Duality in complex analysis

Let $D$ be a domain in ${\mathbf{C}}^n$ and denote by $A(D)$ the space of all functions holomorphic in $D$ with the topology of uniform convergence on compact subsets of $D$ (the projective limit topology). Let $K$ be a compact set in ${\mathbf{C}}^n$. Similarly, let $A(K)$ be the space of all functions holomorphic on $K$ endowed with the following topology: A sequence $\{f_m\}$ converges to a function $f$ in $A(K)$ if there exists a neighbourhood $U\supset K$ such that all the functions $f_m,f\in A(U)$ and $\{f_m\}$ converges to $f$ in $A(U)$ (the inductive limit topology).

The description of the dual spaces $A(D{)}^{\ast }$ and $A(K{)}^{\ast }$ is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.

Grothendieck–Köthe–Sebastião e Silva duality.[edit]

Let $D$ be a domain in the complex plane ${\mathbf{C}}^1$ and let

$$A_0(\bar{\mathbf{C}}\mathrm{\setminus }D)=\{f:f\in A(\bar{\mathbf{C}}\mathrm{\setminus }D),f(\mathrm{\infty })=0\}.$$

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

$$A(D{)}^{\ast }\simeq A_0(\bar{\mathbf{C}}\mathrm{\setminus }D),$$

defined by

$$\begin{array}{}\text{(a1)}& F(f)=F_{\varphi }(f)={\int }_{\mathrm{\Gamma }}f(z)\varphi (z)dz,\end{array}$$

where $\varphi \in A_0(\bar{\mathbf{C}}\mathrm{\setminus }D)$. Here, $\varphi \in A_0(Q)$, where $\bar{\mathbf{C}}\mathrm{\setminus }D\subset Q$ for some domain $Q$; and the curve $\mathrm{\Gamma }\subset D\cap Q$ separates the singularities of the functions $f\in A(D)$ and $\varphi$. The integral in (a1) does not depend on the choice of $\mathrm{\Gamma }$.

Duality and linear convexity.[edit]

When $n>1$, the complement of a bounded domain $D\subset {\mathbf{C}}^x$ is not useful for function theory. Indeed, if $f\in A_0({\bar{\mathbf{C}}}^n\mathrm{\setminus }D)$, then $f\equiv 0$. However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.

A domain $D\subset {\mathbf{C}}^x$ is called linearly convex if for any $\zeta \in \partial D$ there exists a complex hyperplane through $\zeta$ that does not intersect $D$. A compact set $K\subset {\mathbf{C}}^n$ is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of $\alpha$ is $2n-2$.

Some examples:

1) Let $D$ be convex; then for any point $\zeta$ of the boundary $\partial D$ there exists a hyperplane of support $\beta$ of dimension $2n-1$ that contains the complex hyperplane $\alpha$.

2) Let $D=D_1\times \dots \times D_n$, where $D_j\subset {\mathbf{C}}^1$, $j=1,\dots ,n$, are arbitrary plane domains.

Let $D$ be approximated from within by the sequence of linearly convex domains $\{D_m\}$ with smooth boundaries: ${\bar{D}}_m\subset D_{m+1}\subset D$, where $D_m=\{z:{\mathrm{\Phi }}^m(z,\overline{z})<0\}$, ${\mathrm{\Phi }}^m\in C^2({\bar{D}}_m)$, and $\mathrm{grad}{\mathrm{\Phi }}^m{|}_{\partial D_m}\ne 0$. 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 $D_j$ is non-convex.

If $0\in D$, one has the isomorphism

$$A(D{)}^{\ast }\simeq A(\tilde{D}),$$

where $\tilde{D}=\{w:w_1{z}_1+\dots +w_n{z}_n\ne 1,z\in D\}$ is the adjoint set (the generalized complement) defined by

$$\begin{array}{}\text{(a2)}& F(t)=F_{\varphi }(f)={\int }_{\partial D_m}f(z)\varphi (w)\omega (z,w).\end{array}$$

Here, $f\in A(D)$, $\varphi \in A(\tilde{D})$,

$$w_j=\frac{{\mathrm{\Phi }}^{\prime z_j}}{⟨{\mathrm{grad}}_z\mathrm{\Phi },z⟩},j=1,\dots ,n,$$

$$dz=d{z}_1\bigwedge \dots \bigwedge d{z}_n,⟨a,b⟩=a_1{b}_1+\dots +a_n{b}_n.$$

The index $m$ depends on the function $\varphi$, which is holomorphic on the larger compact set ${\tilde{D}}_m\supset \tilde{D}$. The integral in (a4) does not depend on the choice of $m$.

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

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

Duality based on regularized integration over the boundary.[edit]

L. Stout obtained the following result for bounded domains $D\subset {\mathbf{C}}^x$ with the property that, for a fixed $z\in D$, the Szegö kernel $K(z,\zeta )$ is real-analytic in $\zeta \in \partial D$. Apparently, this is true if $D$ is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism

$$\begin{array}{}\text{(a3)}& A(D{)}^{\ast }\simeq A(\bar{D})\end{array}$$

is defined by the formula

$$F(f)=F_{\varphi }(f)={\lim }_{ϵ\to 0}{\int }_{\partial D_{ϵ}}f(z)\overline{\varphi (z)}d\sigma ,$$

where $D_{ϵ}=\{z:z\in D,\rho (z,\partial D)>ϵ\}$, $f\in A(D)$, $\varphi \in A(\bar{D})$ (see [a12], [a13]).

Nacinovich–Shlapunov–Tarkhanov theorem.[edit]

Let $D$ be a bounded domain in ${\mathbf{C}}^n$ with real-analytic boundary and with the property that any neighbourhood $U$ of $\bar{D}$ contains a neighbourhood $U^{\prime }\subset U$ such that $A(U^{\prime })$ is dense in $A(D)$. This is always the case if $D$ is a strictly pseudo-convex domain with real-analytic boundary.

For any function $\varphi \in A(\bar{D})$ there exists a unique solution of the Dirichlet problem

$$\{\begin{array}{ll}\mathrm{\Delta }v=0& \text{in}{\mathbf{C}}^n\setminus \bar{D},\\ v=\varphi & \text{ on }\partial D,\\ |v|\le \frac{c}{|z{|}^{2n-2}}.\end{array}$$

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

$$\begin{array}{}\text{(a4)}& F(f)=F_{\varphi }(f)={\int }_{\partial D_m}f(z)\sum _{k=1}^n(-1{)}^{k-1}\frac{\partial \bar{v}}{\partial z_k}d\bar{z}[k]\bigwedge dz.\end{array}$$

The integral is well-defined for some $m$ (where $\{D_m\}$ is a sequence of domains with smooth boundaries which approximate $D$ from within) since the function $v$, which is harmonic in ${\mathbf{C}}^n\mathrm{\setminus }\bar{D}$, can be harmonically continued into for some $m$ because of the real analyticity of $\partial D$ and $\varphi {|}_{\partial D}$. The integral in (a4) does not depend on the choice of $m$.

Duality and cohomology.[edit]

Let $H^{n,n-1}({\mathbf{C}}^n\mathrm{\setminus }D)$ be the Dolbeault cohomology space

$$H^{n,n-1}=Z^{n,n-1}/B^{n,n-1},$$

where $Z^{n,n-1}$ is the space of all $\bar{\partial }$-closed forms $\alpha$ that are in $C^{\mathrm{\infty }}$ on some neighbourhood $U$ of ${\mathbf{C}}^n\mathrm{\setminus }D$ (the neighbourhood depends on the cocycle $\alpha$) and $B^{n,n-1}$ is the subspace of $Z^{n,n-1}$ of all $\bar{\partial }$-exact forms $\alpha$ (coboundaries).

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

$$A(D{)}^{\ast }\simeq H^{n,n-1}({\mathbf{C}}^n\mathrm{\setminus }D),$$

defined by the formula

$$\begin{array}{}\text{(a5)}& F(f)=F_g(f)={\int }_{\partial D_m}fg,\end{array}$$

where $f\in A(D)$, $g\in H^{n,n-1}({\mathbf{C}}^n\mathrm{\setminus }D)$. Here, for some $U\supset {\mathbf{C}}^n\mathrm{\setminus }D$ one has $g\in H^{n,n-1}(U)$; $\{D_m\}$ is a sequence of domains with smooth boundaries approximating $D$ from within. Although $m$ depends on the choice of $U$, the integral in (a5) does not depend on $m$ (given (a5), the same formula is valid for larger $m$ as well).

A new result [a17] consists of the following: Let $D$ be a bounded pseudo-convex domain in ${\mathbf{C}}^n$ that can be approximated from within by a sequence of strictly pseudo-convex domains $\{D_m\}$; and let $A$ be the subspace of $Z^{n,n-1}$ consisting of the differential forms of type

$$\begin{array}{}\text{(a6)}& g_u=\sum _{k=1}^n(-1{)}^{k-1}\frac{\partial u}{\partial z_k}d\overline{z}[k]\wedge dz,\end{array}$$

where $u$ is a function that is harmonic in some neighbourhood $U$ of ${\mathbf{C}}^n\mathrm{\setminus }D$ (which depends on $u$) such that

$$|u(z)|\le \frac{C}{|z{|}^{2n-2}}.$$

Let $B$ be the space of all forms of type (a6) such that the harmonic function $\bar{u}$ is representable for some $m$ by the Bochner–Martinelli-type integral (cf. also Bochner–Martinelli representation formula)

$$\bar{u}(z)={\int }_{\partial D_m}w(\zeta )\frac{\sum _{k=1}^n(-1{)}^{k-1}({\bar{\zeta }}_k-{\bar{z}}_k)d\bar{\zeta }[k]\wedge d\zeta }{|\zeta -z{|}^{2n}},$$

where $z\in {\mathbf{C}}^n\mathrm{\setminus }{\bar{D}}_m$ and $\omega (\zeta )\in C(\partial D_m)$. Then one has an isomorphism

$$A(D{)}^{\ast }\simeq A/B;$$

it is defined by the formula

(a7)

where $f\in A(D)$ and $g_u\in A/B$. 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 $m$.

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[edit]