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        General Integral Formulas of(0,q)(q>0)Differential Forms on the Analytic Varieties

        2017-03-14 02:46:22

        (School of Mathematical Sciences,Xiamen University,Xiamen 361005,Fujian,China)

        §1.Introduction

        Let?be a holomorphic domain inCn,we consider the following bounded domain??.We assume that the boundary?D(here and for the rest of this paper we always assume that?Dis piecewise smooth)of a bounded domainDin spaceC(n)consists of a chain of slit spaces,and?Dcan be expressed as

        Hatiziafratis[2]obtained an explicit Koppelman type integral formula for the bounded domain with smooth boundary in analytic varieties.On the analytic varieties,in order to get the integral representation formulas of(0,q)(q>0)differential forms must be used to get the integral representation formulas of(0,q)(q=0)differential forms(i.e.differentiable functions)with different methods.That we need to get the following lemma 2 and its corollary,as well as lemma 3.This present paper, firstly using different method and technique we derive the corresponding integral representation formulas of(0,q)(q>0)differential forms for the two types of the bounded domains in complex submanifolds with codimension-mi.e.in the complexn?m(0≤m<n)dimensional analytic varieties.Secondly we obtain the unified integral representation formulas of(0,q)(q>0)differential forms for the general bounded domain in complex submanifolds with codimension-mi.e.in the complexn?m(0≤m<n)dimensional analytic varieties,which include Hatziafratis formula,i.e.Koppelman type integral formula for the bounded domain with smooth boundary in analytic varieties.In particular,whenm=0,we obtain the unified integral representation formulas of(0,q)(q>0)differential forms for general bounded domain inCn,which are the generalization and the embodiment of Koppelman-Leray formula.

        We also note:(a)From the formulas of this paper is easy to get the solution of theˉ?-equation.And we can then consider solution estimation problem[7]of the described(To save space,this paper is not detailed).(b)The integral formulas of this paper can be used as a local results on Stein manifold,which can be extended to the Stein manifold.

        §2.Some Lemmas

        Becauseν(z)is arbitrary,by(2.11)we get the expression(2.10).

        The papers of Hatziafratis[1-2]and the author[6]are most relevant referens to above lemmas.

        §3.Main Theorems

        Theorem 1IfDis a bounded domain with piecewise smooth boundaries inC(n),then for(0,q)(q>0)differential formu(z)onˉDandz∈?D,we have the extension of the Koppelman-Leray formula on the complex submanifolds i.e.on the complexn?m(0≤m<n)dimensional analytic varieties

        Lettingr→0 and taking the limits of the two side of(3.3),by lemma 3 we get(3.1).

        Remark 1Using the properties of the determinant,which includes the exterior differential forms(cf.[4]),and Laplace theorem of the determinant,Hatziafratis formula[2]can be easily obtained from the theorem 1(only justQ0=/|??z|2).

        Theorem 2Let the boundary?Dof a bounded domainDinC(n)be defined by(1.1),and?be defined by(1.4)(wherek=1).Letτm=B0∩Z(Φ1,···,Φm)and?τm=ε0(β),and=?(Λ(β?1)×τm).Assuming thatis a chain,withand?as the boundary chain,then for the(0,q)(q>0)differential formu(z)onandz∈,we have the integral representation on the complex submanifolds for the bounded domain of the II-type?

        Therefore,we have

        On the other hand,since

        It is derived from(3.7)and(3.10)that

        By(3.11)and(3.1)we obtain(3.4).

        Assume thatDis a bounded domain and the boundary?DofDcan be represented aswhereqji(?,z)are holomorphic in(?,z)∈?D×DandFj(?,z)/=0 at?∈σjfor any fixedz∈D.Then we denoteFj(?,z)∈A.

        Theorem 3Let the boundary?Dof a bounded domainDinC(n)be defined by(1.1),and?be defined by(1.4),then for the(0,q)(q>0)differential formu(z)onandz∈we have

        Then by(3.1 8),(3.1 9)and Stokes formula we have

        Taking it in to account that

        From(3.2 0),(3.2 1)and(3.1 3)we obtain

        wherel=1,2,···,k?1.

        Using(3.22)repeatedly,we obtain

        By(3.14),we have

        It is follows from(3.23),(3.24)and(3.4)that(3.15).

        Thus whenβ=1,taking account of(3.25),then(3.15)translates into

        which is the integral representation of the(0,q)(q>0)differential forms on the complex submanifolds for the bounded domainDof I-type.

        Theorem 4Let the boundary?Dof a bounded domainDinC(n)be defined by(1.1),and?be defined by(1.4),then for the(0,q)(q>0)differential formu(z)onandz∈,we have the following formulas:(a)Whenl>1,

        Thus,we have

        By(3.31),we have

        whered=??+?ˉ?+dλ+dμ.

        For the sake of simplicity,

        is denoted byX,Y,Zrespectively,then?X=Y+Z?Stokes formula we have

        By computing we obtain the left hand side of(3.33)=0.

        This implies that

        Especially,whenl=1,by(3.35)we have

        Thus whenl>1,it is derived from(3.35)and(3.15)(wherelis used insteadk)that(3.27).Whenl=1,it is derived from(3.36)and(3.4)that(3.28).

        Remark 3Whenl=k>1,thenT=.Thus(3.27)translates into(3.15).In particular,whenl=k=1,then(3.18)translates into(3.4).

        Remark 4Whenl=k>1 andβ=1,we haveT==Q0and(3.27)translates into that(3.26).

        It is derived from remark 3 and remark 4 that(3.27)and(3.28)are the unified integral representations of(3.4),(3.15)and(3.26).

        Remark 5Whenm=0,then(ζ)=(ζ)=n!ω(ζ),? and?=?D,which is defined by(1.1).From above formulas we can obtain the unified integral formulas of the(0,q)(q?0)differential forms for the bounded domain inCn,i.e.we have the following results:

        (a)By(3.1),we have Koppelman-Leray formula

        where

        (b)By(3.4),we have

        which is an extension and an embodiment of Koppelman-Leray formula.

        (c)By(3.26),we have

        which is an extension and an embodiment of Koppelman-Leray formula.

        (d)By(3.27)and(3.28),we have respectively:

        Forl>1,

        where

        (3.41)and(3.42)are the unified integral formulas of(3.37),(3.39)and(3.40),and make Koppelman-Leray formula generalize and embody.

        [1]HATZIAFRATIS T E.Integral representation formulas on analytic varieties[J].Pacific J Math,1986,123:71-91.

        [2]HATZIAFRATIS T E.An explicit Koppelman type integral representation for muls on analytic varieties[J].Michigan Math J,1986,33:335-341.

        [3]φVRELID V.Integral representation formulas andestimates forequation[J].University of Math Scund,197129:137-160.

        [4]SOMMER F.Ueber die Integralformeln in der Funktionentheorie mehrerer Komplexer Veranderlichen[J].Math Ann,1952,125:172-182.

        [5]RANGE R M,SIU Y T.Uniform extimates for the-equation on domains with piecewise smooth strictly pseudoconvex boundaries[J].Math Ann,1973,206:325-3.

        [6]CHEN Shujin.General integral representation of holomorphic functions on the analytic subvariety[J].Publ RIMS Kyoto Univ,1993,29:511-533.

        [7]RANGE R M.A pointwise A-prori estimate for theNeumann problem on pseudoconvex domains[J].Pacific J Math,2015 275:409-432.

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