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- W87551685 abstract "In this paper we shall prove that any holomorphic Lp function on V, (1≦p 0 small enough, there exists c2>0 such that (1) 2Re(~-z, 7(~, z));~-p(z)+p(~) +c2~[(~k k ' )Izh-~k 12+ Iz ~ 12~ J 2** 2+ ~2~ ~2 k=1 for (~, z) e l~X D~: Moreover, they obtained the following lemmas: Extension of Holomorphic Functions from Subvarieties to Convex Domains 17 LEMMA1.Fo7q>0,s=O o71,j=s,s十1,._。,伽4Aρos吻6,0Jos6渉00, ゐ1姻(燃寝陵隔・一膿父)’”、 初4砂6%46n!‘ゾte(一R,R). LEMMA2.Eo7q〉0,j≧1,伽4Aヵos伽6,6Jos6渉00, ゐ、d(藩旱菱1翠鵠一{9継)il濃、 初4の6,z46曜6ゾte(一R,R),側h6名6r=l z lニ(x2十y2)112, Weset Q一激醤)鴫 Then by Bemdtsson[1],we have the following: PR・P・SITI・N2.五6∫f伽h・励ゆh轍吻漉Ws魏の虹lfldσ 0 and F(z)= fa f*(~)K(~, z) for z e D. av Then F(z) is holomorphic in D. Let z=x+iy e Bo' Then aF (3) a (p(z)'F(z))= ep(z)'~1 ap (z)F(z)+p(z)' (z). ax j ax j ax j Since a-ax~ is a sum of terms f*(~)~l(~, z) A a~yh. A d~*, f*(~)~2(~, z) /¥ a~7k. A d~*, fv i=1 i=1 fv i=1 i=1 (~-z, 7(~, z))k ' (~-z, 7(~, z))k+1 where ~1(~, z) is a smooth (O, 1), form and ~2(~, z) is a smooth function. Since Lj, (j= 1, ......, k-1), form a base for the (O, 1) tangential vector fields, we have to estimate the following integrals: A Lj7j A d~*. A Lj7jAd~t A d~*, f i=1 i=1 i=1 i=1 f avng, (~-z, 7(~, z))k ' ~ (~-z, 7(~, z))k ' av n 8 k-1 k A Ljrj /¥ d~*, f i=1 i=1 avnB' (~-z, r(~,z))k+1 By applying lemmas 1, 2, and inequalities (1), (2), we have k-2 k A Ljrj A d~*,Ad~t f i = I i = I d rkId(1k_ p(~lk+1 ~c. (6) I Since the integrand of ll is less singular than that of 12, we shall show that 12~c. e>0 sufficiently small, we set U.={~ e D : f p(~) I p(~) l By the same method as the proof of theorem 1, we obtain N f I ~~c [ Iog (1 p(~) I + ~: (~2n z+~~~'-2) f zj-~j 1 2 It.'1~R j=k+1 ' t'~ ~R + I zj~j 1 2~') I dt 2k+1 'dt2N' we set A= max mi, and we introduce polar coordinates. Then we have I ~~;cjCR I Iog ( I p(~) f ~ rA) I rdr~c f p(~) f 1/2A= 1, we write F(z) in the following form F(z) = fwf (z)T( ~, z)d(1( ~). Let q be such that ~ + ~ I Then, by applying H6lder's inequality, we have I F(z) I p~( fw I f(~) I p I T(~, z) I d(;(~))( fw I T(~, z) I do(~))p/q By the same method as the case p=1, we obtam 19" @default.
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- W87551685 date "1988-02-29" @default.
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- W87551685 title "Extension of Holomorphic Functions from Subvarieties to Convex Domains" @default.
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