On Functions in the Ball Algebra

Abstract

We show that there exists a function in a ball algebra such that almost every slice function has a series of Taylor coefficients divergent with every power p < 2. In ?7.2 of [3] W. Rudin gives some examples of boundary behavior of holomorphic functions in complex balls of dimension 2 and 3. It was observed in [4, Remark 1.10], that using Theorem 1.2 of [4] such examples can be constructed in arbitrary dimension. In the present note we further pursue this idea. It is well known that in one variable there exists a function f(z) = E' anZ n analytic for lzl K 1 such that E Ian lP oo for all p < 2. Our theorem generalizes this fact to functions of several variables. In this note, B will always denote the unit ball in the complex d-dimensional space Cd, S will stand for the unit sphere and a for the rotation invariant probability measure on S. A(B) will denote the ball algebra of all functions analytic in B and continuous in B. For 0 < P : oox, IIflp denotes (fs If(?)IPdor())I/P. If f is a holomorphic function in B then it has a unique homogeneous expansion as f 00 0 fn, fn is an analytic polynomial homogeneous of degree n. It was shown in [4, Theorem 1.2], that there exist polynomials (Pn) homogeneous of degree n on B such that (*) JlPnil 1 and IlPnloo < X. (X depends only on the dimension of the ball, it can be taken X 2d/l.) Those polynomials will be crucial in our further considerations. In particular, we will use the following inequality (cf. [4, Proposition 1.6]) 00 oo0( ? 1/2 (**) ~ ~~crnP2~-(Ian I ) , O< P<oo. -P ~k=O J We will also use the fact that Cauchy Integral C[M] maps continuously measures on S into all Hp(B), p < 1 (cf. [3, 6.2]). The book [3] is an excellent source of information about the function theory in B. PROPOSITION. The operator T: A(B) -+ 2 defined by T(f) ((f,p2n))??o i a surjective map. PROOF. This map is clearly continuous. By duality it is enough to show that T*: f2 -4 A(B)* is an isomorphic embedding i.e. IIT*(an)IIA(B)? C(Q Ian12)1/2. Received by the editors July 21, 1981. 1980 Mathematics Subject Claasfication. Primary 32A40; Secondary 32A05, 32E25. (D 1982 American Mathematical Society 0002-9939/81/0000-1074/801 .75 184 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:53:16 UTC All use subject to http://about.jstor.org/terms FUNCTIONS IN THE BALL ALGEBRA 185 One easily checks that T*(an)(f) = (f, E'=0 anp2n). Let jt be a measure on S such that, for f E A(B), T*(aln)(f) f f dc4 and 11u1 = IIT*(an)llI Then obviously C[ji], the Cauchy Integral of A, equals E' anp2n, so (by properties of C[ * ] and