Holomorphy, criteria for

criteria for analyticity

The natural criteria for holomorphy (analyticity) of a $C ^ { 1 }$ (or continuous) function $f$ in a domain $\Omega$ of the complex plane are "infinitesimal" (cf. Analytic function), namely: power series expansions, the Cauchy-Riemann equations, and even the Morera theorem, since it states that

\begin{equation*} \int _ { \Gamma } f ( z ) d z = 0 \end{equation*}

for all Jordan curves $\Gamma$ such that $\Gamma \cup \text { int } ( \Gamma ) \subset \Omega$, is a necessary and sufficient condition for $f$ being analytic in $\Omega$. The condition (and the usual proofs) depend on the fact that $\Gamma$ can be taken to be arbitrarily small.

The first "non-infinitesimal" condition is due to M. Agranovsky and R.E. Val'skii (see [a2] and [a6] for all relevant references): Let $\gamma$ be a piecewise smooth Jordan curve, then a function $f$ continuous in $\mathbf{C}$ is entire (analytic everywhere) if and only if for every transformation $\sigma \in M ( 2 )$ and $k = 0,1,2 , \dots$ it satisfies

\begin{equation*} \int _ { \sigma ( \gamma ) } f ( z ) d z = 0. \end{equation*}

(Recall that $\sigma \in M ( 2 )$ means that $\sigma ( z ) = e ^ { i \theta } z + a$, $\theta \in \mathbf{R}$, $a \in \mathbf{C}$.)

A generalization of this theorem and of Morera's theorem which is both local and non-infinitesimal is the following Berenstein–Gay theorem [a3].

Let $\Gamma$ be a Jordan polygon contained in $B ( 0 , r / 2 )$ and $f \in C ( B ( 0 , r ) )$; then $f$ is analytic in $B ( 0 , r )$ if and only if for any $\sigma \in M ( 2 )$ such that $\sigma ( \Gamma ) \subseteq B ( 0 , r )$,

\begin{equation*} \int _ { \sigma ( \Gamma ) } f ( z ) d z = 0. \end{equation*}

This theorem can be extended to several complex variables and other geometries (see [a2], [a5], and [a6] for references).

A different kind of conditions for holomorphy occur when one considers the problem of extending a continuous function defined on a curve in $\mathbf{C}$ (or in a real $( 2 n - 1 )$-manifold in $\mathbf{C} ^ { n }$) to an analytic function defined in a domain that contains the curve (or hypersurface) on its boundary. This is sometimes called a CR extension. An example of this type, generalizing the moment conditions of the Berenstein–Gay theorem, appears in the work of L. Aizenberg and collaborators (cf. also Analytic continuation into a domain of a function given on part of the boundary; Carleman formulas): Let $D$ be a subdomain of $B ( 0,1 ) \subseteq \mathbf C$, bounded by an arc of the unit circle and a smooth simple curve $\Gamma \subseteq B ( 0,1 )$ and assume that $f \in C ( \Gamma )$. Then there is a function $F$, holomorphic inside $D$ and continuous on its closure, such that $F | _ { \Gamma } = f$ if and only if

\begin{equation*} \operatorname { limsup } _ { k \rightarrow \infty } \left| \int _ { \Gamma } \frac { f ( \xi ) } { \xi ^ { k + 1 } } d \xi \right| ^ { 1 / k } \leq 1. \end{equation*}

A boundary version of the Berenstein–Gay theorem can be proven when regarding the Heisenberg group $H ^ { n }$ (cf. also Nil manifold) as the boundary of the Siegel upper half-space

\begin{equation*} S _ { n + 1 } = \left\{ z \in \mathbf{C} ^ { n + 1 } : \operatorname { Im } z _ { n + 1 } > \sum ^ { n _ { j = 1 } } | z _ { j } | ^ { 2 } \right\}, \end{equation*}

but the boundary values are restricted to be in $L ^ { p } ( H ^ { n } )$, $1 \leq p \leq \infty$, [a1]. Related analytic extension theorems from continuous boundary values have been proven by J. Globevnik and E.L. Stout, E. Grinberg, W. Rudin, and others (see [a2] for references) in the bounded "version" of $H ^ { n }$, namely the unit ball $B$ of $\mathbf{C} ^ { n + 1}$, or, more generally, for bounded domains, by essentially considering extensions from the boundary to complex subspaces. An example is the following Globevnik–Stout theorem, [a4].

Let $\Omega$ be a bounded domain in $\mathbf{C} ^ { n + 1}$ with $C ^ { 2 }$ boundary. Let $1 \leq k \leq n$ and assume $f \in C ( \partial \Omega )$ is such that

\begin{equation*} \int _ { \Lambda \bigcap \partial \Omega} f \beta = 0 \end{equation*}

for all complex $k$-planes intersecting $\partial \Omega$ transversally, and all $( k , k - 1 )$-forms $\beta$ with constant coefficients. Then $f$ is a $C R$-function, i.e. has an extension as an analytic function to $\Omega$.

References

[a1]	M. Agranovsky, C. Berenstein, D.C. Chang, "Morera theorem for holomorphic $H _ { p }$ functions in the Heisenberg group" J. Reine Angew. Math. , 443 (1993) pp. 49–89
[a2]	C. Berenstein, D.C. Chang, D. Pascuas, L. Zalcman, "Variations on the theorem of Morera" Contemp. Math. , 137 (1992) pp. 63–78
[a3]	C. Berenstein, R. Gay, "Le probléme de Pompeiu local" J. Anal. Math. , 52 (1988) pp. 133–166
[a4]	J. Globevnik, E.L. Stout, "Boundary Morera theorems for holomorphic functions of several complex variables" Duke Math. J. , 64 (1991) pp. 571–615
[a5]	L. Zalcman, "Offbeat integral geometry" Amer. Math. Monthly , 87 (1980) pp. 161–175
[a6]	L. Zalcman, "A bibliographic survey of the Pompeiu problem" B. Fuglede (ed.) et al. (ed.) , Approximation by Solutions of Partial Differential equations , Kluwer Acad. Publ. (1992) pp. 185–194