By Takeo Ohsawa

ISBN-10: 0821820982

ISBN-13: 9780821820988

One of many methods to the research of services of numerous advanced variables is to take advantage of equipment originating in actual research. during this concise e-book, the writer provides a lucid presentation of the way those tools produce quite a few worldwide lifestyles theorems within the conception of features (based at the characterization of holomorphic features as susceptible suggestions of the Cauchy-Riemann equations). Emphasis is on fresh 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 ebook covers subject matters on the intersection of the speculation of analytic services of a number of variables and mathematical research. it truly is assumed that the reader has a easy wisdom of complicated research on the undergraduate point. The publication could make a very good supplementary textual content for a graduate-level direction on complicated research.

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**Extra resources for Analysis of Several Complex Variables**

**Example text**

In this section, we will mention general properties that such sets possess. Let fl be a domain in C", namely, a connected open set. 15. 14) (al + (1 - (zl al), ... }. (" - (zn - an)) E 12 for any z E fl and any C E (8:,)". 14) holds for any z E fl and any (EA". 3. REINHARDT DOMAINS 19 Polydiscs and complex open balls are examples of complete Rein- hardt domains. Clearly, a complete Reinhardt domain contains the center in it. Let D be a complete Reinhardt domain. Assume that the center of D is 0 for simplicity.

In recent years a remarkable result on the approximation of plurisubharmonic functions has been obtained. For the present, let us introduce the statement of this result while putting off its proof. Let V) be a plurisubharmonic function on Q. x0 log Iz - xol Recall that log If I E PSH(Q) for f E A(Q). 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.

Then this holomorphic extension f will satisfy the above condition. 3 For brevity, we call the condition H°'9 (Q)=0 for all 1

### Analysis of Several Complex Variables by Takeo Ohsawa

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