Ingo Lieb

Beschränkte Lösungen der Cauchy-Riemannschen Differentialgleichungen auf q-konkaven Gebieten

AbstractGeneralising a result of M. Hortmanns we show that the Cauchy-Riemann equations

Holomorphic maps: Geometric aspects

\bar \partial

Non-elementary functions

have bounded solutions on any strictly q-concave domain G, provided f is an exact bounded (o,r)-form with 1≦r≦ dim G-q-1. The proof requires the construction of suitable Cauchy-Fantappie-Kernels and L2-estimates.

Analysis in the complex plane

Bevor wir uns der Integration von Differentialformen zuwenden, mussen einige formale Vorbereitungen getroffen werden. Zunachst stellen wir eine Reihe algebraischer Begriffe zusammen.

Integral formulas, residues, and applications

We study holomorphic, in particular biholomorphic, maps between domains in c and, in one case, in c n , n > 1. These maps are for n = 1 conformal (angle and orientation preserving); so we shall use the terms biholomorphic and conformal interchangeably in this case. For n > 1 we consistently use biholomorphic. Automorphisms of domains, i.e. biholomorphic self-maps, are determined for disks resp. half-planes, the entire plane, and the sphere: they form groups consisting of Mobius transformations (VII.1). The proof of this fact relies on an important growth property of bounded holomorphic functions: the Schwarz lemma 1.3. Because the automorphism group of the unit disk (or upper half plane) acts transitively, it gives rise – according to F. Kleins Erlangen programme – to a geometry, which turns out to be the hyperbolic (non-euclidean) geometry (VII.2 and 3). The unit disk is conformally equivalent to almost all simply connected plane domains: Riemanns mapping theorem, proved in VII.4. For n > 1 even the immediate generalisations of the disk – the polydisk and the unit ball – are not biholomorphically equivalent (VII.4). Riemanns mapping theorem can be generalised to the general uniformization theorem (VII.4): a special but exceedingly useful case of this is the modular map λ which we introduce in VII.7. Its construction uses tools that are also expedient for other purposes: harmonic functions (with a solution of the Dirichlet problem for disks) and Schwarzs reflection principle (VII.5 and 6). The existence of λ finally yields two important classical results: Montels and Picards “big theorems”.

The fundamental theorems of complex analysis

This chapter presents the analogues of the Mittag-Leffler and Weierstrass theorems for functions of several complex variables. To this end it develops fundamental methods of multivariable complex analysis that reach far beyond the applications we are going to give here. – Meromorphic functions of several variables are defined as local quotients of holomorphic functions (VI.2); the definition requires some information on zero sets of holomorphic functions (VI.1). After introducing principal parts and divisors we formulate the main problems that arise: To find a meromorphic function with i) a given principal part (first Cousin problem) ii) a given divisor (second Cousin problem); iii) to express a meromorphic function as a quotient of globally defined holomorphic functions (Poincare problem). These problems are solved on polydisks – bounded or unbounded, in particular on the whole space – in VI.6–8. The essential method is a constructive solution of the inhomogeneous Cauchy-Riemann equations (VI.3 and 5) based on the one-dimensional inhomogeneous Cauchy formula – see Chapter IV.2. Along the way, various extension theorems for holomorphic functions are proved (VI.1 and 4). Whereas the first Cousin problem can be completely settled by these methods, the second requires additional topological information which is discussed in VI.7, and for the Poincare problem one needs some facts on the ring of convergent power series which we only quote in VI.8.

Functions on the plane and on the sphere

The theory of the preceding chapters permits the construction and investigation of new transcendental functions. The Γ-function, interpolating the factorials, is perhaps the most important nonelementary function (V.1). Riemanns ζ-function (V.2,3) and its generalizations play an eminent role in number theory and algebraic geometry. It is the main tool in most proofs of the prime number theorem (V.2). Elliptic functions, i.e. functions with two independent periods, and their connection with plane cubic curves is a classical theme (V.4,5) with applications in many areas, e.g. mathematical physics and cryptography.

Integral Representations for the% MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafyOaIyRbae% baaaa!376B!\bar \partial-Neumann Problem

The fundamental concept of holomorphic function is introduced via complex differentiability in section I.1. The relation between real and complex differentiability is then discussed, leading to the characterization of holomorphic functions by the Cauchy-Riemann differential equations (I.2). Power series are important examples of holomorphic functions (I.3); we here apply real analysis to show their holomorphy, although Chapter II will open a simpler way. In particular, the real exponential and trigonometric functions can be extended via power series to holomorphic functions on the whole complex plane; we discuss these functions without recurring to the corresponding real theory (I.4). Section I.5 presents an essential tool of complex analysis, viz. integration along paths in the plane. In I.6 we carry over the basic theory to functions of several complex variables.