RANDOM WALK AND BOUNDARY BEHAVIOR OF FUNCTIONS IN THE DISK. MICHAEL D. O'NEILL Abstract. Simple martingale proofs of some results of Rohde 9, 10] on the boundary behavior of Bloch functions are presented, making clear their connection with random walk in the plane. 1991 mathematics subject classication: 30D45, 30D50. if 1. Introduction A function f , dened and analytic in the unit disk, is called a Bloch function kf kB = sup(1 ; jz j2)jf 0 (z )j < 1: z2D We write f 2 B. The function is said to be in the little Bloch space B0 if (1 ; jz j2)jf 0 (z )j ! 0 jz j ! 1 z 2 D : The following proposition, which establishes a close connection between Bloch functions and conformal mappings, is well known.(See 1, 2]) Proposition 1.1. If g is a univalent function in D and f = log g0 then f 2 B and kf kB 6. Conversely, if kf kB 1 then there exists a univalent function g such that f = log g0 . In this note we use the device of Bloch martingales, developed by Makarov in 6] and briey outlined below, to give simple proofs of the following two theorems of S. Rohde. Theorem 1.2 (Rohde). An inner function in the little Bloch space which is not a nite Blaschke product has, for each 2 D , the non-tangential limit on a set E T with Hausdor dimension 1. Theorem 1.3 (Rohde). Let fQng be a sequence of squares in the plane with pairwise disjoint interiors, edges parallel to the coordinate axes and of length 1 2 MICHAEL D. O'NEILL a > 0, and such that Qn is adjacent to Qn+1 for all n. There is a universal constant K > 0 such that if g 2 B has nontangential limits almost nowhere then there exists a set E with dim E 1 ; a;1 K kgkB such that, for each 2 E we can nd rn ! 1 with g(r ) Qn Qn+1 for rn r rn+1 : Given an arc I D let zI = (0) where is the conformal self mapping of the disk which maps @ D fRe z > 0g onto I . The theorems in section 2 will follow from the next lemma of Makarov. (See 6].) Lemma 1.4 (Makarov). If b 2 B and I is an arc on @ D then we have () 1 Z n n jI j (b( ) ; b(zI )) d Cn!kbkB I n 1: Here, the integral is dened as the limit of the integrals of (b(r ) ; b(zI ))n as r ! 1. Proof. We have n Z1 Z0 Z0 x 1 + t e n log 1 ; t dt = 2 x (1 + ex )2 dx 2 xn ex dx 2n! : 0 If b(0) = 0 and I = inequality ;1 i ; i e 2 e 2 ] T ;1 then () follows by twice integrating the jb0 (z )j 1 k;bkjzBj2 on the imaginary axis from ;i to i and applying the Cauchy integral theorem. In general, let g(z ) = b( (z )) ; b( (0)) where is the self mapping of D used to dene the point zI . We then have Z and I (b( ) ; b(zI ))n d = Z 1 ;1 (g(iy))n 0 (iy) dy (1 ; jz j2 )jg0 (z )j = (1 ; j j2 )jb0 ( )j = (z ): RANDOM WALK AND BOUNDARY BEHAVIOR OF FUNCTIONS IN THE DISK. Now since 3 2 j 0 (z )j = j11 ;+ jzzIzjj2 C jI j I () follows from the rst case considered, and the proof is complete. We remark that if b 2 B0 then we may replace the inequality in the above proof by jb0 (z )j 1 k;bkjzBj2 ; jz j) jb0 (z )j (1 1 ; jz j for some function = (), determined by b, for which ! 0 as ! 0 and obtain the inequalities Z 1 n n n jI j (b( ) ; b(zI )) d Cn! (jI j)kbkB n 1: I For any b 2 B let br (z ) = b(rz ) for all 0 < r < 1 and dene 1 Z b ( )j d j: bI = rlim r !1 jI j I this limit exists by Cauchy's theorem because, as shown in the proof above, b is integrable on radii. Let Fn denote the increasing sequence of sigma algebras generated by the partition points e2i2;k , 1 k 2n on the probability space (T d2 ). Dening the Fn measurable function Mn by Mn jI = bI it is easily checked that M = (Mn Fn ) is a martingale. By lemma 1.4, and the denition of the Bloch space, we have for all n 1. jMn ( ) ; b((1 ; 2;n ) )j C kbkB 2 @ D 2. jMn ( )j C kbkB 2 @ D We dene a Bloch martingale to be a real dyadic martingale such that there exists b 2 B with Re bI = Sn jI jI j = 22;n for each dyadic interval I . The following lemma from 6], which we record without proof, characterizes Bloch martingales. Lemma 1.5 (Makarov). A dyadic martingale S is a Bloch martingale if and only if the dierences (S jI ; S jJ ) are uniformly bounded for all adjacent pairs of dyadic intervals (I J ) with jI j = jJ j. 4 MICHAEL D. O'NEILL To make dimension estimates in the next section we will use Lemma 1.6 (Hungerford). Fix 0 < < c < 1. Let E0 = T = I00 and for n > 1, En = Ink where Ink are disjoint closed arcs such that for each Ink there is a unique In;1j with 1. Ink In;1j 2. jP Ink j jIn;1j j 3. jIni j cjIn;1j j, where i(j ) runs through all indices such that Ini i(j ) In;1j . T Let E = En . Then with dimE denoting the Hausdor dimension of E , we have n Proofs appear in 4] and 8]. dimE 1 ; log c : log 2. Martingale proofs of some results of Rohde The behavior of Bloch functions at the boundary of the disk is explained by the following lemma, which is a slight renement of lemma5.6] in 6]. Lemma 2.1. Let M = fMng be a complex dyadic martingale determined by a Bloch function b as explained in section 1. Assume that M0 = 0 and that jMn j 1 for all n. Given 0 < < 2 , there exist 0 < C < 1 and a0 > 0 such that for all a > a0 we have m jarg M j < 2 ; where = a = inf fn : jMnj ag. n o > C Proof. By familiar properties of the Fejer kernel, if we are given > 0 we may choose m() so that if 0 (t) 1 is nondecreasing and left continuous on ; t then m m + 1 ; j j Z (t)e;it dt 1 ; =;m m + 1 X ; =) ; 2Z ; 2 + (t) dt > C > 0: RANDOM WALK AND BOUNDARY BEHAVIOR OF FUNCTIONS IN THE DISK. 5 For each dyadic interval I we have by lemma 1.4, () (b ; bI )n ]I C (n)n!kbknB and n X n n k (b ; bI ) ]I = (b )I + (;1) nk (bn;k )I bkI = (bn )I ; bnI + () nX ;2 k=1 k=1 (;1)k n (bn;k ) ; bn;k bk I I I k where the last equality follows from the identity n X (;1)k n = 0: k k=0 n Letting M denote the martingale determined by bn and with 0 n m(), we have and Z Z M dm = 0 Z Mn dm = ; (M ; M )n ] dm Z + nX ;2 k=1 (;1)k nk (M n;k ) ; Mn;k Mk dm: Applying () and () to the terms in square brackets, we have by induction that Z Mn dm C (m) + C 0 (m)an;2 0 n m(): Let (t) ; t be the distribution density of arg M . From the above argument we have Z so that ; m eint (t) dt C (m)a;2 0 n m() m + 1 ; j j Z (t)e;it dt 1 ; C (m)a;2 1 ; =;m m + 1 X ; if a is large enough. By the rst paragraph of the proof this implies that n o m jarg M j < 2 ; > C 6 MICHAEL D. O'NEILL and completes the proof of the lemma. We now use lemma 2.1 to prove the theorems mentioned in the introduction. Recall that there are singular inner functions S in the little Bloch space since there are singular measures with integrals in the little Zygmund class, (7],5],11]). By a theorem of Frostman (see 3]), the functions S; 1 ; S 2D are non-nite Blaschke products except for a set of 2 D with logarithmic capacity zero. Hungerford showed in 4] that the zeros of such products must always accumulate on a set with Hausdor dimension 1. This result was strengthened by Rohde who proved the following theorem 10]. Theorem 2.2 (Rohde). An inner function in the little Bloch space which is not a nite Blaschke product has, for each 2 D , the non-tangential limit on a set E T with Hausdor dimension 1. Proof. Fix 2 D and choose N such that 2;N < 21 (1 ; jj). Let Dn denote the disk of radius 2;(N +n) centered at . Since b is not a nite Blaschke product, given > 0 we can nd a dyadic arc I with jI j < , jbI ; j < and (1 ;jz j2 )jb0 (z )j < for all z with jz j > 1 ; j2Ij . Taking > 0 suciently small and applying lemma 2.1 to the function b;bI , we can nd a dyadic interval E0 = I 0 with bI 0 2 D2 and jI 0 j < . Because jbj ! 1 nontangentially almost everywhere, the set of maximal dyadic subintervals fJj g with bJj 2= D0 has X j jJj j = jI 0 j and given > 0, if > 0 is small enough then since controls (1 ; jz j2 )jb0 (z )j which in turn controls Mn , we have jJj j < jI 0 j for all j . We apply lemma 2.1 in each Jj to obtain 0 < C < 1 and a set of dyadic intervals E1 = fIk1g with bIk1 2 D2 for all k and X k jIk1 j C jI 0 j: We continue the construction in the obvious way, obtaining E0 E1 : : : EN1 . Because (1 ; jz j2 )jb0 (z )j ! 0 as jz j ! 1, with N1 suciently large we may alter the construction by switching to the disks (D1 D3 ) in place of (D0 D2) and keep the same numbers > 0 and C > 0. In general, for n > 1, when Nn is suciently large RANDOM WALK AND BOUNDARY BEHAVIOR OF FUNCTIONS IN THE DISK. 7 we may change the construction by switching to the disks (Dn Dn+2 )Tin place of (Dn;1 Dn+1 ), keeping the same C > 0. By lemma 1.6 the set E = ENk has k C dim E 1 ; log log and by lemma 1.4 and the remark following it, b ! non-tangentially at each point of E . Since > 0 may be chosen arbitrarily small in the above argument, the theorem is proved. Theorem 2.3 (Rohde). Let fQng be a sequence of squares in the plane with pairwise disjoint interiors, edges parallel to the coordinate axes and of length a > 0, and such that Qn is adjacent to Qn+1 for all n. There is a universal constant K > 0 such that if b 2 B has nontangential limits almost nowhere then there exists a set E with dim E 1 ; a;1 K kgkB such that, for each 2 E we can nd rn ! 1 with b(r ) Qn Qn+1 for rn r rn+1 : Proof. We may assume that kbkB 1. The theorem is then interesting for large values of a. Let Q0n denote the square of edge length a2 concentric with Qn and with parallel edges. Let Dn denote the disk of radius 20a concentric with the square Qn . Consider any two adjacent squares Qn and Qn+1 . Let Rn denote the smallest rectangle containing Q0n Q0n+1 . For suciently large a > 0 independent of b 2 B, if bI 2 Dn then by the assumption on the function b, nitely many applications of lemma 2.1 show that there exists a universal constant 0 < C < 1 P and a collection fIj g of dyadic subintervals of I with jIj j C jI j, such that for each j we have BIj 2 Dn+1 and bJ 2 Rn for all dyadic J with Ij J I . Because kbkB 1 we have jIj j c2;a]jI j for all j . Again, by the assumption on b, we nd after nitely many applications of lemma 2.1, a dyadic interval with bI0 2 D0 . Constructing the appropriate nested sequence of intervals contained in I0 , the proof is completed by applying lemma 1.6 and lemma 1.4 with n = 1. References 1] J. Becker. Lownersche di erentialgleichung und schlichtheitskriterien. Math.Ann., B.202:S.321{335, 1973. 2] J. Becker and Ch. Pommerenke. Schlichtheitskriterien und Jordangebiete. J.reine angew.Math., B.354:S.74{94, 1984. 3] J. Garnett Bounded Analytic Functions. Academic Press, San Diego, CA, 1981. 8 MICHAEL D. O'NEILL 4] G. Hungerford. Thesis. Caltech, Pasadena, CA, 1988. 5] J.P. Kahane. Trois notes sur les ensembles parfaits lineaires. Enseignement Math., 15:185192, 1969. 6] N.G. Makarov. Probability methods in the theory of conformal mappings. Algebra i Analiz, 1:3{59, 1989. (Russian) English transl.: Leningrad Math J. 1 (1990) pp. 1-56 ]. 7] G. Piranian. Two monotonic, singular, uniformly almost smooth functions. Duke Math. J., 33:255{262, 1966. 8] Ch. Pommerenke. Boundary Behavior of Conformal Maps. Springer-Verlag, Berlin, 1991. 9] S. Rohde. The boundary behavior of Bloch functions. J. London Math. Soc., 48(2):488{499, 1993. 10] S. Rohde. On functions in the little Bloch space and inner functions. Trans. Amer. Math. Soc.,348:2519{2531, 1996. 11] D.Sarason. Linear and Complex Analysis Problem Book:199 research problems. V.P Havin, S.V. Hruscev, N.K. Nikolskii, editors. Lecture notes in mathematics (Springer Verlag) no.1043. Department of Mathematics, UTEP, El Paso, TX 79968 E-mail address : [email protected]