Zhihong WENGuantie DENGCuiqiao WANGFeifei QU

1 Introduction

Concerning the decomposition,we have known that the decomposition of Lp(R)into the sum of H+(R)and H−(R)is obtained(see[8,11,13]),at least for range 1

Theory of boundary value problems for analytic functions(see[3,10])is one of the most important branches of complex analysis(see[4,9]).It has wide applications because many practical problems in mechanics,physics and engineering may be transformed to such problems or singular integral equations which are closely related to the boundary problems.In the third section,we will discuss the relationship between boundary values of holomorphic functions and distributions(see[1])in n-dimensional complex space.The same conclusions were established when n=1 in[2].First we introduce some notations before we state our main results.

A measurable function f is said to belong to

where mkis the number of−1 in σk.The distributionsare called the distributional boundary values of f and fσk,respectively.

Definition 1.2 A function f holomorphic in TΩσkis said to be slow growth if,for every compact subset K of Rn,there exist an integer k and two positive constants ǫ and C such that

Definition 1.3 Let X⊆Cnbe an open set.A linear formµon the vector space C∞(X)is called to be continuous if there is a compact set K⊆X,a constant c≥0 and a nonnegative integer N such that,for all φ ∈ C∞(X),

2 Decomposition of Lp(∂Da)Space

As previously mentioned in introduction,we have known that the functions of Lp(R)for all 0

then for almost everywhere z∈∂Da,we have

where K is a constant satisfying

Lemma 2.4 Suppose thatis a rational function whose poles are contained in,then there exist two rational functions P and Q such that

Proof For the case 0

Therefore,there exists a number ϕ ∈ [−π,π]such that

It follows from Lemma 2.1 that

almost everywhere,and(1.4)–(1.5)hold.

An argument similar to that used in the proof of the case 1

3 Boundary Value of Holomorphic Function in the Sense of n-Dimensional Distributions

This section presents several theorems as well as a key lemma,which is the main part of this section.

Theorem 3.1 Let Γ = Ωσ1be the first octant in Rn.If f is a holomorphic function of slow growth in TΓ,then it admits a boundary value in the sense of distributions.

Proof Let K be a compact subset of Rn.For any ǫ>0,and a fixed pointTΓ,we denote,···the successive primitives of f in TΓ,vanishing at z0,

Under the condition of Theorem 3.1,let us denote bythe function(and its associated distribution)defined inas follows:

Remark 3.1 If we start with fσkbeing a holomorphic function of slow growth in TΩσk,for any σkin Sn,we obtain

Theorem 3.4 Every distribution T inis a boundary value of a holomorphic function of slow growth in TΩ.

AcknowledgementThe authors are very grateful to the referee for the insightful comments and suggestions,which greatly improved the exposition of the manuscript.

[1]Barros-Neto,J.,An Introducton to the Theory of Distributions,Marcel Dekker Inc.,New York,1973.

[2]Berenstein,C.A.and Gay,R.,Complex Variables:An Introduction,Springer-Verlag,Berlin,1991.

[3]Constantinescu,F.,Boundary values of analytic functions,Comm.Math.Phys.,7,1968,225–233.

[4]Deng,G.T.,Complex Analysis,Beijing Normal University Press,Beijing,2010.

[5]Deng,G.T.and Qian,T.,Rational approximation of functions in Hardy spaces,2014.Complex Anal.Oper.Theory,10(5),2016,903–920.

[6]Ding,Y.,Modern Analysis Foundation,Beijing Normal University Press,Beijing,2008.

[7]Fei,M.B.,Hpspace on Tube,PhD thesis of Beijing Normal University,2014.

[8]Garnett,J.B.,Bounded Holomorphic Functions,Academic Press,1987.

[9]Krantz,S.G.,Function Theory of Several Complex Variables,Wiley,New York,1982.

[10]Lu,J.K.,Boundary value problems for analytic functions,Series in Pure Mathematics,16,World Scientific Publishing Co.,Inc.,River Edge,NJ,1993.

[11]Qian,T.,Characterization of boundary values of functions in Hardy spaces with applications in signal analysis,J.Integral Equations Appl.,17,2005,159–198.

[12]Schwartz,L.,Theorie des Distributions,Hermann,Paris,1978.

[13]Stein,E.M.and Weiss,G.,Introduction to Fourier Analysis on Euclidean Space,Princeton University Press,Princeton,New Jersey,1971.

[14]Stein,E.M.,Weiss,G.and Weiss,M.,Hpclass of holomorphic functions in tube domains,Proc.Nat.Acad.Sci.U.S.A.,52,1964,1035–1039.

[15]Wen,Z.H.and Deng,G.T.,Further discussion on Hardy space in an angular domain,Journal of Beijing Normal University(Natural Science),50,2014,581–586.