# Duality in complex analysis

Let be a domain in and denote by the space of all functions holomorphic in with the topology of uniform convergence on compact subsets of (the projective limit topology). Let be a compact set in . Similarly, let be the space of all functions holomorphic on endowed with the following topology: A sequence converges to a function in if there exists a neighbourhood such that all the functions and converges to in (the inductive limit topology).

The description of the dual spaces and is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.

## Grothendieck–Köthe–Sebastião e Silva duality.

Let be a domain in the complex plane and let

Then one has the isomorphism (see [a1], [a2], [a3])

defined by

(a1) |

where . Here, , where for some domain ; and the curve separates the singularities of the functions and . The integral in (a1) does not depend on the choice of .

## Duality and linear convexity.

When , the complement of a bounded domain is not useful for function theory. Indeed, if , then . However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.

A domain is called linearly convex if for any there exists a complex hyperplane through that does not intersect . A compact set is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of is .

Some examples:

1) Let be convex; then for any point of the boundary there exists a hyperplane of support of dimension that contains the complex hyperplane .

2) Let , where , , are arbitrary plane domains.

Let be approximated from within by the sequence of linearly convex domains with smooth boundaries: , where , , and . Such an approximation does not always exist, unlike the case of usual convexity. For instance, this approximation is impossible in Example 2) if at least one of the domains is non-convex.

If , one has the isomorphism

where is the adjoint set (the generalized complement) defined by

(a2) |

Here, , ,

The index depends on the function , which is holomorphic on the larger compact set . The integral in (a4) does not depend on the choice of .

A similar duality is valid for the space as well (see [a4], [a5], [a6]).

A. Martineau has defined a strongly linearly convex domain to be a linearly convex domain for which the above-mentioned isomorphism holds. It is proved in [a7] that a domain is strongly linearly convex if and only if the intersection of with any complex line is connected and simply connected (see also [a8], [a9], [a10], [a11]).

## Duality based on regularized integration over the boundary.

L. Stout obtained the following result for bounded domains with the property that, for a fixed , the Szegö kernel is real-analytic in . Apparently, this is true if is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism

(a3) |

is defined by the formula

## Nacinovich–Shlapunov–Tarkhanov theorem.

Let be a bounded domain in with real-analytic boundary and with the property that any neighbourhood of contains a neighbourhood such that is dense in . This is always the case if is a strictly pseudo-convex domain with real-analytic boundary.

For any function there exists a unique solution of the Dirichlet problem

Here, the isomorphism (a3) can be defined by the formula (see [a14])

(a4) |

The integral is well-defined for some (where is a sequence of domains with smooth boundaries which approximate from within) since the function , which is harmonic in , can be harmonically continued into for some because of the real analyticity of and . The integral in (a4) does not depend on the choice of .

## Duality and cohomology.

Let be the Dolbeault cohomology space

where is the space of all -closed forms that are in on some neighbourhood of (the neighbourhood depends on the cocycle ) and is the subspace of of all -exact forms (coboundaries).

If is a bounded pseudo-convex domain, then one has [a15], [a16] an isomorphism

defined by the formula

(a5) |

where , . Here, for some one has ; is a sequence of domains with smooth boundaries approximating from within. Although depends on the choice of , the integral in (a5) does not depend on (given (a5), the same formula is valid for larger as well).

A new result [a17] consists of the following: Let be a bounded pseudo-convex domain in that can be approximated from within by a sequence of strictly pseudo-convex domains ; and let be the subspace of consisting of the differential forms of type

(a6) |

where is a function that is harmonic in some neighbourhood of (which depends on ) such that

Let be the space of all forms of type (a6) such that the harmonic function is representable for some by the Bochner–Martinelli-type integral (cf. also Bochner–Martinelli representation formula)

where and . Then one has an isomorphism

it is defined by the formula

(a7) |

where and . Note that (a7) gives a more concrete description of the duality than does (a5). The integral in (a7) is also independent of the choice of .

Other descriptions of the spaces dual to spaces of holomorphic functions for special classes of domains can be found in [a18], [a19], [a20], [a21], [a22], [a23], [a24], [a10].

#### References

[a1] | A. Grothendieck, "Sur certain espaces de fonctions holomorphes" J. Reine Angew. Math. , 192 (1953) pp. 35–64; 77–95 |

[a2] | G. Köthe, "Dualität in der Funktionentheorie" J. Reine Angew. Math. , 191 (1953) pp. 30–39 |

[a3] | J. Sebastião e Silva, "Analytic functions in functional analysis" Portug. Math. , 9 (1950) pp. 1–130 |

[a4] | A. Martineau, "Sur la topologies des espaces de fonctions holomorphes" Math. Ann. , 163 (1966) pp. 62–88 |

[a5] | L. Aizenberg, "The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in " Soviet Math. Dokl. , 7 (1966) pp. 198–202 |

[a6] | L. Aizenberg, "Linear convexity in and the distributions of the singularities of holomorphic functions" Bull. Acad. Polon. Sci. Ser. Math. Astr. Phiz. , 15 (1967) pp. 487–495 (In Russian) |

[a7] | S.V. Zhamenskij, "A geometric criterion of strong linear convexity" Funct. Anal. Appl. , 13 (1979) pp. 224–225 |

[a8] | M. Andersson, "Cauchy–Fantappié–Leray formulas with local sections and the inverse Fantappié transform" Bull. Soc. Math. France , 120 (1992) pp. 113–128 |

[a9] | S.G. Gindikin, G.M. Henkin, "Integral geometry for -cohomologies in -linearly concave domains in " Funct. Anal. Appl. , 12 (1978) pp. 6–23 |

[a10] | S.V. Znamenskij, "Strong linear convexity. I: Duality of spaces of holomorphic functions" Sib. Math. J. , 26 (1985) pp. 331–341 |

[a11] | S.V. Znamenskij, "Strong linear convexity. II: Existence of holomorphic solutions of linear systems of equations" Sib. Math. J. , 29 (1988) pp. 911–925 |

[a12] | L. Aizenberg, S.G. Gindikin, "The general form of a linear continuous functional in spaces of holomorphic functions" Moskov. Oblast. Ped. Just. Uchen. Zap. , 87 (1964) pp. 7–15 (In Russian) |

[a13] | E.L. Stout, "Harmonic duality, hyperfunctions and removable singularities" Izv. Akad. Nauk Ser. Mat. , 59 (1995) pp. 133–170 |

[a14] | M. Nacinovich, A. Shlapunov, N. Tarkhanov, "Duality in the spaces of solutions of elliptic systems" Ann. Scuola Norm. Sup. Pisa , 26 (1998) pp. 207–232 |

[a15] | J.P. Serre, "Une théorème de dualité" Comment. Math. Helvetici , 29 (1955) pp. 9–26 |

[a16] | A. Martineau, "Sur les fonctionelles analytiques et la transformation de Fourier–Borel" J. Anal. Math. , 9 (1963) pp. 1–164 |

[a17] | L. Aizenberg, "Duality in complex analysis" , Israel Math. Conf. Proc. , 11 (1997) pp. 27–35 |

[a18] | H.G. Tillman, "Randverteilungen analytischer funktionen und distributionen" Math. Z. , 59 (1953) pp. 61–83 |

[a19] | S. Rolewicz, "On spaces of holomorphic function" Studia Math. , 21 (1962) pp. 135–160 |

[a20] | L. Aizenberg, B.S. Mityagin, "The spaces of functions analytic in multi-circular domains" Sib. Mat. Zh. , 1 (1960) pp. 153–170 (In Russian) |

[a21] | L. Aizenberg, "The spaces of functions analytic in -circular domains" Soviet Math. Dokl. , 2 (1961) pp. 75–82 |

[a22] | L.J. Ronkin, "On general form of functionals in space of functions, analytic in semicircular domain" Soviet Math. Dokl. , 2 (1961) pp. 673–686 |

[a23] | S.G. Gindikin, "Analytic functions in tubular domains" Soviet Math. Dokl. , 3 (1962) |

[a24] | S.D. Simonzhenkov, "Description of the conjugate space of functions that are holomorphic in the domain of a special type" Sib. Math. J. , 22 (1981) |

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Duality in complex analysis.

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