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Carleman formulas

From Encyclopedia of Mathematics
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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
How to Cite This Entry:
Carleman formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_formulas&oldid=50276
This article was adapted from an original article by L. Aizenberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article