New PDF release: Analysis of Several Complex Variables

By Takeo Ohsawa

ISBN-10: 0821820982

ISBN-13: 9780821820988

One of many ways to the research of capabilities of numerous advanced variables is to take advantage of tools originating in actual research. during this concise ebook, the writer offers a lucid presentation of ways those tools produce a number of worldwide life theorems within the concept of services (based at the characterization of holomorphic capabilities as susceptible ideas of the Cauchy-Riemann equations). Emphasis is on contemporary effects, together with an $L^2$ extension theorem for holomorphic capabilities, that experience introduced a deeper realizing of pseudoconvexity and plurisubharmonic capabilities. in accordance with Oka's theorems and his schema for the grouping of difficulties, the booklet covers themes on the intersection of the speculation of analytic services of a number of variables and mathematical research. it really is assumed that the reader has a easy wisdom of advanced research on the undergraduate point. The e-book might make an excellent supplementary textual content for a graduate-level path on advanced research.

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It is easy to see that v(log If I, xo) = sup{k E Z+ I (a) < k implies f (a) (xo) = 0}. 46 3. R), and set 2m. log X00 L IQ1I2. -P. Demailly, 1992). There exist constants CI and C2, independent of m, which satisfy the following conditions: a. '(z) - CI fin v,,,(z) and r < bn(z). b. v(,, z) - m Sup K-zI_ c} is an analytic subset of Q.

1. l always extends to a holomorphic mapping from 02 to P. 2. C is Hartogs pseudoconvex. PROOF. It suffices to show that the restriction mapping A(02) A(TE) is surjective. Let f E A(TE). 1 which is convergent on 02(0, (e,1)). After changing the order of the summation, let us observe the range on which the following equation holds: f (z) _ > (Eckzz) Z . By setting Ck(Z2) := E ck,tz2 for Iz21 < 1, we regard the right hand side as a series with terms of holomorphic functions ck(z2)zi. 1) Ick(z2)I < sup If (zI, z2)Ir-k for 0 < r < 1.

Let f11 C Cm and f12 C C" be open sets, and let F: f11 - f12 be a holomorphic mapping. Then, for any t' E PSH(f22). one has that v o F E PSH(f11). PROOF. If ,IL E C2 (02) n PSH, then tai o F E PSH(f11) follows from taking its derivatives. 10. 12. Hartogs pseudoconvexity and pseudoconvexity are equivalent to each other. Pseudoconvexity: As PROOF. Hartogs pseudoconvexity take I z 12 for the case f1 = C', and 1z12 - log 5n otherwise. 4. 13. For an increasing sequence {flk}k I of pseu- x doconvex open sets, U flk is pseudoconvex.

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Analysis of Several Complex Variables by Takeo Ohsawa

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