Traces of Certain Classes of Holomorphic Functions on Finite Unions of Carleson Sequences

We give a method allowing the generalization of the description of trace spaces of certain classes of holomorphic functions on Carleson sequences to ®nite unions of Carleson sequences. We apply the result to di€erent classes of spaces of holomorphic functions such as Hardy classes and Bergman type spaces. 0. Introduction. Let D={z2C : jzj<1} be the unit disk and Hol(D) the space of holomorphic functions on D. For a space X Hol(D) we de®ne the trace space Xj ˆ ff j : f 2 Xg …1† and the sequence space X… † ˆ f… f …ln††n 1 : f 2 Xg: …2† Using a decomposition method for , we shall generalize the description of Xj if is a Carleson sequence to ®nite unions of Carleson sequences. In order to do this we will need a certain stability condition that will be introduced in the ®rst section. There we also give the characterization of the trace space of X on a ®nite union of Carleson sequences under the additional condition H1X X, where H1 ˆ f f 2 Hol…D† : k f k1ˆ supz2D j f …z†j <1g is the Hardy space of bounded analytic functions on D. In the second section we shall apply the general characterization obtained in the ®rst section to various classes of spaces of analytic functions on D. We thus obtain a new approach to the characterization of the traces of Hardy spaces on ®nite unions of Carleson sequences (cf. [6] and [1]). Also we give a hitherto unknown description of the traces of Bergman type spaces on ®nite unions of Carleson sequences. 1. The general result. We introduce the pseudohyperbolic metric with the aid of MoÈ bius transformations. For l2D put bl…z† ˆ jlj l lÿz 1ÿ lz ; z 2 D, and de®ne the pseudohyperbolic distance by …l; † ˆ jbl… †j for , 2D (cf. e.g.[5]). The corresponding neighbourhood ( , ) for 2D and 0< <1 is given by …l; † ˆ fz 2 D : jbl…z†j < g: Glasgow Math. J. 41 (1999) 103±114. # Glasgow Mathematical Journal Trust 1999. Printed in the United Kingdom at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/BA6B568D5BA12FF3CC73C7BFCA45940A Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 10 Sep 2017 at 14:06:15, subject to the Cambridge Core terms of use, available We say that a sequence ={ln}n 1 D satis®es the Carleson condition, if inf l2 Y 2 6ˆ l jb …l†j ˆ > 0: In this case we write 2(C) and call a Carleson sequence. In what follows we will refer to as the Carleson constant. It is well known (cf.[3]) that H 1j ˆ l1 if and only if 2(C). Here l 1 is the space of bounded functions on . In this paper we are concerned with ®nite unions of Carleson sequences and for these we have the following result (cf.[14] and [6]). Proposition 1.1. If ˆSNiˆ1 i, where i2(C), then for every 0< <1 there exists a partition ˆ Sn 1 n with the following properties. (i) supn 1jsnj N, where jEj denotes the cardinal of a set E. (ii) jbl( )j< if , 2sn, n 1. (iii) There exists >0 such that for every choice 0={ln}n 1 with an arbitrary ln2sn, n 1, we have 02(C) and 0 . (iv) There exists a sequence …Dn†n 1 H1 such that Dn…l† ˆ 1 if l 2 n; 0 if l 2 n n; …1† X n 1 jDn…z†j M; z 2 D: …2† Remark. The ®rst three statements have been proved by V.I. Vasyunin. (See [14]; see also [6], where a di€erent proof of this fact was given.) The construction of the family (Dn)n 1 was given in [6]. We mention that they generalize P. Beurling's functions which allow the construction of a linear operator of interpolation in the case of a single Carleson sequence. Let us number sn=fln,kg nj kˆ1 and set k ˆ fln;kgn 1;k j nj, (k=1,. . .,N). For technical reasons we will also de®ne ~ i ˆ fln;ign 1, where we set ln,i=ln;j nj if i jsnj. In view of statement (iii) of Proposition 1.1 these sequences are also Carleson sequences whose Carleson constant is bounded below by d. We now de®ne a sequence space by l0=X( 1). This de®nition of l0 depends a priori on the choice of the element ln,12sn, n 1. For this reason we need the following de®nition of stability which is motivated by that of [9]. Definition 1.2. Let X Hol(D). The space X is called (C)-stable if for all pairs of Carleson sequences ={ln}n 1 and ˆ f~ lngn 1 satisfying supn 1jbln…~ ln†j < 1; …3† we have X… † ˆ X… ~ †: …4† 104 ANDREAS HARTMANN at https://www.cambridge.org/core/terms. https://www.cambridge.org/core/product/BA6B568D5BA12FF3CC73C7BFCA45940A Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 10 Sep 2017 at 14:06:15, subject to the Cambridge Core terms of use, available A sequence ~ satisfying (3) with ˆ supn 1jbln …~ ln†j will be called -shifted with respect to . (See also [19].) Remember that the sequences in X( ) are indexed by the natural numbers (cf. (2) of the Introduction) and thus there will be no confusion in interpreting equality (4). With this de®nition it is clear that if a space X is (C)-stable then the corresponding sequence space l0 does not depend on the choice of ln,12sn. Remarks. (1) It is easy to see that if we do not have (3) then, in general, we need not have equality (4). Take for example X ˆ Hp…D† ˆ ff 2 Hol…D† : Np…f† ˆ sup0< r< 1 1 2 … ÿ jf…reit†jpdt <1g; 0 < p <1; the Hardy space on the unit disk. For 1 p<1 this is a Banach space with norm k f kpp=Np(f) and for 00}. Take now X=LA the space of Fourier transforms on C+ of L -functions: LA p={F2Hol(C+) :F(z)=(F f )(z), f2Lp(0,1)}, where …F f†…z† ˆ „1 0 f …t†eitzdt. It was mentioned in [16] that for the sequences ={i2}n 1 and ~ ={i+2}n 1, we have X… † ˆ l ; X… ~ † ˆ l ; which are obviously not isomorphic. Here