##
**Pluripotential theory.**
*(English)*
Zbl 0742.31001

Although many books about functions of several complex variables contain some work on plurisubharmonic (PSH) functions, there has hitherto been only one book dealing with recent developments in the rapidly growing theory of such functions, U. Cegrell’s “Capacities in complex analysis” (Wiesbaden, 1988; Zbl 0655.32001). The present book is even more up-to-date and the author takes a point of view different from Cegrell’s. He points out that the relationship between PSH functions and pluriharmonic (PH) functions is not completely analogous to the relationship between subharmonic and harmonic functions. Most importantly, in a natural sense, maximal subharmonic functions are harmonic, but in domains of \(\mathbb{C}^ n\) \((n\geq 2)\) not all maximal PSH (MPSH) functions are PH. Thus, in some contexts at least, the proper generalizations of harmonic functions are not the PH functions but the MPSH functions, and the author therefore emphasises the role of MPSH functions. (In domains in \(\mathbb{C}\), harmonic \(\equiv\) PH, subharmonic \(\equiv\) PSH, and hence harmonic \(\equiv\) MPSH \(\equiv\) PH.) The part played by the Laplace operator in the classical theory is played by the Monge-Ampère operator in the theory of MPSH functions, although the latter operator must be understood in a distributional sense since there exist non-smooth, even discontinuous, MPSH functions.

The book is at an advanced level: some knowledge of measure theory, distributions, functional analysis and one-dimensional complex analysis is assumed, and previous exposure to holomorphic functions of several complex variables and classical potential theory would be very advantageous, although fundamental results in these areas are covered in the text.

Part I of the book consists of two introductory chapters. Chapter 1 on complex differentiation introduces important concepts in the differential calculus of several complex variables. The greater part of Chapter 2 develops fundamental results in the classical theory of harmonic and subharmonic functions in \(\mathbb{R}^ n\) \((n\geq 2)\) and includes applications to holomorphic functions on \(\mathbb{C}^ n\); at the end of Chapter 2, PSH functions are introduced.

The four chapters in Part II present the theory of MPSH functions. The result that a \(\mathbb{C}^ 2\) MPSH function satisfies the Monge-Ampère equation comes early in Chapter 3. Then there is a development of the technical machinery required for the introduction of the generalized Monge-Ampère operator which is needed in the study of non-smooth PSH functions. The rest of the chapter gives the theory of this operator and includes results on its continuity properties, a notion of capacity defined in terms of PSH functions, and the quasi-continuity of PSH functions.

In Chapter 4 the generalized Dirichlet problem for the Monge-Ampère operator on an open Euclidean ball in \(\mathbb{C}^ n\) is shown to have a unique solution in the case where the boundary function is finite-valued and continuous. This result is then used to complete the proof of the fundamental theorem that a locally bounded PSH function is maximal if and only if it satisfies the generalized Monge-Ampère equation. The chapter concludes with a detailed study of pluripolar and plurithin sets.

Chapter 5 is “Maximal functions of logarithmic growth”. The main object of study here is the \(L\)-extremal function \(V_ E\) of a set \(E\subset\mathbb{C}^ n\), which has important applications; it is defined on \(\mathbb{C}^ n\) by \(V_ E(z)=\sup\{u(z): u\in L\), \(u\leq 0\) on \(E\}\), where \(L\) is the class of PSH functions \(u\) on \(\mathbb{C}^ n\) satisfying \(u(z)- \log\| z\|\leq O(1)\) as \(\| z\|\to\infty\).

In Chapter 6, “Maximal functions with logarithmic singularities”, the pluricomplex Green function \(g_ \Omega(\cdot,a)\) of a domain \(\Omega\subset\mathbb{C}^ n\) with pole at \(a\in\Omega\) is defined on \(\Omega\) as the supremum of all negative PSH functions \(u\) on \(\Omega\) satisfying \(u(z)-\log\| z-a\|\leq O(1)\) as \(z\to a\). In the case where \(n=1\) and \(\Omega\) is hyperconvex, \(-g(\cdot,a)\) is the classical Green function. However, in domains in \(\mathbb{C}^ n\) \((n\geq 2)\) the pluricomplex Green function lacks some of the important properties, such as symmetry and differentiability, of the classical Green function. The pluricomplex Green function is used to obtain PSH versions of the Schwarz lemma and the Poisson-Jensen formula. This monograph will surely become a standard work both for mathematicians wishing to learn about pluripotential theory and for those who are already actively researching in this field.

The book is at an advanced level: some knowledge of measure theory, distributions, functional analysis and one-dimensional complex analysis is assumed, and previous exposure to holomorphic functions of several complex variables and classical potential theory would be very advantageous, although fundamental results in these areas are covered in the text.

Part I of the book consists of two introductory chapters. Chapter 1 on complex differentiation introduces important concepts in the differential calculus of several complex variables. The greater part of Chapter 2 develops fundamental results in the classical theory of harmonic and subharmonic functions in \(\mathbb{R}^ n\) \((n\geq 2)\) and includes applications to holomorphic functions on \(\mathbb{C}^ n\); at the end of Chapter 2, PSH functions are introduced.

The four chapters in Part II present the theory of MPSH functions. The result that a \(\mathbb{C}^ 2\) MPSH function satisfies the Monge-Ampère equation comes early in Chapter 3. Then there is a development of the technical machinery required for the introduction of the generalized Monge-Ampère operator which is needed in the study of non-smooth PSH functions. The rest of the chapter gives the theory of this operator and includes results on its continuity properties, a notion of capacity defined in terms of PSH functions, and the quasi-continuity of PSH functions.

In Chapter 4 the generalized Dirichlet problem for the Monge-Ampère operator on an open Euclidean ball in \(\mathbb{C}^ n\) is shown to have a unique solution in the case where the boundary function is finite-valued and continuous. This result is then used to complete the proof of the fundamental theorem that a locally bounded PSH function is maximal if and only if it satisfies the generalized Monge-Ampère equation. The chapter concludes with a detailed study of pluripolar and plurithin sets.

Chapter 5 is “Maximal functions of logarithmic growth”. The main object of study here is the \(L\)-extremal function \(V_ E\) of a set \(E\subset\mathbb{C}^ n\), which has important applications; it is defined on \(\mathbb{C}^ n\) by \(V_ E(z)=\sup\{u(z): u\in L\), \(u\leq 0\) on \(E\}\), where \(L\) is the class of PSH functions \(u\) on \(\mathbb{C}^ n\) satisfying \(u(z)- \log\| z\|\leq O(1)\) as \(\| z\|\to\infty\).

In Chapter 6, “Maximal functions with logarithmic singularities”, the pluricomplex Green function \(g_ \Omega(\cdot,a)\) of a domain \(\Omega\subset\mathbb{C}^ n\) with pole at \(a\in\Omega\) is defined on \(\Omega\) as the supremum of all negative PSH functions \(u\) on \(\Omega\) satisfying \(u(z)-\log\| z-a\|\leq O(1)\) as \(z\to a\). In the case where \(n=1\) and \(\Omega\) is hyperconvex, \(-g(\cdot,a)\) is the classical Green function. However, in domains in \(\mathbb{C}^ n\) \((n\geq 2)\) the pluricomplex Green function lacks some of the important properties, such as symmetry and differentiability, of the classical Green function. The pluricomplex Green function is used to obtain PSH versions of the Schwarz lemma and the Poisson-Jensen formula. This monograph will surely become a standard work both for mathematicians wishing to learn about pluripotential theory and for those who are already actively researching in this field.

Reviewer: D.H.Armitage (Belfast)

### MSC:

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

31C10 | Pluriharmonic and plurisubharmonic functions |

32U05 | Plurisubharmonic functions and generalizations |