) ϕ (z) 1 ϕ(z) 2, so tat ϕ B if and only if lim z 1 τ ϕ (z) =. Our first result sows tat tere is a restriction on te average rate at wic τ ϕ can go to zero wen ϕ is an inner function. 1.1 Teorem. If ϕ is an inner function, ten D τ ϕ (z) 2 da(z) =. 1 z 2 Wile it is clear tat finite Blascke products belong to B, it is not obvious tat B contains oter inner functions as well. Several constructions of suc functions ave appeared in te literature recently; see [Sa], [St], [B1], and [B2]. On te oter and, it is not obvious tat B contains any inner functions at all. In particular, it is easy to see tat if ϕ is an inner function in B, ten te wole unit circle is in te singular set for ϕ, i.e. ϕ does not ave an analytic continuation across any arc in D. Tus B contains no finite Blascke products. Our main result is tat tere are inner functions in B. 1.2 Teorem. Let η be a nonnegative increasing function suc tat 1 η(t) 2 dt = and η(4t) 2η(t), < t < t, t for some t >. Ten tere exists is an inner function ϕ and a constant C suc tat (1.1) τ ϕ (z) Cη(1 z 2 ). From Teorem 1.1, we see tat Teorem 1.2 is, subject to te regularity assumption on η, te best possible result of tis kind. A typical function satisfying te ypoteses of Teorem 1.2 is η(t) = log t 1/2. Te result remains valid wen te regularity assumption tat η(4t) 2η(t) is replaced by η(4t) (4 ɛ)η(t), for some ɛ >. Tis is equivalent assuming tat η is of upper type less tan one; see [J]. It is for clarity of presentation tat te simpler form of tis regularity condition is used ere. Since containment in B caracterizes compact composition operators on B, we get te following corollary to Teorem 1.2.

3 INNER FUNCTIONS IN B Corollary. Tere exists is an inner function ϕ suc tat C ϕ : B B is compact. Te function constructed to prove Teorem 1.2 will be a singular inner function ϕ. A Möbius map σ from D onto D satisfies te identity 1 σ(z) 2 = (1 z 2 ) σ (z), and from tis it is easy to ceck tat τ ϕ = τ σ ϕ. It is well known tat σ can be cosen so tat σ ϕ is a Blascke product, so tere are Blascke products tat satisfy (1.1). It would be interesting to ave a description of te zero sets of Blascke products in B, suc as was given by C. Bisop in [B1] for B. Te singular set of eac suc Blascke product is te full unit circle, as noted above, and so every point on te unit circle is a limit of its zeros. To see wat is involved in te construction of te required singular inner function, let ϕ(z) = exp( F (z)), were 2π e it + z F (z) = e it z dµ(t) and µ is a positive singular measure. Ten (1.2) τ ϕ (z) = (1 z 2 ) F (z) exp( RF (z)) 1 exp( 2RF (z)) (1 z 2 ) F (z), RF (z) since x e x (1 e 2x ) wen x. For ϕ to belong to B, it is only required tat te numerator of tis estimate for τ ϕ (z) tends to as z 1, i.e. F B. Tis is ow an inner function in B was sown to exist by D. Sarason [Sa], wo observed tat F B if te indefinite integral f of µ belongs to te Zygmund class λ. Recall tat a continuous function f is said to belong to λ if te second differences 2 f(x) = f(x + ) 2f(x) + f(x ) are uniformly o() as. Te construction was ten completed by citing constructions of J. -P. Kaane [K], G. Piranian [P] and H. S. Sapiro [Sa] of increasing singular functions in λ. In te present situation, an appropriate lower bound for te denominator of te estimate for τ ϕ in (1.2) is also required. Suc a lower bound will result from a lower bound for te first differences f(x) = f(x + ) f(x) of f. We terefore need a construction of an increasing singular function tat produces appropriate estimates for f and 2 f, from below and above, respectively. Te constructions of Kaane, Piranian and Sapiro, referred to above, do not provide te required lower bounds for f. However, teir metods can be adapted to produce te required function. Te formulation of te resulting teorem and te construction will given in 3 below. It sould be remarked tat te assumptions made on η in Teorem 1.2 are essentially best possible for te existence of a monotone singular function f satisfying 2 f Cη(); see [K] and [Sa]. It also is of interest to express te estimate (1.2) for τ ϕ in terms of µ. Noting tat RF (z) is just te Poisson-Stieltjes integral of µ, it is easy to verify tat (1 z 2 ) F (z) RF (z) = 2π 2e it (e it z) 2 dµ(t) / 2π 1 e it z 2 dµ(t). Tus te positive singular measure µ we construct will ave te property tat sufficient cancellation occurs in te numerator for tis ratio to tend to as z 1. Te proof of Teorem 1.1 will be given in 2, and te construction of f is in 3. We begin 4 by proving a teorem tat sows ow te estimate we get for 2 f gives an

4 4 WAYNE SMITH estimate for te growt of te second derivative of te Herglotz integral of f. Tis is ten applied to prove Teorem 1.2. I would like to acknowledge te ospitality of Micigan State University, were I was visiting wen tis researc was done. In particular, I would like to tank Joel H. Sapiro for many elpful discussions on te material presented ere. 2. Proof of te Teorem 1.1 Te proof of Teorem 1.1 uses te following non-univalent cange of variable formula; see [S, p.398]. If g is a measurable function on D and ϕ : D D is analytic, ten D g ϕ(z) ϕ (z) 2 log(1/ z )da = D gn ϕ da. Here da is area measure on D and N ϕ is te Nevanlinna counting function, defined by N ϕ (w) = log(1/ z ), z ϕ 1 {w} w D \ {ϕ()}. We also recall Littlewood s Inequality, wic asserts tat if ϕ : D D is analytic, ten N ϕ (w) log 1 wϕ() w ϕ() for all w D \ {ϕ()}. Moreover, if ϕ is an inner function, ten equality olds for all w outside a set of logaritmic capacity ; see [L] or [S]. Proof of Teorem 1.1. Te cange of variable formula sows tat D τ ϕ (z) 2 (1 z 2 ) 2 log 1 z da(z) = D ϕ (z) 2 (1 ϕ(z) 2 ) 2 log 1 z da(z) = D N ϕ (w) (1 w 2 ) 2 da(w). Since ϕ is an inner function, equality olds outside a set of area measure in Littlewood s Inequality. Tus N ϕ (w) is comparable to 1 w 2 off tis set, and so tis last integral diverges. Hence te first integral above also diverges, wic finises te proof since log(1/ z ) is comparable to 1 z 2 for 1/2 < z < Construction of te increasing singular function In tis section we construct te increasing singular function f tat as te good estimates for bot f and 2 f described in te introduction. Te construction sould be compared to te constructions by Kaane and Sapiro in [K] and [Sa] of monotone singular functions in te Zygmund class λ. Wile tese constructions do produce te estimate for 2 f tat we need, tey do not provide te required estimate for f. Our construction uses ideas from bot [K] and [Sa]. We begin wit two elementary lemmas.

5 INNER FUNCTIONS IN B Lemma. Suppose tat {b j } is a sequence of positive real numbers suc tat b j 3b j+1, all j J, for some integer J. Ten tere is a constant C suc tat n b j 4 j Cb n 4 n, all n 1. j=1 Proof. Assume first tat n J. If J j n, ten b j 3 n j b n, and so n b j 4 j b n j=1 n j=j 3 n j 4 j Cb n 4 n. After increasing te constant C, tis estimate will old for n < J as well. 3.2 Lemma. Suppose tat te function η satisfies 1 η(t)2 t 1 dt =. Ten tere exists an increasing function ρ suc tat lim ρ(t) =, ρ(4t) 3 1 t + 2 ρ(t), and [η(t)ρ(t)] 2 dt =. t Proof. Let a = 1, and by induction coose a k, k 1, so tat < a k a k 1 4 and ak 1 η(t) 2 a k t dt 1. Now define ρ(t) = 1/ k + 1, a k+1 < t a k. It is easy to verify tat tis function as te stated properties. 3.3 Teorem. Let η be a nonnegative increasing function suc tat, for some t >, 1 η(t) 2 dt = and η(4t) 2η(t), < t < t, t and let ρ be te associated function from Lemma 3.2. Ten tere exists an increasing singular function f defined on [, 2π] and a positive constant C suc tat, for >, (3.1) f(x + ) 2f(x) + f(x ) Cη()ρ() and (3.2) f(x + ) f(x) C 1 ρ(),

6 6 WAYNE SMITH provided [x, x + ] [, 2π]. Proof. Following Sapiro, te required function f will be presented as a limit of functions {f n }, were eac function is constructed from its predecessor using te basic building block g(x) = sin x 2 sin 2x. 4 Note tat g, g, and g all vanis at and 2π, and furter g(x) 3 4, g (x) 1, g (x) < 3 2. Let ρ be te function from Lemma 3.2 and define, for integers n, Note tat if 4 n < t, ten b n = η(4 n )ρ(4 n ) and c n = 2η(1) η(4 n ). (3.3) b n = η(4 n )ρ(4 n ) 2η(4 n 1 ) 3 2 ρ(4 n 1 ) = 3b n+1, and also (3.4) b 2 n = n= [η(4 n )ρ(4 n )] 2 1 log 4 n= n= 4 n [η(t)ρ(t)] 2 4 t n 1 dt =. By multiplying ρ by a constant, we may assume tat b = 1. Ten, using tat η and ρ are increasing and lim t + ρ(t) =, we see (3.5) 1 b n b n+1, b n c n b n+1 c n+1, and 2 = c c n. Let f (x) = x, and, for n 1, suppose f n 1 (x) as been defined so tat it is increasing and twice differentiable. Divide [, 2π] into 4 n equal intervals {I n,k : 1 k 4 n } of lengt δ n = 2π 4 n, and set m n,k = min{f n 1(x) : x I n,k }. For x I n,k, we now define { fn 1 (x), if m n,k b n c n, f n (x) = f n 1 (x) + b n ψ n (x), if m n,k > b n c n, were ψ n (x) = 4 n g(4 n x). Since ψ n(x) 1 and c n 2, f n is increasing. Writing u n = f n f n 1, we see tat (3.6) u n (x) b n 4 n and u n(x) < 3b n 4 n /2. Tus f n = f + n j=1 u j converges uniformly to a nondecreasing function wic we denote f. Also, by (3.3) we may apply Lemma 3.1 to get n 1 f n 1(x) 3 < 2 b j4 j Cb n 4 n, j=1

7 INNER FUNCTIONS IN B 7 and so (3.7) m n,k f n 1(x) m n,k + Cb n 4 n δ n < m n,k + Cb n, for x I n,k. Here and below, C denotes a constant wose value may cange from line to line, but is independent of any parameters suc as n in te inequality above. To prove tat f is singular, consider a point t for wic f (t) exists and is positive. Write d n,k (G) = (G((k + 1)δ n ) G(kδ n ))δn 1 for te difference quotient of a function G over te interval I n,k. Ten d n,k (f) = d n,k (f n 1 ), since u j is zero at te endpoints of I n,k for all j n. Tus, coosing k(n) so tat t I n,k(n), f (t) = lim n d n,k(n)(f) = lim n d n,k(n)(f n 1 ) = lim n f n 1(x n ), were x n I n,k(n) is cosen to satisfy d n,k(n) (f n 1 ) = f n 1(x n ). By (3.7), f n 1(t) f n 1(x n ) < Cb n, and since b n from (3.5), it follows tat lim f n(t) = f (t). Also, since f (t) >, it follows from (3.7) tat lim inf m n,k(n) >. Recalling te definition of f n and tat b n c n from (3.5), we now see tat f n (t) = f n 1 (t) + b n ψ n (t) for all n sufficiently large. Since lim f n(t) = f (t) and f n(t) f n 1(t) = b n ψ n(t) for all large n, te series bn ψ n(t) = 2 1 b n (cos(4 n t) cos(2 4 n t)) is convergent. But b 2 n = from (3.4), and so tis lacunary trigonometric series diverges off a set of measure ; see [Z, p.23]. Tus f (x) = a.e. and f is singular. We now turn to te proofs of (3.1) and (3.2). We write G(x) = G(x + ) G(x) and 2 G(x) = G(x + ) 2G(x) + G(x ) for te first and second differences of a function G. Note tat bounds for te second difference are 2 G(x) 4 sup G and, wen G is twice differentiable, 2 G(x) 2 sup G. Tus, using (3.6), 2 f 2 = lim f n n k=1 2 u k b k min( k, 4 k ). k=1 We coose p to satisfy 4 p 1 < 4 p and estimate from Lemma 3.1. Also, 1 p 4 k b k 4 p Cb p 4 p Cb p, k=1 k=p+1 since te sequence {b k } is decreasing. Tus 2 f b k 4 k 4p+1 b p+1 4 p 4b p, Cb p = Cη(4 p )ρ(4 p ) Cη()ρ(), from te definition of b k, te coice of p, and te regularity properties of η and ρ. Tis proves (3.1).

8 8 WAYNE SMITH To get a lower bound for f, we first sow by induction tat (3.8) f n(x) b n c n /2 for all x and n. Since f (x) = x, b 1 and c = 2, tis is true for n =. Now assume tat (3.8) as been establised for n 1, and consider x I n,k. If m n,k b n c n, ten by te induction ypotesis and (3.5), f n(x) = f n 1(x) b n 1 c n 1 /2 b n c n /2, as required. Te oter possibility is tat m n,k > b n c n, in wic case f n(x) = f n 1(x) + b n ψ n(x) m n,k b n > b n (c n 1) b n c n /2, since c n 2. Tus (3.8) olds. Now let (, 1) and x be given, and coose te integer q to satisfy 4 q 1 < /(4π) 4 q. Recalling tat u j vanises at te endpoints of I q+1,k for all j q + 1, from (3.8) we get f((k + 1)δ q+1 ) f(kδ q+1 ) = f q ((k + 1)δ q+1 ) f q (kδ q+1 ) b q c q δ q+1 /2. Since > 2δ q+1, tere are at least /(3δ q+1 ) terms in te sum below, and so f(x) I q+1,k [x,x+] f((k + 1)δ q+1 ) f(kδ q+1 ) b qc q 6 = η(1)ρ(4 q ) 3 3η(1)ρ(). 4 Tus (3.2) olds and te proof is complete. 4. Proof of Teorem 1.2 Let f be a function on [, 2π] wit continuous periodic extension, and let G(z) = π π e it + z e it z f(t)dt be te Herglotz integral of f. Before proving Teorem 1.2, we need a preliminary result sowing ow an estimate of 2 f gives an estimate on te growt of G. It is well known tat f λ, i.e. 2 f = o() as, if and only if G B, or equivalently (1 z 2 ) G (z) = o(1) as z 1; see [Z, p.263]. Since our goal is to construct a ϕ B, and (1 z 2 ) G (z) will just provide an estimate for te numerator of τ ϕ (z), we need an estimate for te rate at wic tis goes to zero. Te following teorem gives te estimate we need. Te proof uses te same ideas used in te classical case tat f λ ; c.f. [Z, p.19]. We note tat in te teorem te assumption f is continuous is made since tere are nonmeasurable functions f tat satisfy 2 f. 4.1 Teorem. Suppose tat ω(t) is positive and nondecreasing for t >, and ω(4t) 3ω(t), for all t sufficiently small. If f is continuous and satisfies 2 f ω(), for >, ten tere is a constant C suc tat (1 z 2 ) G (z) Cω(1 z 2 ).

9 Proof. Let U(r, θ) = RG(re iθ ), so tat INNER FUNCTIONS IN B 9 U(r, θ) = π π f(t)p (r, t θ)dt, were P (r, t) = (1 r 2 ) e it r 2 is te Poisson kernel. Noting tat P (r, t) is even in t and integrates to, were P (r, t) represents te second derivative wit respect to t, we see tat π U θθ (r, θ) = f(t) 2 π θ 2 P (r, t θ)dt = 1 π 2 t f(θ)p (r, t)dt. 2 π Te ypotesized bound for 2 t f terefore gives U θθ (r, θ) π tω(t) P (r, t) dt. One can ceck tat P (r, t) < for < t < τ and P (r, t) > for τ < t < π, were τ = τ(r) is asymptotic to (1 r)/ 3 as r 1; see [Z, p.19]. Te assumption tat ω(4t) 3ω(t) implies tat sup ω(t)t 1 sup ω(t)t 1 3ω(τ)τ 1. τ t π τ t 4τ Using tis and te assumption tat ω(t) is increasing, we get Integrating by parts, we ave Similarly, π τ τ U θθ (r, θ) ω(τ) tp (r, t)dt + 3 ω(τ) π t 2 P (r, t)dt. τ τ τ tp (r, t)dt = τp (r, τ) + P (r, τ) P (r, ) τp (r, τ) C τ. t2 P (r, t)dt C, and so U θθ (r, θ) Cω(τ)/τ Cω(1 r)/(1 r), since τ is asymptotic to (1 r)/ 3 and ω(t) is increasing. Now set ρ = (1 + r)/2, were r = z, and let H = G θθ. Ten H(z) = π π Differentiating wit respect to z, tis gives π H (z) 2 π ρe it + z ρe it z U θθ(ρ, t)dt. U θθ (ρ, θ + t) ω(1 ρ) π ρ 2 dt C 2ρr cos t + r2 (1 ρ) π dt ρ 2 2ρr cos t + r 2 dt.

10 1 WAYNE SMITH Tis last integral is equal to 2π/(ρ 2 r 2 ), and so, using tat ω(t) is increasing, we get te estimate H (z) Cω(1 r)/(1 r) 2. Integrating H to get H = G θθ, we now see tat G θθ (z) G θθ () + C r ω(1 t) 1 (1 t) 2 dt C 1 r Coosing p so tat 4 p 1 < 1 r 4 p and using Lemma 3.1, ω(t) t 2 dt. (4.1) 1 1 r ω(t) t 2 dt p k= 4 k ω(t) p 4 t 2 dt C k 1 k= 4 k ω(4 k ) C4 p ω(4 p ω(1 r) ) C (1 r). Putting tese estimates togeter, we get tat ω(1 r) G θθ (z) C (1 r). A computation sows tat G θθ (z) = zg (z) z 2 G (z). Also G B since te assumptions on ω imply it is bounded, and so f Λ. Also ω(1) 3 k ω(4 k ), wic implies 1 Cω(t)t 1, and so G (z) C log z 2 = C 1 r 2 dt 1 t C 1 r 2 were (4.1) was used to get te last inequality. Tus ω(t) ω(1 r) dt C t2 (1 r), G (z) z 1 G (z) + z 2 ω(1 r) G θθ (z) C (1 r), for 1/2 < z < 1, and tis completes te proof. Proof of Teorem 1.2. Let µ be te positive singular measure on [, 2π] wit indefinite integral equal to te singular function f from Teorem 3.3, and let F (z) = 2π e it + z e it z dµ(t) be te Herglotz integral of µ. Recall from (1.2) tat te associated singular inner function ϕ(z) = exp( F (z)) ten satisfies (4.2) τ ϕ (z) (1 z 2 ) F (z). RF (z) Integration by parts sows tat F (z) = izg (z) 2πK, were G(z) = 2π e it + z e it (f(t) + Kt)dt. z

11 INNER FUNCTIONS IN B 11 We set K = (2π) 1 (f() f(2π)), so tat te periodic extension of f(t)+kt is continuous and Teorem 4.1 can be applied. Ten 2 [f(t) + Kt] = 2 f(t), and so te bound 2 f Cρ()η() from Teorem 3.3 along wit Teorem 4.1 (applied wit ω(t) = Cρ(t)η(t)) gives an upper bound for G (z). Since G (z) G () + z max{ G (w) : w z }, we terefore get tat (4.3) (1 z 2 ) F (z) (1 z 2 ) zg (z) + (1 z 2 ) G (z) Cη(1 z 2 )ρ(1 z 2 ), for all z sufficiently close to 1. Let z = z e iθ, were θ [, 2π), and assume first tat 2π / [θ, θ + (1 z 2 )). Te denominator of te upper bound for τ ϕ in (4.2) is just te Poisson-Stieltjes integral of µ, and so an estimate for it is (4.4) RF (z) = 2π 1 z 2 e it z 2 dµ(t) C 1 (1 z 2 ) 1 µ[θ, θ + (1 z 2 )] = C 1 (1 z 2 ) 1 (1 z 2 )f(θ) C 1 ρ(1 z 2 ), were te estimate for f from Teorem 3.3 was used for te last inequality. If 2π [θ, θ+(1 z 2 )), ten µ[θ, θ+(1 z 2 )] = 1 f(θ)+ 2 f(), were = 1 z 2. Tus max{ 1, 2 } (1 z 2 )/2, and it follows tat (4.4) olds in tis case as well. Inequalities (4.2), (4.3), and (4.4) now combine to give and te proof is complete. τ ϕ (z) Cη(1 z 2 ), References [B1] Bisop, C.J., Bounded functions in te little Bloc space, Pacific J. Mat. 142 (199), [B2] Bisop, C.J., An indestructible Blascke product in te little Bloc space, Publ. Mat. 37 (1993), [G] Garnett, J.B., Bounded Analytic Functions, Academic Press, New York, [J] Janson, S., Generalizations of Lipscitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Mat. J. 47 (198), [K] Kaane, J.-P., Trois notes sur les ensembles parfaits linéaires, Enseign. Mat. 15 (1969), [L] Littlewood, J.E., On inequalities in te teory of functions, Proc. London Mat. Soc. 23 (1925), [MM] Madigan, K. and Mateson, A., Compact composition operators on te Bloc space, Trans. Amer. Mat. Soc. 347 (1995), [P] G. Piranian, Two monotone singular, uniformly almost smoot functions, Duke Mat. J. 33 (1966), [Sa] Sarason, D. E., Blascke products in B, in Linear and complex analysis problem book, ed. by V. P. Havin, S. V. Hruščëv and N.K. Nikol skii, Lecture Notes in Mat., vol. 143, Springer-Verlag, Berlin, [Sa] Sapiro, H. S., Monotone singular functions of ig smootness, Micigan Mat. J. 15 (1968), [S] Sapiro, J.H., Te essential norm of a composition operator, Annals of Mat. 127 (1987), [Sm] Smit, W., Composition operators between Bergman and Hardy spaces, Trans. Amer. Mat. Soc. 348 (1996),

12 12 WAYNE SMITH [SZ] Smit, W. and Zao, R., Composition operators mapping into te Q p spaces, preprint. [St] Stepenson, K., Construction of an inner function in te little Bloc space, Trans. Amer. Mat. Soc. 38 (1988), [Y1] Yamasita, S., Hyperbolic Hardy class H 1, Mat. Scand. 45 (1979), [Y2] Yamasita, S., Hyperbolic Hardy classes and yperbolically Diriclet-finite functions, Hokkaido Mat. J. 1 (1981 Sp), [Z] Zygmund, A., Trigonometric series, Second Edition, Vol. I, Cambridge University Press, New York, Department of Matematics, University of Hawaii, Honolulu, Hawaii address: wayne@mat.awaii.edu