By Takeo Ohsawa

One of many techniques to the examine of capabilities of numerous complicated variables is to exploit tools originating in actual research. during this concise e-book, the writer supplies a lucid presentation of ways those equipment produce quite a few international life theorems within the concept of capabilities (based at the characterization of holomorphic capabilities as vulnerable options 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 features. according to Oka's theorems and his schema for the grouping of difficulties, the publication covers subject matters on the intersection of the speculation of analytic services of numerous variables and mathematical research. it's assumed that the reader has a simple wisdom of complicated research on the undergraduate point. The publication might make a very good supplementary textual content for a graduate-level path on complicated research.

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**Example text**

A domain of holomorphy is Hartogs pseudoconvex. PROOF. Let f be a holomorphic mapping from TE to P. 3, f has a holomorphic extension f : 02 _ C. 1. PSEUDOCONVEXITY OF DOMAINS OF HOLOMORPHY 37 true that f (A2) fl, then, letting T. be the maximum among connected open sets U that contains T. and satisfies f (U) C fl, it would follow that 8TE n A2 54 0. The maximum property of TE implies f (p) E 89 for a given point p E 8T£ fl A2. 15, choosing a sequence of points p in TE fl A2 that converges to p, there exists an element g in A(fl) such that lim 1g(f(pµ))J = oo.

11) liminf E-2m({z E U I inf Iz - wI < E}) < oo. wEE C-0 Then A2(fl \ E) = A2(f2). PROOF. Set dE(Z) := ti E Iz - wI. Also, take a C°° function p : R --+ [0,1] such that PI (- oo, 2) = 1 and pI(1, oo) = 0, and define a function Xf on fl by (dE(z)) Xe(Z) := E where E > 0. dE(z) is almost everywhere differentiable, since it is Lipschitz continuous. 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.

For the sake of brevity, we call a complex 1-dimensional hyperplane a complex line and a hyperplane of complex codimension 1 simply a hyperplane. It is self-evident that when X is the intersection of a complex mdimensional hyperplane with fl, the above definition of holomorphic function coincides with the usual definition, in which X is regarded as an open set in C1. Putting L := {z E C' I z1 = = zn_m = 0}, let us convert the surjectivity problem of the restriction map A(fl) -A(flnL) into the D equation.