Function theory in the unit ball of \(\mathbb{C}^n\). (English) Zbl 0495.32001

Rudin, Walter

Grundlehren der mathematischen Wissenschaften, Bd. 241. New York-Heidelberg-Berlin: Springer-Verlag. xiii, 436 p. DM 79.50; $ 46.90 (1980).

The book constitutes an excellent introduction into one of the most active research field of the complex analysis. The author – himself one of the promoter of this field – chooses the unit ball \(B = B_n\) of \(\mathbb C^n\) as the frame of this work since the ball is the prototype of the strictly pseudoconvex domains and of the bounded symmetric ones, and the existence of a transitive group of automorphisms simplifies essentially the technics. As the author emphasizes, the principal ideas can be presented clearly and explicitly in the ball, specific theorems can be quickly proved and the reader arrives sooner and easier to understand and dominate a large area of facts, methods and problems at the research level. Extension to a general frame will be not difficult then.
Chapter 1 introduces the fundamental notions, tools and notations that will be used in the study of the \(\mathbb C\)-algebra \(H(B)\) of the holomorphic functions in \(B\): Cauchy formula in polydiscs and consequences (in particular the global power series representation for \(f\in H(B)\)), composition of holomorphic maps and as application the slice functions, differentiation results, rotation-invariant measures (in particular \(\sigma\) – the positive Borel measure on \(S = \partial B\), with \(\sigma(S)= 1\)), integration formulae and integrals that will occur later, homogeneous expansions.
Chapter 2 begins with Cartan’s uniqueness theorems for bounded and circular bounded regions in \(\mathbb C^n\) , and is concerned with \(\operatorname{Aut}(B)\). Every \(\psi\in\operatorname{Aut}(B)\) is decomposed into \(\psi = U \circ \varphi_a\), where \(U\in\mathcal U\) the unitary group, \(\psi(a) = 0\), and \(\varphi_a\in\operatorname{Aut}(B)\) is an involution which interchanges \(a\) and \(0\). The fixed point sets of the automorphisms of \(B\) are thoroughly studied. Their connection with the affine subsets of \(B\), the Hayden-Suffridge theorem, examples obtained by means of the Cayley transform and by considering the \(\varphi_a\) maps clarify the subject. -
The Bergman, the Cauchy and the invariant Poisson kernels and integrals are discussed in chapter 3. Among the numerous results let us remark the Cauchy and the invariant Poisson formula for functions of the ball algebra \(A(B)\), properties of the Poisson transform \(P[f]\) for \(f\in C(S)\) and \(f\in L^p(S)\), \(1\le p\le \infty\), or \(P[\mu]\) for \(\mu\) a complex Borel measure on \(S\), the invariant mean value property for \(P[\mu]\), the Möbius (\(\mathcal M\)-)invariance of the class of all functions with this property.
Chapter 4 is devoted to the invariant Laplacian \(\tilde\Delta\) and its eigenfunctions, especially the \(\mathcal M\)-harmonic functions. It presents connections between \(\mathcal M\)-harmonic functions and Poisson integrals, criteria for \(\mathcal M\)-harmonicity, pluriharmonic functions, especially Forelli’s results.
The boundary behaviour of the invariant Poisson integrals forms the theme of chapter 5, which contains Korányi’s extension of Fatou’s theorem to these integrals (the main tools being given by the maximal functions and the Marcinkiewicz interpolation) and the Calderón-Privalov-Plessner theorem for \(\mathcal M\)-harmonic and holomorphic functions in \(B\). The Nevanlinna class \(N(B)\), the Hardy spaces \(H^p(B)\), \(0< p< \infty\), as well as the general \(H_{\varphi}(B)\) spaces for \(\varphi: [-\infty,\infty) \rightarrow [0,\infty)\), \(\varphi\not\equiv 0\), nondecreasing and convex, are studied in connection with \(\mathcal M\)-harmonic majorants, K-limits, Poisson integrals of their K-limits. The spaces \(A(S)\) and \(H^p(S)\) complete this chapter.
The boundary behaviour of Cauchy integrals of measures, of \(L^p\)-functions and of Lipschitz functions is treated in chapter 6. We mention the author’s simplified version of the Korányi-Vagi theorem, the author’s and Ahern-Schneider’s \(n\)-dimensional extension of the Hardy-Littlewood theorem (with the radial derivative in the growth condition and the Lip conditions only on the slices), Stein’s phenomenon of additional smoothness in the complex-tangential directions, Toeplitz’ operators and as applications the author’s result on \(H^\infty(S) +C(S)\) and Gleason’s problem for \(A(B)\), \(H^p(B)\), \(1\le p\le \infty\), \(A(B)\cap \operatorname{Lip} \alpha\), \(0<\alpha\le 1\) and other classes of functions. The solution of this problem is used in the study of the homomorphisms of \(H^\infty(B)\).
Chapter 7 is devoted to \(L^p\)-topics. It begins with projections of Bergman type, which have according to Forelli and the author the reproducing property on \((L^p\cap H)(B)\). Further it contains relations between \(H^p(B_n)\) and \((L^p\cap H)(B_{n-1})\), growth and boundary behaviour properties and significant examples. Quantitative global properties of the zero-varieties and of the determinant sets of subspaces of \(H(B)\) are presented starting from the Blaschke condition. The counting functions are defined using slices and a Jensen inequality is proved in the general frame of the spaces \(H_{\varphi}(B)\), whence Weyland’s necessary condition for the zero-varieties of functions in \(H_{\varphi}(B)\), in particular in \(N(B)\) and in \(H^p(B)\), \(0<p<\infty\), are established. The Lumer-Hardy spaces \((LH)^p(\Omega)\), \(\Omega\) a region of \(\mathbb C^n\), \(0<p<\infty\), are introduced by means of pluriharmonic majorants and their zero-varieties determined (for \(\Omega\) simply connected). The author’s results put in evidence the pathological features of \((LH)^p(B)\) and permit the description of the isometries of \(H^p(B)\).
The base of chapter 8 consists in the Schwarz lemma for holomorphic maps between balanced regions, in particular between \(B_n\) and \(B_n\). It follows the author’s theorem on the fixed-point set of a holomorphic map \(B\rightarrow B\), and as a consequence the Suffridge result on the holomorphic retracts, a norm-preserving \(H^\infty\) extension problem with respect to a holomorphic map \(B_n\rightarrow B_m\), \(1\le n<m\), the Lindelöf-Čirka theorem and the extension of the Julia-Carathéodory theorem to holomorphic maps \(B_n\rightarrow B_m\), both last theorems being largely exemplified.
Chapter 9 contains Henkin’s measure on \(S\) and its decomposition by Valskii. After a fascinating analysis of the Glicksberg-König-Seever generalization of Lebesgue’s decomposition theorem for Hausdorff compact spaces, one proves a generalization of the F. and M. Riesz theorem and the Cole-Range theorem. All these powerful results are applied to the spaces \((LH)^p(B)\) and \(A(B)^*\).
In chapter 10 the author proves that a compact set in \(S\) is simultaneously a zero set, a peak set, an interpolation set, a peak-interpolation set ((PI)-set), a nullset and a totally null set. The development of this subject is followed from the case \(n = 1\) to Bishop’s moment, when the natural frame of function algebras far beyond the holomorphy became clear, till Chollet’s recent work on strictly pseudo-convex domains. The theorem of Varopoulos, Bishop, Davie-Øksendal are presented and sufficient criteria for being a (PI)-set obtained. Further the author deals with smooth sets in \(S\), putting in evidence the role of the complex tangentiality for \(C^1\)-curves or higher dimensional \(C^1\)-manifolds that are (PI)-sets. Then determining sets for \(A(B)\) (e.g. Pinchuk’s theorem) and peak-sets for \(A^\infty(B)\) are considered. Interesting examples, open problems, conjectures are indicated.
Chapter 11 studies the boundary behaviour of a function \(f\in H^\infty(B)\) on smooth curves \(\gamma\) in \(S\) and its dependence on the fact whether \(\gamma\) are complex-tangential or not (results of Nagel and the author). It contains also other properties of non complex tangential curves or higher dimensional manifolds, and connections for a complex regular Borel measure \(\mu\) on \(S\) between the existence of radial limits of \(f\) (1) pointwise a.e. \([\vert \mu\vert]\), (2) in the weak\(^*\)-topology of \(L^\infty(\vert \mu\vert)\)), and (3) the property of \(\mu\) to be a Henkin measure.
Chapters 12 and 13 consist mostly in joint results of Nagel and the author and of the author alone. Chapter 12 deals with a subject of harmonic analysis – the unitarily invariant function spaces – but Nagel-Rudin’s theorem, which characterizes the closed \(\mathcal U\)-invariant subspaces of \(X= C(S)\) or \(L^p(\sigma)\), \(1\le p<\infty\), will be essential in the classification of the \(\mathcal M\)-invariant closed subspaces of \(X\) in chapter 13. A description of the \(\mathcal U\)-invariant subalgebras of \(C(S)\) by means of the algebra patterns is also included in chapter 12 and applied to \(C_0(B)\) and \(C(\bar B)\) in chapter 13.
Chapter 14 gives an introduction to analytic varieties starting with the Weierstrass preparation theorem. By means of a projection theorem for analytic subvarieties the finiteness of every compact subvariety in \(\mathbb C^n\) is established. Hausdorff measures permit to derive topological properties of analytic varieties from size estimates.
Chapter 15 deals with proper holomorphic maps, starting with their basic covering properties in the case \(f: \Omega\rightarrow\Omega'\), \(\Omega\) a region in \(\mathbb C^n\) and \(\Omega'\) one in \(\mathbb C^k\) (surjectivity, \(n\le k\), multiplicity, regular and critical values, openness when \(\Omega' = \mathbb C^n\), invariance of subvarieties). Then facts specific to the case \(n>1\) are given: the nonexistence of proper holomorphic maps from polydiscs to balls and conversely, justified first by means of the local peak points and then by a theorem of Alexander, local theorems asserting that some holomorphic maps defined near points in \(S\) must extend to automorphisms of \(B\), and Alexander’s theorem that the only proper holomorphic maps \(B\rightarrow B\) are the automorphisms of \(B\). Domains having \(C^2\)-boundary and strictly pseudoconvex are studied and Rosay’s characterization of the ball by the transitivity of the group \(\operatorname{Aut}(B)\) is proved.
In chapter 16, the \(\bar\delta\)-problem (1) with compact support, (2) in convex domains, and (3) in \(B\) is considered successively. For the sake of completeness, the case \(n=1\) – the Cauchy-Pompeiu formula and Pompeiu integral – are also detailed, and then generalized for a bounded region \(\Omega\) in \(\mathbb C^n\) which has \(C^2\)-boundary. One uses first a universal kernel, e.g. the Bochner-Martinelli one, and further, assuming that \(\Omega\) is also convex, the holomorphic kernel of Koppelman (proof for the solution of the \(\bar\delta\)-problem according to Ovrelid). Discussions on \(\bar\delta\)-problems, an example of a globally unsolvable one, and finally the explicit solution in \(B\) conclude the chapter.
A slight modification of this solution will be used in chapter 17 to prove the Henkin-Skoda theorem, one of the most impressive results of the theory, characterizing the zero-varieties for the functions in \(N(B)\). Blaschke’s condition given in chapter 7 by the counting functions is reformulated now using Jensen’s formula or the existence of an \(\mathcal M\)-harmonic majorant in \(B\) for \(\log \vert f\vert\). By an approximation argument the author simplifies the proof of Henkin-Skoda’s theorem, working only with smooth plurisubharmonic functions. Blaschke’s condition further is viewed geometrically as a growth condition on the area of the zero-variety.
Chapter 18 is concerned with the tangential Cauchy-Riemann operators: extension theorems for solutions of such equations from the boundary to a domain (in particular from a spherical cap to its convex hull, but also a theorem of this type according to Hörmander including a classical result of Bochner, as well as a theorem in terms of Levi’s form); Lewy’s locally unsolvable equation and the similar phenomenon by the adjoints of the tangential Cauchy-Riemann operators, Audibert’s results on pluriharmonic functions.
This succinct description of the content of the book illustrates its richness and variety, the vast and recent bibliography used, the great systematization work carried on by its author. His specific clarity and dynamic style stimulate the reader. The mathematics lives in the book: main ideas of theorems and proofs, essential features of the subjects, lines of further developments, problems and conjectures are continuously underlined. The whole chapter 19 is devoted to the discussion of open problems. Numerous examples threw light on the results as well as on the difficulties.
There are several unifying principles in the book. One of them emphasizes the difference between the classical case \(n=1\) and the case of several variables, which gives rise to interesting comments and opens a large perspective on the field.
The book is accessible to a wide circle of readers requiring only advanced calculus, basic function theory of one complex variable, Lebesgue measure and integration and a little functional analysis. It is also extremely valuable to the specialist who finds in it a considerable amount of bibliographic material presented for the first time in a book which brings at the same time many contributions of the author. We think the book will strongly influence the further development of this area of complex analysis.

Reviewer: C. Andreian Cazacu

MSC:

32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32A10	Holomorphic functions of several complex variables
32A40	Boundary behavior of holomorphic functions of several complex variables
32A38	Algebras of holomorphic functions of several complex variables
32A22	Nevanlinna theory; growth estimates; other inequalities of several complex variables
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A35	\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32E35	Global boundary behavior of holomorphic functions of several complex variables
32U05	Plurisubharmonic functions and generalizations
32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators
32M05	Complex Lie groups, group actions on complex spaces
32H99	Holomorphic mappings and correspondences

Keywords:

unit ball; zero-varieties; algebra of holomorphic functions; uniqueness theorem; invariant kernels; invariant Laplacian; growth; boundary behaviour; Hp; Schwarz lemma; peak; interpolation; null set; Bochner- Martinelli kernel; delta equation; d-bar-equation; tangential Cauchy Riemann operators; harmonic functions; analytic varieties; PI-set