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.
Contents
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 ![]() |
[a6] | L. Aizenberg, "Linear convexity in ![]() |
[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 ![]() ![]() ![]() |
[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 ![]() |
[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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