Hartogs theorem

2020 Mathematics Subject Classification: Primary: 32-XX [MSN][ZBL]

The term is used for different fundamental theorems in the theory of holomorphic functions of several complex variables, all proved by F. Hartogs. The term Hartogs' lemma is sometimes used also for a useful property of sequences of subarhominc functions.

Hartogs' theorem on separate analyticity

The following theorem goes under the name Hartogs' theorem on separate analyticity or Hartogs' fundamental theorem.

Theorem 1 If $U\subset \mathbb C^n$ is open and the function $f:D \to \mathbb C$ is holomorphic at every point $\zeta\in D$ with respect to each variable $z_k$, then it is holomorphic with respect to all variables.

In the theorem above holomorphic with respect to the variable $z_k$ means that, if $\zeta = (\zeta_1, \ldots, \zeta_n)$, then the map \[ z \mapsto f (\zeta_1, \ldots, \zeta_{k-1}, z, \zeta_{k+1}, \ldots, \zeta_n) \] is an holomorphic function of one complex variable in its domain of definition. Holomorphicity with respect to all variables means that $f$ is complex differentiable on $U$ (see Analytic function).

There exist many generalizations of Theorem 1 to include cases when some of the variables are real, when not all points of the domain $U$ are used or when some singularities of $f$ are permitted. For example:

Theorem 1a If $U := \{(z,w)\in \mathbb C^k\times \mathbb C^n: |z|< R_1, |w|< R_2\}$ and a function $f:U\to \mathbb C$ is holomorphic in the domain $\{|z|<r_1, |w|<R_2\}$ for some $r_1$ and is holomorphic in the ball $B_{R_1}$ with respect to the variables $z$ for any fized $w\in B_{R_2}$, then $f$ is holomorphic in $U$.

Theorem 1b If a function $f: \mathbb C^n \to \bar{\mathbb C}$ (where $\bar{\mathbb C}$ denotes the Riemann sphere $\mathbb C \cup \{\infty\}$) is rational with respect to each variable, then it is a rational function.

Hartogs' extension theorem

The term refers to another fundamental result in the theory of holomorphic functions of several complex variables, which indeed establishes a sharp contrast between the latter and the classical theorem of holomorphic function of one variable.

Theorem 2 If $U\subset \mathbb C^n$ is a connected open set, with $n\geq 2$, $K\subset U$ is a compact set which does disconnect $U$ and $f: U\setminus K \to \mathbb C$ is an holomorphic function, then $f$ can be extended holomorphically to the whole domain $U$.

The theorem is also known as Osgood-Brown lemma. In particular it implies that an holomorphic function of more than one complex variable cannot have an isolated singularity, in contrast with holomorphic functions of one complex variable (for instance $z\mapsto \frac{1}{z}$ is holomorphic on $\mathbb C \setminus \{0\}$ but cannot be extended to an holomorphic function of the whole complex plane).

Another version of Hartogs' extension theorem goes sometimes under the name of Hartogs' theorem on the continuous distribution of singular points and it states the following

Theorem 3 Assume $U\subset \mathbb C^n$ is an open set, $r$ a positive real number and $\{a_k\}\subset \mathbb C^{n-1}$ a sequence with the following properties:

• the disks $D_k := \{(z_1, a_k): |z_1|< r\}$ are all contained in $U$;
• $a_k \to a\in \mathbb C^{n-1}$ and the circle $S_\infty:= \{(z_1, a): |z_1|=r\}$ is contained in $U$.

Then $f$ can be extended holomorphically to a neighborhood of the disk $D_\infty:=\{(z_1, a): |z_1|<r\}$.

Hartogs' lemma on subharmonic functions

This lemma states that a sequence of subharmonic functions which is pointwise bounded is in fact locally uniformly bounded.

Theorem 4 If $U\subset \mathbb R^n$ and $f_k: U \to \mathbb R$ is a sequence of subharmonic functions such that \[ \limsup_{k\to\infty} f_k (x) \leq C < \infty\, , \] then for every compact set $K\subset U$ and for every $\varepsilon > 0$ there is a $k_0\in \mathbb N$ such that \[ f_k (x) \leq C + \varepsilon \qquad \forall k\geq k_0, \forall x\in K\, . \]

Hartogs' theorem on the analyticity of the singular set

This term is only used by some authors to refer to the results proved by Hartogs on the analyticity of the set of singularities of holomorphic functions of several complex variables in [Ha2]. An example is given by the following theorem (see also [BG], p. 684).

Theorem 5 Let $U := D_1 \times D_2 = \{|z|<r_1\}\times \{|w|<r_2\} \subset \mathbb C^2$, $\eta: D_1 \to D_2$ a continuous function and $f: U\setminus {\rm Gr}\, (\eta)$ an holomorphic function (where ${\rm Gr} (\eta) = \{((z, \eta(z)): z\in D_1\}$). Assume that each point $\eta (z)$ is a true singularity of the map $w\mapsto f_z (w):= f(z, w)$, namely that $f_z$ cannot be extended holomorphically to the whole $D_2$. Then $\eta$ is an holomorphic function.

References

[BM]	S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948) MR0027863 Zbl 0041.05205
[BG]	U. Bottazzini, G. Gray, "Hidden harmony - geometric fantasies", Springer Verlag, 2013.
[Fu]	B.A. Fuks, "Theory of analytic functions of several complex variables" , 1–2 , Amer. Math. Soc. (1963–1965) (Translated from Russian) MR0188477 MR0174786 MR0168793 MR0155003 MR0037915 MR0027069 Zbl 0146.30802 Zbl 0138.30902 Zbl 0040.19002
[Ha]	F. Hartogs, "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten" Math. Ann. , 62 (1906) pp. 1–88
[Ha2]	F. Hartogs, "Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde" Acta Math., 32 (1908) pp. 57-79
[Ho]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001
[Kr]	S.G. Krantz, "Function theory of several complex variables" , Wiley (Interscience) (1982) MR0635928 Zbl 0471.32008
[Sh]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001