Interpolation sets for Hardy-Sobolev spaces on the boundary

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Interpolation sets for Hardy-Sobolev spaces on the boundary of the unit ball of C n .Jaume Gudayol ?April 23,1998Abstract We study the interpolation sets for the Hardy-Sobolev spaces de?ned on the unit ball of C n .We begin by giving a natural extension to C n of the condition that is known to be necessary and su?tient for interpolation sets lying on the boundary of the unit disk.We show that under this condition the restriction of a function in the Hardy-Sobolev space to the set always exists,and lies in a Besov space.We then show that under the assumption that there is an holomorphic distance function for the set,there is an extension operator from these Besov spaces to the Hardy-Sobolev ones.1Introduction In this work we study the boundary interpolation sets for Hardy-Sobolev spaces de?ned on the unit ball of C n .The study of interpolation sets for di?erent spaces is one of the classical subjects of S.C.V.analysis.But in the previous works there are serious restrictions:one considers either sets contained in varieties or sets that have dimension less than one.In this work we study sets not having such restrictions.Even though there are other kinds of restrictions,we believe that one can ?nd here a (perhaps small)step towards the general case.The study of interpolation sets was begun by Carleson and Rudin (See

[Rud,80],chapter 10,for references).They showed (independently)that,for n =1,interpolation sets for the ball algebra were precisely those of zero Lebesgue 71b1b63083c4bb4cf7ecd126ter,and also for n =1,interpolation sets for A ∞(D )were described by Alexander,Taylor and Williams in [ATW,71].In this case the interpolation sets are those satisfying that for any arc I ?T ,

1d (e it ,E )dt ≤C log

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?Partially supported by MEC grant PB95-0956-c02-01and CIRIT grant GRQ94-2014.

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Interpolation sets for the spaces Aα(D)were caracterized by Dynkin in[Dyn,80]

and Bruna in[Bru,81].We will say that a closed set E?X satis?es the Uniform

Hole Condition(UHC-sets,for short)with respect to X if there exists0

sup{d(y,E),y∈B(x,r)}≥Cr.(1) The UHC as related to interpolation properties was introduced by Kotochigov, but other equivalent de?nitions have been introduced by other authors in di?er-

ent contexts.The de?nition says that a set has holes of a?xed size when looked at at any scale.Dynkin and Bruna proved that,for the spaces Aα(D),E is

an interpolation set i?E is a UHC-set.This characterization was obtained by

Dynkin forα∈N and by Bruna for all0<α<+∞.Later Dynkin([Dyn,84]) proved that a set is an interpolation set for the Hardy-Sobolev spaces i?it is a

UHC-set.

For n>1no caracterization of boundary interpolation sets is known,not

even for the ball algebra.This does not mean that there is no information about

interpolation sets.For the ball algebra,Rudin in[Rud,80]devotes all of chapter 10to these sets,that in this case are the same as peak sets and zero sets.There some examples are given,and one can?nd some background on the problem.

Also for the ball algebra,Nagel in[Nag,76]proved that any subset of a complex-tangential manifold is an interpolation set.On the other hand,Davie and?ksendal(see[Rud,80],section10.5)proved that any set that has,in a sense,dimension less than1is an interpolation set.Both results point out to the fact that an interpolation set can be as large as one wants in the complex-tangential directions,but has to be small in the other ones.

The study of zero sets and interpolation sets for spaces other than the ball

algebra has been done by several authors.In the case of Aα(B n)and Hardy-

Sobolev spaces results concerning sets contained in varieties were given by Bruna and Ortega in[B-O,86],[B-O,91],and[B-O,93].These works have provided us with our main inspiration.Chaumat and Chollet obtained several results for the space A∞(B n)in,for example,[C-C,86],and for the Gevrey classes in[C-C,88].

Our goal was to study interpolation sets for Hardy-Sobolev spaces.But in

this case,the?rst problem was to know,given an f in a Hardy-sobolev space, to which function space de?ned on the set would the restriction belong.This question,which is in most cases trivial,in this case is not so.However,the results in[B-O,86]showed clearly that the space of the restrictions should be some Besov space.But even in the real case,no general result on restrictions of functions to Besov spaces de?ned on arbitrary sets is known(however,Jonsson and Wallin in[J-W,84]and Jonsson in[Jon,94]give some partial results).In this article we give a restriction theorem for a general set E?S.For the restriction to exist,we impose that the Uniform Hole Condition(1)holds.We show that this condition in equivalent to other conditions that will be useful later,and in

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particular,that is equivalent to the fact that the set has,in a sense,dimension less than the dimension of S.

Once we have done that,we show that under some restrictions,there is an extension operator,thus proving that the given set is interpolating.The restriction we impose is that we assume that there is a holomorphic function behaving like the distance to the set.We give some examples of such functions. 2De?nitions and statement of results

The upper dimension of a set

Let(X,ρ)be a compact pseudo-metric space,with diam(X)<+∞(this means thatρsatis?es the triangle inequality with a constant).For x∈X,R>0 and k≥1,let N(x,R,k)be the maximum number of points lying in B(x,kR) separated by a dis …… 此处隐藏：20582字，全部文档内容请下载后查看。喜欢就下载吧 ……