By A. G. Vitushkin
ISBN-10: 0387170049
ISBN-13: 9780387170046
The 1st contribution describes simple strategies, evidence and difficulties of the fashionable concept of whole services of a number of complicated variables. the second one contribution offers with analogies of simple Nevanlinna's theorems concerning the distribution of values within the multidimensional case and diverse functions. The 3rd contribution is dedicated to invariant metrics and volumes and their purposes in difficulties of functionality concept of a number of variables. The fourth contribution touches upon numerous effects in regards to the stress of holomorphic mappings of advanced areas beginnning with classical Liouville's and Picard's theorems. Contribution 5 provides effects pertaining to extension of holomorphic mappings to the bounds of domain names, and effects approximately correspondence of obstacles and equivalence of domain names with admire to biholomorphic mappings. Contribution six dwells at the challenge of biholomorphic equivalence of manifolds during this differential geometric point. The final contribution studies functions of multidimensional complicated geometry in sleek actual theories - supergravitation and supergauge fields. This quantity might be beneficial to complicated analysts and physicists. it really is rounded off by means of an in depth bibliography.
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Extra resources for Several Complex Variables I: Introduction to Complex Analysis
Example text
I D . where dl/, is the element of k-dimensional volume in C”. 2) differ respectively from the orthogonal Szegii Projection S: L2@D)+H2(aD) and the orthogonal Bergman ejection q y, let us B: L2(D)+H2(D) by completeb continuous operators. More precise t ” II. The Method of Integral Representations in Complex Analysis ,::Aff(z) = If(aA~(g,a)dv,,-,(tQ;J _;: :,;. : 4 Y,‘:’:is~~:‘ ! : i T : I. : ‘. dD -‘: :. :. - .. ,’ ‘($*. ‘. , ’ ~II.. >f,, . , :.. : . _! , /pf(@& j(a&g;,z)~pi(~:’ ;. ‘:-:! ;:, ; : : : d where A’((, z)= ZP(c,z)- K”(C,z),‘sLO, 1.
1) wherep = {p,, . . , pk} is a collection of real smooth functions in the domain $2 with the propelety apI A . . A &I, # 0 on M. 1), we have If f is lqcally integrable or even a generalized function on M, then the equation &f = 0 should be understood in the generalized sense: Iwf A 8~ = 0, for any smooth differential form (p of compact support. If the CR-manifold M is real analytic and f a real analytic function on M, then as Tomassini (1966, see [3]) has shown, the equation &,f= 0 is equivalent to the requirement that the function f be the trace on M of some function holomorphic in a nlighbourhood of M.
6 (Kohn [22], [56]). 5 and g a &closed form of class L& l,(D), then a) there exists a unique solution fc L’(D) to the equation af = g which is orthogonal to the space Hz(D); this solution is given by the equation f = 6* Ng; b) moreover fc Lzsk(D) gfge L$:(D). 6 is valid for any: bounded pseudoconvex domain D. Moreover, the following result holds. 7 (Hiirmander [44]). Let D be any pseudoconvex bmain in C” and cp any plurisubharmonic function in D. Then, the equation af = g adyits ). a solution with the following global L2-estimate d(f(z)12e-+‘(‘)(1 + Iz~2)-2dV2,(z) s d Ig(z)12e-*@)dV,,(z).