An indestructible Blaschke product in the little Bloch space

Publicacions Matemátiques, Vol 37 (1993), 95-109 . AN INDESTRUCTIBLE BLASCHKE PRODUCT IN THE LITTLE BLOCH SPACE A bstract CHRISTOPHER J . BISHOP The little Bloch space, 13 0 , is the space of all holomorphic functions f on the unit disk such that lim1 z 1l (f'(z)j(1 - Iz12) = 0 . Finite Blaschke products are clearly in 130, but examples of infinite products in 80 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others) . Stephenson has asked whether 130 contains an infinite, indestructible Blaschke product, Le ., a Blaschke product B so that (B(z) - a)/(1 - QB(z)), is also a Blaschke product for every a E D . In this paper we give an afirmative answer to his question by constructing such a Blaschke product . We also answer a question of Carmona and Cufí by constructing a VMO function, f, so that Ilf J¡ . = 1 and whose range set, R(f, a) = {w : there exists z n , ~ a, f(z~) = w}, equals the open unit disk for every a E T . 1 . Introduction Let D = fiz1 < 1} denote the unit disk . The little Bloch space, B O , is the space of holomorphic functions f on D such that lim  1 f'(z)1(1 - Iz12) = 0 . Iz1-1 Basic facts about B O can be found in [21 . A Blaschke product is a holomorphic function of the form This work was supported by NSF Grant DMS 91-00671 96  C . J . BISHOP B(z) = 11 ~ z n - z Iznl n 1-znz zn , where E(1 - Izn1) G oo . Finite Blaschke products are clearly in B O , but examples of infinite products in BO are not so obvious . Such functions do exist, as is shown in [1], [8], [10] . A more explicit example, as well as a characterization of such products in terms of the zero sequence {z n }, has been given in [3] . That result answers several questions about 130, but does not resolve the following question from [10] : does B o contain an infinite, indestructible Blaschke product, Le ., a Blaschke product B so that _ B(z) - a Ba(z)  1 - úB(z)' is also a Blaschke product for every a ED? The question arises because of Frostman's theorem . An inner function F is a holomorphic function on D with boundary values of absolute value 1 a .e . on T . Any such function can be written as a product _ II  ¡o F(z) = B(z)S(z) = (fl 1 n zn z ~)(exp( - f e ie ± zdp(e)), of a Blaschke product and a singular inner function (M is a finite, positive measure singular to dB) . Frostman's theorem states that for any inner function F on D, Fa  F(z) - a (z) = 1 - úF(z)' is a Blaschke product for every a E D\E where E is an exceptional set of zero logarithmic capacity [5, Theorem 11 .6 .4] . The constructions of Blaschke products in [8], [10] first build an inner function in B o and then apply Frostman's theorem . Stephenson asked if this was unavoidable, e .g ., does 13 o contain any indestructible Blaschke products? The example in [3] is built by constructiog the zero set, so does not use Frostman's theorem . Furthermore, a variant of Stephenson's construction gives a Blaschke product without using Frostman's theorem (see next section) . In this note we expand on this observation to give a "cut and paste" construction of an indestructible Blaschke product in 130 . INDESTRUCTIBLE BLASCHKE PRODUCT IN BO  9 7 One could also try to produce such an example by finding a sufficient condition on the zeros for the product to be indestructible and which includes some sequences satisfying the 13 o condition from [3] . One such sufficient condition for indestructibility is that the sequence be thin, Le ., zk - zn kan 1 - znzk However, this condition is incompatible with the BO condition . An even more ambitious problem is to characterize indestructibility in terms of the zero-set . In [6], Morse has constructed a destructible Blaschke product which becomes indestructible when a single point is deleted from its zero-set . This indicates any characterization of indestructibility in terms of the zero set would be very delicate (and probably very dificult) . A related problem has been solved, however . In some sense, finite products of interpolating Blaschke products are the "conformally invariant" class of Blaschke products . In [7] Nicolau has given a zeroset characterization of those Blaschke products B so that B a is a finite product of interpolating Blaschke products for every a in the disk . Thus he has solved the conformally invariant version of the problem of characterizing indestructibility . Our construction gives a Blaschke product whose singular set (the accumulation set of the zeros) has measure zero . If we could construct an example whose singular set was the entire circle, this function would also have the property that its range set, R(f, a) = {w : there exists z n - a, f (z n ) = w}, equals the whose disk for every a E T . Carmona and Cufí had asked in [4] if there was a function in H°° n BO with this property . I believe the construction can be modified to give such an example, but rather than do this, I will sketch the construction of a function f E H°° n VMO with 11 f jj,, = 1 and R(f, a) = D for every aE T . Since VMO C BO, this is an even stronger result (again answering a question of Carmona and Cufí) . I thank Arturo Nicolau for bringing this question to my attention and our discussions on it . I also thank the referee for his helpful comments . His suggestions have clarified the exposition in several places . 98  C . J . BISHOP 2 . The B O construction The idea is quite simple ; we will build a simply connected Riemann surface by taking copies of the unit disk with slits and "gluing" different copies along the slits . A simple example of this idea is to take infinitely many copies of D\[2,1), and identifying the "top" edge of one copy with the "bottom" edge of the next . See Figure 1 . . .(D (D (D (D ... Figure 1 . A single example s~ Let So be the initial sheet, which we also refer to as the "zeroth sheet" . This sheet contains a point corresponding to zero in the unit disk and we refer to this point as "0" on the surface . Let S  , be nth stage of this construction (the union of copies -n to n) and S = UnSn the limiting surface . For each of there surfaces the point "0" refers to the point 0 on So . In the rest of this paper we shall assume that any Riemann mapping of the unit disk to a constructed surface like S n or S maps 0 in the disk to the point 0 on the surface . Whenever we talk about harmonic on the surface it is the push forward of normalized Lebesgue measure on the circle under such a Riemann mapping, Le ., harmonic measure will always be with respect to the point 0 on the zeroth sheet . S is simply connected so there is a Riemann mapping ob : D --> S and there is an obvious holomorphic projection P : S -3 D . We claim that F = P o D must be an inner function because all the harmonic measure for S lives on the part of the boundary above the the unit circle . To prove this we consider S n and show that the harmonic measure of the two radial slits in its boundary are O(ñ) . To do this we mapS  , to a half infinite strip W = {(x, y) : -oo < x < 0, -(2n -f1)7r < y < (2n + 1)7r} by the mapping INDESTRUCTIBLE BLASCHKEPRODUCT IN B o  99 - z --> log( z  2 ) 1-2z which has a well defined branch on S  , . The point 0 on the surface is mapped to -1/2 and the radial edges are mapped to the horizontal edges of the strip . Standard estimates (e .g ., map the strip to a halfplane via sin(z/i) and use the Poisson integral) show that the harmonic measure of the horizontal edges of the strip with respect to the point -1/2 are approximately 1/n . By conformal invariance of harmonic measure the claim about S n is proved . Thus the circular part of OS n has measure >_ 1 - C/n . Taking n --> oo we see that the circular part of OS has full measure, Le, ¡Fl = 1 a .e . on the unit circle . In fact, F must be Blaschke product . To see this, recall that a function f in the unit ball of HI(D) is a Blaschke product iff the least harmonic majorant of log lf 1 is 0 (e .g ., [5, Theorem II .2 .4]) . This says that F is a Blaschke product iff 0 is the least harmonic majorant of log ¡P(z) 1 on S . Let u be the least harmonic majorant of logjP(z)1 on S . Then u restricted to S,,, is harmonic and has boundary values 0 on P -1 (T) and >_ log 2 on the two radial slits in OS, . Thus 0 >_ u(0) >_ 2 log 2 . Since this holds for any n, u(0) = 0 (recall that the "0" in u(0) refers to the designated point on the zeroth sheet) . Since u is nonpositive this implies u - 0 and so F is a Blaschke product . Let a E D and Ta(z) = (z - a)/(1 - áz) . An argument like the one above shows shows that if a :,¿ 2 then F a = Ta o F is a Blaschke product . However, since no point of S covers the point {2 }, F2 is never zero, so must be a singular inner function (in fact, since F is continuous except for one boundary point, up to rotations it must be exp(A i+z ), for some A > 0, Le ., the singular inner function corresponding to a positive point mass) . To build an indestructible Blaschke product we will have to vary the construction, adding sheets which cover the omitted points of earlier generations, and in particular, so that the least harmonic majorant of log 1To,(P(z))1 on S is 0 for any choice of a E D . This says that not only is each point covered infinitely often, but there is some sense in which it is "frequently" covered . To get F into the little Bloch space imposes another constraint : given any e > 0 only finitely many of the sheets we attach may contain a disk of radius e . This arises because of a geometric characterization of ,Cio due to Stegenga and Stephenson [9] . For f analytic on D and a E D, r > 0 they define SZ a (r) to be the component of f -1 (D (f (a), r)) containing a, F a (r) = Og a (r) n T and r f(a) = sup{r : F a (r) = 0} . Then fE Bo iff r f(a) = o(1) as ¡al --> 1 . In 10 0  C . J . BISHOP particular, if the Riemann surface of f is obtained by identifying copies of D along slits, then the endpoints of any such slit are in the ideal boundary of the surface . Therefore f will be in the little Bloch space if for every E > 0, these endpoints of pasted edges are E-dense in D for all but finitely many sheets . We will inductively construct a sequence of positive numbers {E n ,} tending to zero, a sequence of finite point sets {E ,}, a collection of radial line segments T  ,, and two sequences of integers {gn}, {h  } tending to infinity . The sets {Ej, {T  } will satisfy (1) Tn C Tra+l, EnC En+1, EnC T  . (2) The endpoint of each segment in T,, is in E,,, . (3) E  ,/E  ,+1 is an even integer . (4) Adjacent points of & en a segment of T,, are at distante E , from each other . (5) SupzED dist(z, E n ,) G lOE  . See Figure 2 . An "edge" I of T  denotes a subinterval en T  , connecting two adjacent points of E,,, Le ., a component of T  ,\E n . We let en , denote the number of edges in T  , . In the construction below each such edge will be treated as two separate pieces of the boundary of the domain R n = D\T,, corresponding to its two sides . One side will be pasted te a sheet of previous generation, the other pasted te one or more sheets in the next higher generation . Figure 2 . E  , T  ,, R n INDESTRUCTIBLE BLASCHKE PRODUCT IN 13p  10 1 Given an edge I in the boundary of R  , we can either attach another copy of R n ,, or divide the edge into m = En/En+1 edges in T . .+1 ( since E n C En+1) and attach m copies of R,+1 . Given a sequence of integers {gi} we could build a Riemann surface as follows . Start with one copy of R1 and attach 2e1 copies of of R1 along (both sides of) each edge of T 1 . Call this 5 1 . Then attach more copies of R1 along each edge in OS, to obtain S2and continuing for g 1 generations, obtaining a nested sequence of surfaces S1 C S 2 C . . . C Sgl . The term "generations" refers to the fact that to connect the point 0 in the zeroth sheet So to any of the unpasted edges of Sk a path must pass though at least k + 1different sheets (i .e ., copies of R1) belonging to So, S1\So, ... , Sk\Sk_1 . We have obtained Sgl by pasting together identical sheets, Le ., copies of Rl . To get the next surface, Sgl+1, we attach to each unpasted edge of Sgl El/E2 copies of the sheet R2 . We obtain Sgl+2 by pasting a copy of R2 to each unpasted edge of Ssl+1 . We continue in this way for 92 generations, obtaining a surface S91+92 . The next surface Sg1+g2+1, is constructed by attaching copies of R3 to the unpasted edges of Sg1+g2 . Thus given the sequence of integers {gk} (which tells us for how many generations to attach copies of Rk) and continuing in the obvious manner, we obtain an increasing, nested sequence of simply connected surfaces, {S n } . Then S = UnSn, is a simply connected connected Riemann surface . If -P : D - S is the Riemann map (mapping 0 to 0 on So), and P : S --> D the projection then F = P o <P is a holomorphic function on the unit disk which we claim is an infinite Blaschke product in 130, if the parameters are chosen correctly . This is essential Stephenson's construction in [10] . The fact that F E BO follows from the characterization of Stegenga and Stephenson mentioned earlier . If the sequence {gi} grows quickly enough, Stephenson shows the mapping F is an inner function . If dist(o,T n ) >_ En then F is actually a Blaschke product (again if gn / oo fast enough) . To prove this, consider the least harmonic majorant u of log IP(z)j restricted to SN = Sgl+ . . .+g" The boundary breaks into two pieces aS N = 01SN U a2SN corresponding respectively to P -1 (T) and the radial edges . Then u has boundary values 0 on 8 1 SN and u >_ 109 En on á2SN . The set a2SN can be made to have as small harmonic measure as we wish by taking gn large enough, so we may take 0>- U> - W(ó2SN)109E n >- 1 - n if gn is large enough (recall that as before, harmonic measure refers to the harmonic measure with respect to the point 0 on the zeroth sheet 10 2  C . J . BISHOP So) . Thus F is a Blaschke product, but it cannot be indestructible since it only takes values in each E n finitely often . As Stephenson points out, this example shows the exceptional set in Frostman's theorem may be dense in the unit disk . To make F indestructible, we modify the construction slightly . Associated to each E,, define another set F n , by replacing each zE E nby a point w E En,+1 with Iz - w i = á Era and such that w is on same radius as z . The sets F  , satisfy approximately the same density conditions as the E n (with E n replaced by 2c,  ,) . Our idea is to modify the construction by alternating the use of the sets E  , and F  in the construction . Since E  , f1 F n , = 0 this means our surface will cover the whole disk and since max(dist(z, En), dist(z, Fn)) >_ c  ,/4 for every point z in the disk, we should be able to prove our function is indestructible by estimating harmonic measure either on the "E  ,-sheets" or 'T,,-sheets" (depending on whether z is far from E  , or far from Fn) . However since E", f1 F,, = 0, we need some further modifications to to able to attach an "F  ,-sheet" to an " En - sheet" . This is how we attach a F  ,-sheet to an E n ,-sheet . Consider a component interval I of T n with endpoints in E  , . Let T n , be the analogue of T  , for the set F  , and let R,, = D\Tn, . Assume (without loss of generality) that F  , has been chosen so T  , C T  , . Let {ao, al, . . . . an} = I n E  ,+1, listed in order (e .g ., ao, ara are the endpoints of I) . Let F  ,j = F n U {aj , a j+ 1} . Along each interval (a j , aj+1) attach a copy of R n . To this sheet we attach copies of Ñ,, along all component intervals of t,,\F j . We continue in this way, attaching copies of R n along intervals of Tn\Fn, except for those sheets reached by either looping around a j or around aj+1, in which case we are forced to attach copies of R n along intervals of the form T,,\F n U{a j } (or Tn\F  ,U{aj+1}) . Some of these identifications are illustrated in Figure 3 . More precisely, Figure 3 shows regions on four sheets, labeled I, II, III, IV . Sheet Iis pasted to sheet II along the edge [aj, aj + 1] . Sheet II is pasted to sheet III along edge [q, aj ] and to sheet IV along the edge [p, q], where p, q are points of F,,, adjacent to aj . The solid and doted curves illustrate paths from sheet I to sheets III and IV respectively which (must) pass through sheet III . Notice that the point A E E n in the ideal boundary of sheet I is covered when sheets II and IV are pasted along [p, q] . Similarly ao E E n is covered when II is pasted to III along [q, aj] (assuming j 0 0 ; otherwise it would be covered by some sheet attached to sheet IV) . INDESTRUCTIBLE BLASCHKE PRODUCT IN B O  103 4 4 1 4 aj Figure 3 . Modifications to cover E n Suppose we have already constructed a surface S  , whose boundary consists of arcs covering T or edges of T,, . To each component interval I of T , ,n\En we attach copies of R  , as described above . Do this for g,+1 generations . The resulting sheets cover E n , (the only sheets which do not cover every point of E n , are those attached along subintervals of the form (ao, al) or (an_I, an) in the construction above) . We call the resulting surface S n , . To the boundary of S, z attach copies of R n+ 1 = D\Tn+1 along component intervals of Tn+1\En+1 for hn+1 generations (this poses no difficulties since E n , F n and all points of the form a j in the previous stage of construction were in En+1 ; thus every radial interval in the boundary of S n has endpoints in En+l) . The resulting surface is called Sn+I and satisfies the induction hypothesis . The union over n of those (nested) surfaces is denoted S and we obtain the desired function by mapping the disk to S and then projecting back to the disk . All that remains is to choose the sequences {En}, {gn} and {hn} so that the harmonic measure estimates hold . We will first choose gn , then En+1 and then hn+1 Let u be the least harmonic majorant of log ITa o P(z) 1 .  We want