# Analysis of Several Complex Variables by Takeo Ohsawa

By Takeo Ohsawa

One of many techniques to the research of capabilities of a number of advanced variables is to take advantage of tools originating in actual research. during this concise ebook, the writer supplies a lucid presentation of the way those equipment produce numerous international lifestyles theorems within the thought of capabilities (based at the characterization of holomorphic features as susceptible options 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 figuring out of pseudoconvexity and plurisubharmonic features. in keeping 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 features of numerous variables and mathematical research. it truly is assumed that the reader has a simple wisdom of advanced research on the undergraduate point. The booklet could make a great supplementary textual content for a graduate-level path on complicated research.

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**Example text**

1) E fkgk = 1, k=1 then fl = Spec,, A(fl). PROOF. This is clear from the paracompactness of fl and the definition of Spec, A(0). O 23 2. ' The proof of the converse requires a good amount of preparation, and is deferred until Chapter 5. 1) as a solution of the 8 equation with some restraints. We can assume that the series h a rIfkl2 k=1 converges uniformly on compact sets by replacing the given sequence of functions fk with Ekfk (Ek # 0) if necessary. 2) and the Ascoli-Arzela theorem imply h E CI (R), and h has no zero point by assumption.

For this family. 18) yields A(t,1') E PSH(fl). 11. 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.

This chapter begins with the definition of Hartogs pseudoconvexity and verifies that a domain of holomorphy is Hartogs pseudoconvex. Secondly, we show that some canonical function on a Hartogs pseudoconvex open set expressed by a distance function is plurisubharmonic. In consequence of this, it is derived that a domain of holomorphy is pseudoconvex. Hartogs and Oka's discovery of this relevancy gives a unique vitality to the theory of analytic functions of several variables. In Chapter 4, in order to show that a pseudoconvex open set is a domain of holomorphy, some kind of differentiable plurisubharmonic functions will be needed.