By Takeo Ohsawa

One of many methods to the examine of services of a number of advanced variables is to take advantage of equipment originating in actual research. during this concise booklet, the writer provides a lucid presentation of ways those tools produce a number of international life theorems within the concept of capabilities (based at the characterization of holomorphic features as susceptible recommendations of the Cauchy-Riemann equations). Emphasis is on fresh effects, together with an $L^2$ extension theorem for holomorphic services, that experience introduced a deeper figuring out of pseudoconvexity and plurisubharmonic features. according to Oka's theorems and his schema for the grouping of difficulties, the publication covers themes on the intersection of the speculation of analytic capabilities of a number of variables and mathematical research. it really is assumed that the reader has a easy wisdom of complicated research on the undergraduate point. The booklet could make a very good supplementary textual content for a graduate-level direction on complicated research.

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**Example text**

Accordingly, so is XF, and (z) i9x < E I P (t)I s sup almost everywhere on Q. Suppose that f E A2(f2 \ E). dV=0fori=1,... ,n . 2. CAUCHY-RIEMANN EQUATIONS 17 For this purpose, divide the left hand side of the above equation into fn f (XEu)dV +Jnf i((1-XE)u)dV. IFi First, since D f = 0 on Si \ E, integration by parts implies a f ((1 - XE)u) dV = 0. 12) On the other hand, r r Jf(XeU)dVLH J n azi azi The first term on the right hand side satisfies 2 n fL'-6udVI < E-2 supIU12 suPIP I2 -m(Eu,,)J "J IfI2dV , u Eu,E := {z I dE(z) < e} fl supp u.

Take a sequence a j of points in D that converges to a and satisfies aj E Dj+1 \Dj. Let mad (z) be such a monomial as above that is determined for each domain Dj and point aj. 15) sup Ickmak (Z) "k I < zEDk Then the series 1 . 2k 00 E cjma, (z)-' j=1 converges uniformly on compact sets in D. 6 implies F E A(D). 15), we clearly have IF(ak)I > k - 1 for each k. This results in lim IF(z)j = oo. 3. " General open sets that possess this property will be discussed later. It seems interesting, at this point, to study the convergence m range of a series > fk whose terms are homogeneous polynomials fk k=0 of degree k as a slight generalization of a power series, but we do not do so in this book.

Q(fl) = {0} for all q with 1 <_ q <- n - m. This is called Serre's criterion. For the proof, we need one lemma. 6. The restriction homomorphism a : HP,q(fl) -, HP,q(fl n L1) is a surjection if Hp,q+l(fl) = {0}. In particular. {0} if HP,q+k(fl) = {0} for all k = 0,1. n Ln_1) _ ZIm overlaps the notation for the imaginary part of a complex number, but there should not be any confusion. 28 2. FUNCTION RINGS AND 8 COHOMOLOGY PROOF. fl n L,,_1) n Ker B. fl such that av = 0 on some neighborhood of ,R n L,,_1.