Carleman formulas

Let $D$ be a bounded domain in $\mathbf{C} ^ { n }$ with piecewise smooth boundary $\partial D$, and let $M$ be a set of positive $( 2 n - 1 )$-dimensional Lebesgue measure in $\partial D$.

The following boundary value problem can then be posed (cf. also Boundary value problems of analytic function theory): Given a holomorphic function $f$ in $D$ that is sufficiently well-behaved up to the boundary $\partial D$ (for example, $f$ is continuous in $\overline{ D }$, $( f \in H _ { c } ( D ) )$, or $f$ belongs to the Hardy class $H ^ { 1 } ( D )$) how can it be reconstructed inside $D$ by its values on $M$ by means of an integral formula?

Three methods of solution are known, due to:

1) Carleman–Goluzin–Krylov;

2) M.M. Lavrent'ev; and

3) A.M. Kytmanov. See [a1].

The following are some very simple solutions:

a) $n = 1$. If $M = \Gamma$ is a smooth arc connecting two points of the unit circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and lying inside $\gamma$ and $D$ is the domain bounded by a part of $\gamma$ and the arc $\Gamma$, with $0 \notin \overline { D }$, then for $z \in D$ and $f \in H ^ { 1 } ( D )$ the following Carleman formula holds:

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \frac { 1 } { 2 \pi i } \int _ { \Gamma } f ( \zeta ) \left( \frac { z } { \zeta } \right) ^ { m } \frac { d \zeta } { \zeta - z }. \end{equation*}

b) $n > 1$. Let $\Omega$ be a circular convex bounded domain (a Cartan domain) with $C ^ { 2 }$-boundary and let $\Gamma$ be a piecewise smooth hypersurface intersecting $\Omega$ and cutting from it the domain $D$, with $0 \notin \overline { D }$. Then there exists a Cauchy–Fantappié formula for the domain $D$ with kernel holomorphic in $z$. Let $\Omega = \{ \zeta : \rho ( \zeta ) < 0 \}$, $\rho \in C ^ { 2 } ( \overline { \Omega } )$, and $\gamma = ( \partial D ) \backslash \Gamma$. Assume that there exists a vector-valued function (a "barrier" ) $w = w ( z , \zeta )$, $z \in D$, $\zeta \in \Gamma$, such that $\langle w , \zeta - z \rangle \neq 0$, $w \in C _ { \zeta } ^ { 1 } ( \Gamma )$, and $w$ smoothly extends to $\rho ^ { \prime }$ on $\gamma \cap \Gamma$, where $\rho ^ { \prime } = \operatorname { grad } \rho = ( \partial \rho / \partial \zeta _ { 1 } , \dots , \partial \rho / \partial \zeta _ { n } )$. Then for every function $f \in H _ { c } ( D )$ and $z \in D$, the following Carleman formula with holomorphic kernel is valid (see [a2]):

\begin{equation*} f ( z ) = \end{equation*}

\begin{equation*} = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \left[ \operatorname{CF} ( \zeta - z , w ) - \sum _ { k = 0 } ^ { m } \frac { ( k + n - 1 ) } { k ! } \phi _ { k } \right]; \end{equation*}

here, $\operatorname {CF}$ is the Cauchy–Fantappié differential form (see [a3])

\begin{equation*} \operatorname{CF} ( \zeta - z , w ) = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } w _ { k } d w [ k ] \wedge d \zeta } { \langle w , \zeta - z \rangle ^ { n } }, \end{equation*}

where $d w [ k ] = d w _ { 1 } \wedge \ldots \wedge d w _ { k - 1 } \wedge d w _ { k + 1 } \wedge \ldots \wedge d w _ { n }$, $d \zeta = d \zeta _ { 1 } \wedge \ldots \wedge d \zeta _ { n }$, $\langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }$,

\begin{equation*} \phi _ { k } = \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \left\langle \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } ( \zeta ) , \zeta \rangle } , z \right\rangle ^ { k } \sigma, \end{equation*}

\begin{equation*} \sigma = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \rho ^ { \prime } d \rho ^ { \prime } [ j ] \bigwedge d\zeta . \end{equation*}

c) Now, let $\Omega$ be an $n$-circular domain (a Reinhardt domain); then

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) [ \operatorname{CF} ( \zeta - z , w ) + \end{equation*}

\begin{equation*} \left. - \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \sum _ { | \alpha | = 0 } ^ { m } \frac { ( | \alpha | + n - 1 ) ! } { \alpha _ { 1 } ! \ldots \alpha _ { n } ! } \left( \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } , \zeta \rangle } \right) ^ { \alpha } z ^ { \alpha } \sigma \right], \end{equation*}

where $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.

If $\Omega = \{ z : | z | < r \}$ is a ball, then

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \times \end{equation*}

\begin{equation*} \times \left[ \operatorname {CF} ( \zeta - z , w ) - \frac { ( n - 1 ) ! ( | \zeta | ^ { 2 m } - \langle \overline { \zeta } , z \rangle ^ { m } ) ^ { n } } { [ 2 \pi i | \zeta | ^ { 2 m } \langle \overline { \zeta } , \zeta - z \rangle ] ^ { n } } \sigma _ { 0 } \right], \end{equation*}

where

\begin{equation*} \sigma _ { 0 } = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \overline { \zeta } _{j} d \overline { \zeta } [ j ] \bigwedge d \zeta . \end{equation*}

In all the above Carleman formulas the limits are understood in the sense of uniform convergence on compact subsets of $D$. A description of the class of holomorphic functions representable by Carleman formulas is given in [a4]. In [a1] applications of Carleman formulas in analysis and in mathematical physics can be found as well.

References

[a1]	L. Aizenberg, "Carleman's formulas in complex analysis" , Kluwer Acad. Publ. (1993)
[a2]	L. Aizenberg, "Carleman's formulas and conditions of analytic extendability" , Topics in Complex Analysis , Banach Centre Publ. , 31 , Banach Centre (1995) pp. 27–34
[a3]	L. Aizenberg, A.P. Yuzhakov, "Integral representation and residues in multidimensional complex analysis" , Amer. Math. Soc. (1983) (In Russian)
[a4]	L. Aizenberg, A. Tumanov, A. Vidras, "The class of holomorphic functions representable by Carleman formula" Ann. Scuola Norm. Pisa , 27 : 1 (1998) pp. 93–105