By E. Bedford, J.P. D'Angelo, Steven G. Krantz, R.E. Greene
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Extra info for Several Complex Variables and Complex Geometry (Proceedings of Symposia in Pure Mathematics, Vol.52) (Pt. 2)
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Scuola Norm. Sup. Pisa (3) 15, (1961), 97—142. (in teorema d'annuiamento deli acoomologia, Atti Convegno Internaz Geometna Algebrica (Torino, 1961). 57—52, Rauero, Turin, 1962. (AnV3) —, Disuguaglianze di Carleman sopra una variela complessa, Atti. Accad. Maz. Lincei Rend. Cl. Sd. Fis. Mat. Natur (8) 35 (1963), 431—434. [AnV4J —, Carlenian estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Etudes Sci. PubI. Math. 25 (1965), 81—130. (Ao) K. Aomoto, L'analyse harmonique sur les espaces Riemanniens a courbure Riemanniene negative.
LimkD(y(t), We say that a function f: D —' C has restricted K-limit L at x (with respect to the given projection device) if f(y(i)) —. L as I —' 1 along any special restricted x-curve. 2. device at x E 3D. limf(y°(t)) f—si exists. Then f has restricted = L K-limit L at x. should be remarked that the same theorem holds (with a very similar proof) for functions which are normal in the sense of Cima and Krantz [CKI. In this paper we shall need three different projection devices. We already introduced the first one: let D c C'1 be a strongly convex C2 domain.
We shall say that a function f: D C has n-restricted K-limit L at for all n-restricted n-special curves y in L as I D ending at x. 4) holomorphic function f: D — C has limit L along a n-restricted curve in D ending at x E o D, then f has n-restricted K-limit L at x. It is worth noticing that this is only one kind of Lindelöf principle: a large class of such statements has been introduced in [A3]; see also § 1. The second new feature a several variables versãon of the Julia-Wolff-Carathéodory theorem depends on a much more simple-minded fact: in one variable one has only one derivative, whereas in several variables one has a whole bunch of them (a—jacobian—matrix of them), and they have no reason to display all of them the same behavior near the boundary—and indeed they do not: the behavior of the normal component of the differential is quite different from the one of the complex tangential components.