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On Grothendieck-type duality for spaces of holomorphic functions of several variables Yu. A. Khoryakova, A. A. Shlapunov Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, Russia
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   § 1. IntroductionOne of the first types of duality in spaces of holomorphic functions was independently discovered in the 1950s by Grothendieck [1], Köthe [2] and Sebastião e Silva [3], who described the strong dual space $({\mathcal O} (D))^*$ of the space of holomorphic functions ${\mathcal O} (D)$ (with the standard Fréchet topology) in a bounded simply connected domain $D\subset{\mathbb C}$:
$$
\begin{equation}
({\mathcal O} (D))^* \cong{\mathcal O} (\widehat{\mathbb C} \setminus D),
\end{equation}
\tag{1.1}
$$
where ${\mathcal O} (\widehat{\mathbb C} \setminus D)$ is the space of holomorphic functions in a neighbourhood of the closed set ${\mathbb C} \setminus D$ that vanish at infinity, which is endowed with the standard topology of an injective limit. Unfortunately, Hartogs’s theorem on the removal of compact singularities and Liouville’s classical theorem prohibit the direct analogue of this result from holding for holomorphic functions of several variables (although some generalizations in this direction are known for linearly convex multidimensional domains: see [4] and [5]). Other types of duality were established by Aizenberg and Gindikin [6], Aizenberg and Mityagin [7], and Stout [8]:
$$
\begin{equation}
({\mathcal O} (D) )^*\cong{\mathcal O} (\overline D)
\end{equation}
\tag{1.2}
$$
for (pseudoconvex for $n>1$) bounded domains $D \subset{\mathbb C}^n$ with real analytic boundaries; different duality relations were used in different cases (namely, ones induced by the inner products in Bergmann and Hardy spaces, respectively; see also [9] and [10] for other duality relations). Many generalizations of the duality relations mentioned above are known in the theory of Dolbeault cohomology [11]–[13] and in the theory of elliptic systems of partial differential equations; see [9] and [14]–[19]. One of the most general results describing duality in spaces of solutions of elliptic systems with the topology of local uniform convergence is due to Grothendieck: see [15], Theorems 3 and 4; in a certain sense this description is quite analogous to (1.1). Unfortunately, a full description exists only for operators that have a regular two-sided fundamental solution (see [15], Theorem 4), which cannot be applied to such overdetermined elliptic systems as the multidimensional Cauchy–Riemann operator. On the other hand Theorem 3 in [15] can still be applied to spaces of solutions of overdetermined elliptic operators that have a regular left fundamental solution: it gives an answer in terms of spaces of solutions of the relevant formally adjoint underdetermined operators. Spaces of this type are too large to fit for a topological isomorphism, that is, this theorem only ensures the existence of a surjective linear map, and to construct an isomorphism we must take a further quotient, which reduces significantly the effectiveness of the whole scheme of establishing duality. Another general scheme for establishing duality for (determined and overdetermined alike) elliptic systems was proposed in [19]. It uses the concept of Hilbert spaces with reproducing kernels, and the corresponding duality relations are closely connected with the inner products in these Hilbert spaces. However, whether this scheme can be used depends on fairly delicate information about the properties of reproducing kernels, which is not always available. To state the central result in our paper let $D$ denote a bounded domain with Lipschitz boundary in ${\mathbb R}^{2n}$, $n>1$, and let ${\mathcal H} (\widehat{\mathbb R}^{2n} \setminus D)$ be the space of harmonic complex-valued functions on the closed set ${\mathbb R}^{2n} \setminus D$ which vanish at infinity, that is, functions such that We endow this space with the standard topology of an inductive limit for harmonic functions on closed subsets. To describe this space we need the Cauchy–Riemann tangential conditions on $\partial D$: for instance, see [20]. More precisely, now let $\Sigma ( \widehat{\mathbb C}^n \setminus D)$ denote the closed subspace of ${\mathcal H} (\widehat{\mathbb R}^{2n} \setminus D)$ consisting of functions $v\in{\mathcal H} (\widehat{\mathbb R}^{2n} \setminus D)$ satisfying the Cauchy–Riemann tangential conditions on $\partial D$. We present alternative descriptions of $\Sigma ( \widehat{\mathbb C}^n \setminus D)$ in Corollary 1 below.Theorem 1. Let $n> 1$, and let $D$ be a bounded domain in ${\mathbb C}^n$ with connected complement and Lipschitz boundary such that the space $H^1 (D)\cap{\mathcal O} (D) $ of holomorphic functions in $D$ belonging to the Sobolev class $H^1 (D)$ is dense in ${\mathcal O} (D) $. Then (topologically) We describe the duality relation for (1.4) in § 3; see (3.1). Its idea was inspired by Aizenberg’s paper [13], where he refined Serre’s duality (see [11]). Also note that for bounded domains $D \Subset{\mathbb C}^n$ with real analytic boundaries such that the space ${\mathcal O}(\overline D)$ is dense in ${\mathcal O}(D)$ Theorem 1 can be extracted from the results in [9], where the isomorphism (1.2) was established by means of a certain quite special duality relation. Of course, as $\mathcal O(\overline D) \subset H^1 (D)\cap{\mathcal O} (D)$, it follows from the well-known Oka–Weil theorem that the approximation property assumed in Theorem 1 holds for strictly pseudoconvex domain in ${\mathbb C}^n$, $n>1$; for instance, see [21], Ch. 1, §§ F and G. We show in § 4 that this approximation property is also necessary for duality in Theorem 1 to hold. Note also that for the original duality (1.1) of spaces of holomorphic functions of one variable no restrictions on the smoothness of the curve $\partial D$ are needed. Finally, it is worth noting that Theorem 1 can apparently be generalized to a wide class of overdetermined elliptic operators with regular left fundamental solutions which induce nontrivial tangential conditions on hypersurfaces. § 2. Preliminary factsLet ${\mathbb R}^n$, $n\geqslant 2$, denote Euclidean space with coordinates $x=(x_1,x_2, \dots, x_n)$, and let ${\mathbb C}^n \cong{\mathbb R}^{2n}$ be the $n$-dimensional complex space with coordinates $z=(z_1,z_2, \dots, z_n)$, $z_j=x_j+\iota x_{n+j}$, where $\iota$ is the imaginary unit. Also let $D$ denote a bounded domain (an open connected set) in ${\mathbb C}^n$ with Lipschitz boundary $\partial D$. We consider complex-valued functions on subsets of ${\mathbb C}^n$. As usual, for $s\in{\mathbb Z}_+$ we use the notation $C^s (D)$ and $C^s (\overline D)$ for spaces of $s$ times continuously differentiable functions on $D$ and $\overline D$, respectively. We denote the Lebesgue and Sobolev Hilbert spaces on $D$ by $L^2 (D)$ and $H^s (D)$, respectively. Also let $H^{s} (\partial D)$, $0<s<1$, denote the standard Sobolev–Slobodetskii spaces on $\partial D$. Given an open set $U$ in ${\mathbb R}^n$, we let ${\mathcal H} (U)$ denote the space of harmonic functions on $U$ with the topology of local uniform convergence on $U$. It is known that ${\mathcal H} (U)$ is a topological vector Fréchet space ($F$-space): see [22], Ch. II, § 4. Moreover, standard a priori estimates for solutions of elliptic equations (for instance, see [23]) mean that ${\mathcal H} (U) \subset C^\infty (U)$ is in fact a closed subspace of $C (U)$, so that the topology in this space can be defined by the system of seminorms $p_\nu (u)=\max_{x\in K_\nu} | u (x)|$ associated with an arbitrary increasing sequence of compact sets $\{K_{\nu} \} \subset U$ such that $\bigcup_{\nu} K_\nu=U$, or by the system of seminorms
$$
\begin{equation*}
p^{(\alpha)}_\nu (u)= \max_{z\in K_\nu}|\partial ^\alpha u (z)|, \qquad \alpha\in{\mathbb Z}^{n}_+,
\end{equation*}
\notag
$$
where we assume in addition that each set $K_\nu$ is the closure of an open subset of $U$. As each holomorphic function on an open set $U\subset {\mathbb C}^n$ is harmonic, we regard the space ${\mathcal O} (U)$ of holomorphic functions on $U$ as a closed subspace of ${\mathcal H} (U)$. Now, given a closed set $\sigma \subset{\mathbb R}^n$, we denote by ${\mathcal H} ( \sigma)$ the set of functions harmonic in various neighbourhoods of $\sigma$ depending on the function. In fact, ${\mathcal H} (\sigma)$ can be regarded as the space of equivalence classes of harmonic functions on $\sigma$; here two functions are equivalent if there exists a neighbourhood of $\sigma$ in which they coincide. We say that a sequence $\{u_{\nu} \}$ converges in ${\mathcal H} (\sigma)$ if there exists a neighbourhood $\mathcal V$ of $\sigma$ in which all of these functions are defined and converge locally uniformly in $\mathcal V$. Alternatively, we can define the topological space ${\mathcal H} (\sigma)$ as the inductive limit of the spaces ${\mathcal H} (U_{\nu})$, where $\{U_{\nu} \}$ is a decreasing sequence of open sets containing $\sigma$ such that each neighbourhood of $\sigma$ contains some $U_{\nu}$ and each connected component of every $U_{\nu}$ intersects $\sigma$. Thus, the map ${\mathcal H} (U_{\nu}) \to{\mathcal H} (\sigma)$ is bijective. Then ${\mathcal H} (\sigma)$ is a Hausdorff space, usually also called a $\mathrm{DF}$-space; see [22], Ch. II, § 6. Given a bounded domain $D \subset{\mathbb R}^{2n}$, we introduce the space ${\mathcal H} (\widehat{\mathbb R}^{2n} \setminus D)$, $n>1$, as a closed subspace of ${\mathcal H} ({\mathbb R}^{2n} \setminus D)$ whose elements satisfy (1.3). Next recall that a function $w_0\in L^1 (\partial D)$ is called a $\mathrm{CR}$-function on the hypersurface $\partial D$ if its satisfies the tangential Cauchy–Riemann conditions on $\partial D$, that is, for all $(n,n-2)$-differential forms $\psi$ with coefficients in the class $C^1 (\overline D)$ (for instance, see [20] or [24], Ch. 2, § 6); of course, for sufficiently smooth functions on a smooth surface the $\mathrm{CR}$-conditions can be verified pointwise. Then the space $\Sigma (\widehat{\mathbb C}^n \setminus D) $, defined above as the set of elements $v\in{\mathcal H} (\widehat{\mathbb R}^{2n} \setminus D)$ satisfying the tangential Cauchy–Riemann conditions on $\partial D$ is a closed subspace of ${\mathcal H} (\widehat{\mathbb C}^n \setminus D)$ because a sequence convergent in ${\mathcal H} (\widehat{\mathbb C}^n \setminus D)$ is also locally convergent on the compact set $\partial D \subset{\mathbb C}^n$.Let us characterize the spaces $\Sigma (\widehat{\mathbb C}^n \setminus D) $ for domains with Lipschitz boundaries. To do this we let $\Delta_n $ denote the standard Laplace operator $\sum_{j=1}^n {\partial^2}/{\partial x _j^2} $ in ${\mathbb R}^n$. It is well known to have a fundamental solution $\Phi_{n}$ of convolution type:
$$
\begin{equation*}
\Phi_{n} (y-x)= \begin{cases} \dfrac{|y-x|^{2-n}}{(2-n)\sigma_n}, & n\geqslant 3, \\ \dfrac{\ln{|y-x|}}{2\pi}, & n=2, \end{cases}
\end{equation*}
\notag
$$
where $\sigma_n$ is the area of a unit sphere in ${\mathbb R}^n$. Let $\overline \partial$ denote the Cauchy–Riemann operator in ${\mathbb C}^n$, that is, the $n$-column with components
$$
\begin{equation*}
\overline \partial_j= \frac{1}{2}\biggl(\frac{\partial}{\partial x_j}+\iota \,\frac{\partial}{\partial x_{j+n}} \biggr).
\end{equation*}
\notag
$$
Now we set
$$
\begin{equation*}
{\mathfrak U}_n (z,\zeta)=\frac{(n-1)}{(2\pi \iota)^n} \sum_{j=1}^n \frac{(-1)^{j-1}(\overline \zeta_j - \overline z_j)} {|\zeta-z|^{2n}} \,d\overline \zeta[j] \wedge d\zeta, \qquad z,\zeta\in{\mathbb C}^n, \quad z\ne \zeta,
\end{equation*}
\notag
$$
to be the Bochner–Martinelli kernel in ${\mathbb C}^n$; for instance, see [24], § 1. It is known that ${\mathfrak U}_n$ has the representation
$$
\begin{equation}
{\mathfrak U}_n (\zeta,z)=\sum_{j=1}^n (\overline \partial ^*_{j,\zeta} \Phi_{2n} (\zeta,z))(-1)^{j-1}\,d\overline \zeta [j] \wedge d\zeta,
\end{equation}
\tag{2.2}
$$
where $\zeta=(\zeta_1, \dots, \zeta_n )$, $\zeta _j=y_j+\iota y_{j+n}$, and the operators
$$
\begin{equation*}
\overline \partial_j ^*= \frac{1}{2}\biggl(\frac{\partial}{\partial y_j} - \iota \frac{\partial}{\partial y_{j+n}}\biggr)
\end{equation*}
\notag
$$
are components of the formally adjoint operator $\overline \partial ^*=(\overline \partial_1 ^*, \dots, \overline \partial_n ^*)$ of $\overline \partial$; see [24], § 1. In particular, the Martinelli–Bochner kernel is harmonic in $z$ for $z \ne \zeta $ and $n>1$. Of course, for $n=1$ ${\mathfrak U}_n (\zeta,z)$ coincides with the Cauchy kernel
$$
\begin{equation*}
{\mathfrak K} (\zeta,z)= \frac{1}{2\pi \iota} \, \frac{1}{\zeta-z},
\end{equation*}
\notag
$$
so that in this case it is holomorphic in $z$ for $z \ne \zeta $. If $D$ is a bounded domain with Lipschitz boundary, then, given a sufficiently regular function $u_0$ on $\partial D$, we let
$$
\begin{equation*}
M_{\partial D} u_0 (z)=\int_{\partial D}{\mathfrak U}_n (z,\zeta) \, u_0 (\zeta), \qquad z \notin \partial D,
\end{equation*}
\notag
$$
denote its Bochner–Martinelli kernel. Clearly, $M_{\partial D}u_0 (z)$ is well defined for ${u_0\in H^{1/2} (\partial D)}$ as an integral depending on the parameter $z \notin \partial D$. Let $M^-_{\partial D}u_0 $ denote its restriction to $D$ and $M^+_{\partial D}u_0$ denote the restriction to ${\mathbb C}^n \setminus \overline D$. In fact, it can be regarded as a double layer potential, so it defines a continuous linear operator
$$
\begin{equation}
M_{\partial D}^-\colon H^{1/2} (\partial D) \to H^1 (D)
\end{equation}
\tag{2.3}
$$
(for instance, see [24], § 16, or [25], § 2.3.2.5, for smooth domains and see [26] for domains with Lipschitz boundary). On the other hand, for a bounded domain $G \subset{\mathbb C}^n$ with Lipschitz boundary that contains $\overline D$ formula (2.3) means that $M_{\partial D}$ defines a continuous linear operator
$$
\begin{equation*}
M^+_ {\partial D} \colon H^{1/2} (\partial D) \to H^1 (G \setminus \overline D)\cap {\mathcal H} (G \setminus \overline D) .
\end{equation*}
\notag
$$
From the structure of ${\mathfrak U}_n$ we can conclude that
$$
\begin{equation}
|M^+_ {\partial D} u_0 (z)| \leqslant c (\partial D) \|u_0\|_{H^{1/2} (\partial D)} |z|^{1-2n},
\end{equation}
\tag{2.4}
$$
so that $M_{\partial D}$ induces a continuous linear map
$$
\begin{equation}
M^+_ {\partial D} \colon H^{1/2} (\partial D) \to H^{1}_{\mathrm{loc}} ({\mathbb C}^n \setminus D)\cap{\mathcal H} (\widehat{\mathbb C}^n \setminus \overline D) ;
\end{equation}
\tag{2.5}
$$
here $H^{1}_{\mathrm{loc}} ({\mathbb C}^n \setminus D)$ is the space of functions belonging to the classes $H^{1} (G\setminus D)$ for each bounded domain $G$ containing $\overline D$; it carries the topology of convergence in each such class $H^{1} (G\setminus D)$. In the following statement $\mathrm t^-$ and $\mathrm t^+$ denote the continuous trace operators
$$
\begin{equation*}
\mathrm t^-\colon H^1 (D) \to H^{1/2} (\partial D)\quad\text{and} \quad \mathrm t^+\colon H^{1}_{\mathrm{loc}} ({\mathbb C}^n \setminus D) \to H^{1/2} (\partial D).
\end{equation*}
\notag
$$
For domains with smooth boundaries the result in Theorem 2 can be extracted from Theorem 7.1 and Corollary 15.5 in [24]. Theorem 2. Let $D\subset{\mathbb C}^n$ be a bounded Lipschitz domain with connected complement and $w_0$ be a function in $H^{1/2} (\partial D)$. Then the following conditions are equivalent: (1) $w_0$ is a $\mathrm{CR}$-function on $\partial D$; (2) there exists $w\in H^1 (D)\cap{\mathcal O} (D)$ such that $\mathrm t^-(w)= w_0 $ on $\partial D$; (3) $M^{+}_{\partial D} w_0 \equiv 0$ in ${\mathbb C}^n \setminus \overline D$. Proof. This is based on the following result, well known for $C^1 (\partial D)$-functions and domains $D$ with $C^1$-smooth boundary: see [24], Corollary 15.5.
Lemma 1. Let $D\subset{\mathbb C}^n$ be a domain with Lipschitz boundary. Then for $w_0 \in H^{1/2} (\partial D)$ there exists a function $w\in H^1 (D)\cap{\mathcal O} (D)$ such that $\mathrm t^-(w)=v$ on $\partial D$ if and only if $M^{+}_{\partial D} w_0\equiv 0$. Proof. Since the operators $M^{\pm}_{\partial D}$ defined in (2.3) and (2.5) are continuous, by the Bochner–Martinelli formula for holomorphic Sobolev functions (see [24], § 16) we have for each $w\in H^1 (D)\cap{\mathcal O} (D)$. In particular, this means that for $w_0\in H^{1/2} (\partial D)$ we have $M^{+}_{\partial D} w_0 \equiv 0$ in ${\mathbb C}^n \setminus \overline D$, provided that there exists $w\in H^1 (D)\cap{\mathcal O} (D)$ such that $\mathrm t^-(w)=w_0$ on $\partial D$.
Conversely, consider a function $w_0\in H^{1/2} (\partial D)$ such that $M^{+}_{\partial D} w_0 \equiv 0$ in ${{\mathbb C}^n \setminus \overline D}$. Let $\overline \partial_\nu$ denote the so-called complex normal derivative on $\partial D$: for instance, see [24], § 4:
$$
\begin{equation*}
\overline \partial_\nu w=\ \sum_{j=1}^n (\nu _j - \iota \nu_{j+n})\, \overline \partial _j w,
\end{equation*}
\notag
$$
where $\nu (z)=(\nu_1 (\zeta), \dots, \nu_{2n} (\zeta))$ is the unit outward normal to $\partial D$ at $\zeta\in \partial D$. Then by the formulae for jumps of the Bochner–Martinelli integral we obtain
$$
\begin{equation*}
M^-_{\partial D} w_0 - M^+_{\partial D}w_0=w_0 \quad\text{on } \partial D
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\overline \partial_\nu ( M^-_{\partial D} w_0 ) - \overline \partial_\nu(M^+_{\partial D} w_0)=0 \quad\text{on } \partial D;
\end{equation*}
\notag
$$
see [24], Corollary 4.9, for smooth domains and [26] for domains with Lipschitz boundary. In particular, as $M^{+}_{\partial D} w_0\equiv 0$ in ${\mathbb C}^n \setminus \overline D$, it follows that
$$
\begin{equation*}
\mathrm t^- M^{-}_{\partial D} w_0=w_0 \quad\text{on } \partial D\quad\text{and} \quad \overline \partial_\nu ( M^-_{\partial D} w_0 ) ^-=0 \quad\text{on } \partial D.
\end{equation*}
\notag
$$
By construction $M^{-}_{\partial D}w_0$ is harmonic in $D$. Moreover, it follows from (2.3) that $M^{-}_{\partial D} w_0\in H^1 (D)$, so the function $w=M^{-}_{\partial D}w_0$ is an $H^1(D)$-solution of the Cauchy problem
$$
\begin{equation*}
\begin{cases} \Delta_{2n} w=0 &\text{in } D, \\ w=w_0 &\text{on } \partial D, \\ \overline \partial_\nu w=0 &\text{on } \partial D \end{cases}
\end{equation*}
\notag
$$
with data on the whole boundary of $D$. As $D$ has a Lipschitz boundary, there exists a sequence $\{w_k\}\subset C^1 (\overline D)$ approximating $w$ in $H^1 (D)$. By the Gauss–Ostrogradskii formula
$$
\begin{equation*}
\begin{aligned} \, \sum_{j=1}^n \|\overline \partial_j w \|^2 _{L^2 (D)} &=\lim_{k \to+\infty} \sum_{j=1}^n (\overline \partial_j w,\overline \partial_j w_k ) _{L^2 (D)} \\ &= \lim_{k \to+\infty} \int_{\partial D} \overline w_k \, \overline \partial_\nu w \,ds(y)=0, \end{aligned}
\end{equation*}
\notag
$$
because $\overline \partial_\nu w=0$ on $\partial D$. Hence $ \overline \partial w=0$ weakly in $D$, so that by [27], § 24.7, the function $w$ is holomorphic in $D$, that is, $w\in H^1 (D)\cap{\mathcal O} (D)$ satisfies $\mathrm t^- w=w_0$ on $\partial D$, as required.
The proof is complete. It follows immediately from Lemma 1 that conditions (2) and (3) in Theorem 2 are equivalent. Now using Stokes’ formula we observe that (2) implies (1):
$$
\begin{equation*}
\int_{\partial D} w_0 \, \overline \partial \psi =\int_{\partial D} \mathrm t^-w \, \overline \partial \psi =\int_{D} \bigl( \overline \partial w \wedge \overline \partial \psi +w\,\overline\partial (\overline \partial \psi) \bigr) =0
\end{equation*}
\notag
$$
for all $(n,n-2)$-differential forms $\psi$ with coefficients in $C^1 (\overline D)$. Finally, let $w_0\in H^{1/2} (\partial D)$ be a $\mathrm{CR}$-function on $\partial D$. As $D$ is bounded, there exists a ball $B(0,R)$ containing $\overline D$. Then for $z \notin \overline B(0,R)$ the Bochner–Martinelli kernel satisfies
$$
\begin{equation*}
\overline \partial _\zeta{\mathfrak U} (\zeta, z)=0 \quad\text{in } \overline B(0,R).
\end{equation*}
\notag
$$
The Dolbeaux complex is exact on smooth forms in convex domains, so for each $z \notin \overline B(0,R)$ there exists an $(n,n-2)$-form $\psi_z (\zeta)$ with smooth coefficients in $\overline B(0,R)$ such that
$$
\begin{equation*}
\overline \partial _\zeta \psi_z (\zeta)={\mathfrak U} (\zeta, z) \quad\text{in } B(0,R);
\end{equation*}
\notag
$$
see [28]. Hence by (2.1)
$$
\begin{equation*}
(M_{\partial D} w_0)(z)=\int_{\partial D}w_0 (\zeta)\,\overline \partial _\zeta \psi_z (\zeta)=0 \quad\text{for all } z \notin \overline B(0,R).
\end{equation*}
\notag
$$
In particular, as $(M_{\partial D} w_0)(z)$ is harmonic outside $\overline D$, the uniqueness theorem for harmonic functions shows that $M^{+}_{\partial D} w_0 \equiv 0$ in ${\mathbb C}^n \setminus \overline D$ because the latter set is connected. Thus, (1) is equivalent to (2) and (3) (cf. also [24], Theorem 7.1, for domains with smooth boundary), as required. Theorem 2 is proved. Corollary 1. Let $D\subset{\mathbb C}^n$ be a bounded domain with Lipschitz boundary and connected complement, and let $v$ be a function in ${\mathcal H} (\widehat{\mathbb C}^{n} \setminus D) $. Then the following conditions are equivalent: (1) an element $v$ belongs to the space $\Sigma (\widehat{\mathbb C}^{n} \setminus D) $; (2) there exists a function $w\in H^1 (D)\cap{\mathcal O} (D)$ such that $\mathrm t^-(w)= \mathrm t^+(v) $ on $\partial D$; (3) $M^{+}_{\partial D} \mathrm t^+v \equiv 0$ in ${\mathbb C}^n \setminus \overline D$. Now denote by $({\mathcal O} (D))^*$ the dual space of ${\mathcal O} (D)$, that is, the space of continuous linear functionals on ${\mathcal O} (D)$. As usual, we endow $({\mathcal O} (D))^*$ with the strong topology, that is, the topology of uniform convergence on bounded subsets of ${\mathcal O} (D)$; see [22], Ch. IV, § 6. Then for $n=1$ classical results (see [1]–[3]) state that for a simply connected bounded domain $D$ we have the isomorphism (1.1) induced by the duality relation
$$
\begin{equation}
\langle \cdot, \cdot \rangle_{1}\colon {\mathcal O} (D) \times{\mathcal O} ( \widehat{\mathbb C}\setminus D) \to{\mathbb R},
\end{equation}
\tag{2.7}
$$
by means of the curvilinear integral
$$
\begin{equation}
\langle u, h \rangle _{1}=\int_{\partial G} h (z) u(z)\,dz
\end{equation}
\tag{2.8}
$$
for $ u\in{\mathcal O} (D)$ and $h\in{\mathcal O}(\widehat{\mathbb C} \setminus D)$, where $G\Subset D$ and the piecewise smooth curve $\partial G$ lies in the intersection of the domains of $u$ and $h$ (of course, the pairing is independent of $G$ with the properties as above). To engage the general theory of partial differential equations let $S_{\overline \partial ^*} (U)$ denote the space of smooth solutions of the equation $\overline \partial ^* g=0 $ on the open set $U \subset{\mathbb C}^n$; these solutions are $n$-columns $g=(g_1, \dots, g_n)^\top$ of functions $g_j\in C^\infty (U)$ such that
$$
\begin{equation*}
\sum_{j=1}^n \overline \partial_j^* g_j=0 \quad\text{in } U.
\end{equation*}
\notag
$$
For $n=1$ $S_{\overline \partial ^*} (U)$ is precisely the space of antiholomorphic functions, and the space $S_{\overline \partial ^*} (\widehat{\mathbb C} \setminus D)$ has the properties quite analogous to those of ${\mathcal O}(\widehat{\mathbb C} \setminus D)$. In this case it is clear that complex conjugation induces the antilinear isomorphism
$$
\begin{equation*}
{\mathcal O} (\widehat{\mathbb C} \setminus D) \ni h \to \overline h=g\in S_{\overline \partial ^*} (\widehat{\mathbb C} \setminus D).
\end{equation*}
\notag
$$
In particular, for $n=1$ the duality relation
$$
\begin{equation}
\langle u, g \rangle _{\mathrm{Gr}}=\int_{\partial G} \overline{g} (z)u(z) \, dz,
\end{equation}
\tag{2.9}
$$
defined for $ u\in{\mathcal O} (D)$ and $g\in S_{\overline \partial ^*} (\widehat{\mathbb C} \setminus D)$, induces the topological (antilinear) isomorphism (see [15])
$$
\begin{equation}
({\mathcal O} (D))^* \cong S_{\overline \partial ^*} (\widehat{\mathbb C} \setminus D).
\end{equation}
\tag{2.10}
$$
Grothendieck proved that for any elliptic operator $A$ on a smooth manifold $X$ that has a two-sided regular fundamental solution there is a topological isomorphism where we write $\widehat X$ to indicate that we only consider solutions regular at ‘infinity’ with respect to the fundamental solution selected; see [15], Theorem 4. Unfortunately, by Hartogs’s theorem on the removal of compact singularities of holomorphic functions for $n>1$ the Cauchy–Riemann operator $\overline \partial $ has no two-sided fundamental solution, and therefore we cannot apply Theorem 4 in [15] to this case. Nevertheless, the operator $\overline \partial $ has a regular left fundamental solution, for example, the one presented by the Bochner–Martinelli kernel ${\mathfrak U}_n (\zeta,z)$, $n>1$. Then Theorem 3 in [15] means that Grothendieck’s duality relation
$$
\begin{equation}
\langle u, g \rangle _{\mathrm{Gr}}=\int_{\partial G}\sum_{j=1}^n \overline g_j (z)u(z)(-1)^{j-1}\, d\overline z[j] \wedge dz,
\end{equation}
\tag{2.12}
$$
defined for $ u\in{\mathcal O} (D)$ and $g\in S_{\overline \partial ^*} (\widehat{\mathbb C}^n \setminus D)$, where $G\Subset D$ and the piecewise smooth hypersurface $\partial G$ lies in the intersection of the domains of $u$ and $g$, induces a surjective continuous antilinear map
$$
\begin{equation}
S_{\overline \partial ^*} (\widehat{\mathbb C}^n \setminus D) \to ({\mathcal O} (D))^*
\end{equation}
\tag{2.13}
$$
(again, the pairing here is independent of $G$ with the properties indicated). However, the operator $\overline \partial ^*$ is not elliptic for $n>1$, and therefore the space $S_{\overline \partial ^*} (\widehat{\mathbb C}^n \setminus D) $ is too large for (2.13) to be bijective; see the example below. Example 1. Let $n\geqslant 1$, and let $D=B (0,1)$ be the init ball with centre at the origin. To see that the map (2.13) is not injective for $n>1$, we use homogeneous spherical polynomials. Namely, let $\{h^{(j)}_{r} \}$, $r\geqslant 0$, be an orthonormal basis of the Lebesgue space $L^2 (\partial B (0,1))$ on the sphere $\partial B (0,1)$ in ${\mathbb R}^{2n}$, $n\geqslant 1$, which consists of spherical harmonics of degree $r$ (for instance, see [29], Ch. XI), where $j$, $1\leqslant j\leqslant J_{r,2n}$, is the index of the polynomial in the basis, $J_{0,2}=1$, $J_{r,2}=2$ for $r\in \mathbb N$, and
$$
\begin{equation*}
J_{r,2n}=\frac{(2n+2r-2)(r+2n-3)!}{r!\, (2n-2)!}, \qquad n>1.
\end{equation*}
\notag
$$
For each $r$ the harmonic extension of $h_r$ in $D$ produces a homogeneous harmonic polynomial, which we also denote by $h_r$. The harmonic extension of $h_r$ to ${\mathbb R}^{2n} \setminus \overline D$ that vanishes at infinity for $n>1$ and is bounded at infinity for $n=1$ is defined by If $n\geqslant 1$, then any vector-valued function of the form
$$
\begin{equation*}
\overline \partial \biggl(\frac{h_r (x)}{|x|^{2n+2r-2}}\biggr), \qquad r\in{\mathbb Z}_+,
\end{equation*}
\notag
$$
satisfies the equality
$$
\begin{equation}
\overline \partial ^* \biggl(\overline \partial \biggl(\frac{h_r (x)}{|x|^{2n+2r-2}}\biggr)\biggr)=0 \quad\text{in } {\mathbb C}^{n}\setminus \{0\},
\end{equation}
\tag{2.15}
$$
and therefore $\overline \partial ({h_r (x)}/{|x|^{2n+2r-2}})$ belongs to the space $S_{\overline \partial ^*} (\widehat{\mathbb C}^{n} \setminus D)$ for each $r\in{\mathbb Z}_+$. Using the complex structure we can express a homogeneous harmonic polynomial $h_r$ as
$$
\begin{equation*}
h_r (z,\overline z)=\sum_{|p+q|=r} a_{p,q} z^p \overline z^q,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, p=(p_1, \dots, p_n), \qquad q=(p_1, \dots, q_n)\in{\mathbb Z}^{n}_+, \\ z^p=z_1^{p_1} \dotsb z_n^{p_n}, \qquad \overline z^p= \overline z_1^{q_1} \dotsb \overline z_n^{q_n}, \end{gathered}
\end{equation*}
\notag
$$
and the $a_{p,q}$ are appropriate complex coefficients ensuring harmonicity; see [24], Ch. 1, § 5. Then it follows from (2.15) that the space $S_{\overline \partial^*} (\widehat{\mathbb C}^{n} \setminus B(0,1))$ contains series of the type
$$
\begin{equation}
\sum_{r=0}^\infty \sum_{|p+q|=r} a^{(j)}_{p,q} \sum_{j=1}^{ J_{r,2n}} \overline \partial \biggl(\frac{z^p \overline z^q}{|z|^{2n+2r-2}}\biggr)
\end{equation}
\tag{2.16}
$$
with suitable complex coefficients $a^{(j)}_{p,q}$ which converge in a neighbourhood of ${{\mathbb C}^n \setminus B(0,1)}$; moreover, it contains nontrivial elements $g$ annulling the pairing (2.12) for each $u\in{\mathcal O}(B(0,1))$ for $n>1$. In fact, consider the vector
$$
\begin{equation*}
g^{(p,q)}=\overline \partial \biggl(\frac{2\overline z^q z^p}{|z|^{2n+2|q|-2}}\biggr)\in S_{\overline \partial^*} (\widehat{\mathbb C}^{n} \setminus D).
\end{equation*}
\notag
$$
For $n=1$ we obtain
$$
\begin{equation*}
\frac{2\overline z^q z^p}{|z|^{2n+2|q|-2}}= \frac{2\overline z^q z^p}{|z|^{2|q|}}=\frac{2 z^p}{z^q},
\end{equation*}
\notag
$$
so that $g^{(p,q)}\equiv 0$ in ${\mathbb C} \setminus \{0\}$ in this particular case. We continue for $n>1$. As is known, for each smooth domain ${\mathcal D} \subset{\mathbb R}^n$ we have
$$
\begin{equation*}
(-1)^{j-1}\, dx[j]=\nu_j (x)\, ds (x) \quad\text{on } \partial{\mathcal D},
\end{equation*}
\notag
$$
where $\nu (x)=(\nu _1(x), \dots, \nu _n(x))$ is the unit outward normal to $\partial{\mathcal D}$ at the point $x$, and $ds $ is the volume form on $\partial D$. It is clear that for each sphere $S_R$ with centre at the origin and radius $R$ we have $ \nu(x)=x/R$. Therefore,
$$
\begin{equation*}
(-1)^{j-1} \, d\overline z[j] \wedge dz=2^{n-1}\iota^n \frac{z_j}{2R} \, ds (z,\overline z)
\end{equation*}
\notag
$$
on $S$; see [24], Lemma 3.5. Now from Euler’s formula for positively homogeneous functions we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \sum_{j=1}^n \overline g^{(p,q)}_j (z) \, (-1)^{j-1}\, d\overline z[j] \wedge dz &= 2^{n-1}\iota^n \sum_{j=1}^n \frac{z_j}{R} \,\overline{\overline \partial _j \biggl( \frac{\overline z^q z^p}{|z|^{2n+2|q|+2|p|-2}}\biggr)}\,ds \\ \notag &=\frac{2^{n-1}\iota^n}{R} \overline{\sum_{j=1}^n \overline z_j \, \overline \partial _j \biggl(\frac{\overline z^q z^p}{|z|^{2n+2|q|+2|p|-2}}\biggr)} \, ds \\ &=2^{n-1}\iota^n (1-|p|-n) \frac{z^{q} \overline z^p \, ds}{R^{2n+2|q|+2|p|-1}} \end{aligned}
\end{equation}
\tag{2.17}
$$
on the sphere $S_R$. Finally, since Grothendieck’s duality relation $\langle \cdot,\cdot \rangle_{\mathrm{Gr}} $ is independent of the choice of the domain $G$, we can take a ball $G=B(0,R)$ of suitable radius $R$. Then $S_R \subset B(0,1)$, and it follows from (2.17) that for all $q\in{\mathbb Z}_+$ such that $|q|\geqslant 1$ we have
$$
\begin{equation*}
\langle z^s,g^{(0,q)}\rangle_{\mathrm{Gr}}=2^{n-1}\iota^n (1-n)\int_{S_R} \frac{z^{q+s}\,ds}{R^{2n+2|q|-1}}=0 \quad\text{for all } s\in{\mathbb Z}^n_+
\end{equation*}
\notag
$$
by Cauchy’s integral theorem for holomorphic functions. Therefore,
$$
\begin{equation*}
\langle u, g^{(0,q)} \rangle_{\mathrm{Gr}}=0 \quad\text{for all } u\in{\mathcal O}(B(0,1)),
\end{equation*}
\notag
$$
for an infinite number of $g^{(0,q)}$ with $|q|\geqslant 1$, as required. At the end of the example note that by formula (2.14) and Corollary 1 the space $\Sigma (\widehat{\mathbb C}^{n} \setminus B(0,1))$ coincides with the set of series of the form
$$
\begin{equation*}
\sum_{|p|\geqslant 0} \frac{a^{(j)}_{p} z^p}{|z|^{2n+2|p|-2}}
\end{equation*}
\notag
$$
with appropriate coefficients $a^{(j)}_{p}$ that converge in a neighbourhood of ${\mathbb C}^n \setminus B(0,1)$. Of course, the vector
$$
\begin{equation*}
g^{(p)}(z)=\overline \partial \biggl(\frac{z^p}{|z|^{2n+2|p|-2}}\biggr)
\end{equation*}
\notag
$$
lies in the space $S_{\overline \partial^*} (\widehat{\mathbb C}^{n} \setminus D)$. However, by (2.17)
$$
\begin{equation*}
\langle z^p, g ^{(p)}\rangle_{\mathrm{Gr}}= 2^{n-1}\iota^n(1-|p|-n)\int_{S_R} \frac{|z|^{2p_1} \dotsb |z|^{2p_n}\,ds}{R^{2n+2|p|-1}} \ne 0 \quad\text{for all } p\in{\mathbb Z}^n_+,
\end{equation*}
\notag
$$
that is, $g^{(p)}$ does not annul the space ${\mathcal O} (B(0,1))$ in the pairing (2.12). § 3. Proof of Theorem 1The proof follows rather close to the classical scheme tested on (1.1) and (2.11). 3.1. The duality relationFirst recall that elements of the space ${\mathcal H} (U)$ are infinitely differentiable on the open set $U\,{\subset}\,\mathbb R^n$. In particular, for each closed set $\sigma \subset{\mathbb C}^n$ an element of ${\mathcal H} (\sigma)$ has continuous derivatives of all orders in some open set $U\Supset\sigma$. Fix $u\in{\mathcal O} (D) $ and $v\in \Sigma ( \widehat{\mathbb C}^n \setminus D)$. By the above there exists an (unbounded) domain $U_v$ containing ${\mathbb C}^n \setminus D$ such that $v\in{\mathcal H} (U_v) $. Hence $V_v=U_v\cap D$ is an open set in ${\mathbb C}^n$. Moreover, since $D$ has a connected complement, there exists a closed hypersurface $\Gamma \subset V_v$ bounding some domain $G\Subset D$. Now we can define the duality relation between the spaces $\langle\,\cdot\,{,}\,\cdot\,\rangle$ ${\mathcal O}(D)$ and $\Sigma (\widehat{\mathbb C}^n \setminus D)$:
$$
\begin{equation}
\langle u, v \rangle=\int_{\Gamma} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}u(z) \, d\overline z[j] \wedge dz
\end{equation}
\tag{3.1}
$$
for $ u\in{\mathcal O} (D)$ and $v\in \Sigma ( \widehat{\mathbb C}^n \setminus D)$. Note that the vector $w=\overline \partial v$ belongs to $S_{\overline \partial ^*}( \widehat{\mathbb C}^n \setminus D)$ for $v\in \Sigma ( \widehat{\mathbb C}^n \setminus D)$, that is, (3.1) is quite closely connected with Grothendieck’s duality relation (2.12). Of course, the fact that the integral (3.1) involves an $(n,n-1)$-form relates this pairing to the one constructed by Serre in [11]. Now, by Stokes’ formula, for two arbitrary hypersurfaces $\Gamma_1$ and $\Gamma_2$ with the required properties we have
$$
\begin{equation*}
\begin{aligned} \, &\int_{\Gamma_1} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}u(z) \, d\overline z[j] \wedge dz \\ &\qquad\qquad - \int_{\Gamma_2} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}u(z) \, d\overline z[j] \wedge dz \\ &\qquad =\int_{\Omega} \biggl(\sum_{j=1}^n\overline{(\overline \partial_j v)}\, \overline \partial_j u - \frac{1}{4} \Delta_{2n} u\biggr)\,d\overline z\wedge dz=0 \end{aligned}
\end{equation*}
\notag
$$
for all $ u\in{\mathcal O}(D)$ and $v\in\Sigma (\widehat{\mathbb C} ^n\setminus D)$, where $\Omega \subset D$ is an open set bounded by the surfaces $\Gamma_1$ and $\Gamma_2$. We see that the pairing (3.1) is independent of the choice of $\Gamma \subset V_v$ with the properties indicated above. It is obvious that
$$
\begin{equation}
|\langle u, v \rangle| \leqslant C_{\Gamma} \max_{z\in \Gamma} |\nabla v (z)| \max_{x\in \Gamma} |u (z)|
\end{equation}
\tag{3.2}
$$
for some constant $C_{\Gamma}$ independent of $ u\in{\mathcal O}(D)$ and $v\in \Sigma (\widehat{\mathbb C}^n \setminus D)$. So taking the topology of the spaces in question into account we conclude that (3.1) induces a sesquilinear separately continuous map
$$
\begin{equation}
\langle \cdot, \cdot \rangle\colon {\mathcal O}(D) \times \Sigma (\widehat{\mathbb C}^n \setminus D) \to \mathbb C.
\end{equation}
\tag{3.3}
$$
In particular, for each fixed function $v\in \Sigma ( \widehat{\mathbb C}^n \setminus D)$ the functional
$$
\begin{equation}
f_v (u)=\langle u, v \rangle, \qquad u\in{\mathcal O}(D),
\end{equation}
\tag{3.4}
$$
is bounded and linear, that is, $f_v\in ({\mathcal O}(D))^*$. Moreover, by (3.2) the map
$$
\begin{equation}
\Sigma ( \widehat{\mathbb C}^n \setminus D) \ni v \to f_v\in ({\mathcal O}(D))^*
\end{equation}
\tag{3.5}
$$
is continuous and antilinear. 3.2. The injectivity of the mapWe show that the map (3.5) is injective. Let
$$
\begin{equation*}
\langle u, v \rangle=0 \quad\text{for all } u\in{\mathcal O}(D).
\end{equation*}
\notag
$$
By Corollary 1 there exists a function $w\in H^1 (D)\cap{\mathcal O} (D)$ such that $\mathrm t^-(w)=v$ on $\partial D$. In particular, On the other hand $w\in H^1 (D)\cap{\mathcal O} (D)$, and therefore by Stokes’ formula
$$
\begin{equation}
\begin{aligned} \, \notag 0 &=\langle w, v \rangle=\int_{\Gamma} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}w(z)\,d\overline z[j] \wedge dz \\ \notag &=\int_{\partial D}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}v(z) \, d\overline z[j] \wedge dz \\ &=\lim_{R\to+\infty}\biggl(\int_{B (0,R) \setminus \overline D} |\overline \partial v (z)|^2\,dx -\int_{|z|=R}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}v(z) \, d\overline z[j] \wedge dz \biggr). \end{aligned}
\end{equation}
\tag{3.6}
$$
Since $v $ is harmonic in ${\mathbb R}^{2n} \setminus D$ and vanishes at infinity, it follows that
$$
\begin{equation*}
|\partial^\alpha v(z)| \leqslant c_1 |z|^{2-2n-|\alpha|} \quad\text{for } n\geqslant 2\quad\text{and} \quad \alpha\in{\mathbb Z}^{2n}_{+}
\end{equation*}
\notag
$$
(for instance, see [27], § 24.10, formulae (33)–(35)). Hence
$$
\begin{equation}
\lim_{R\to+\infty}\biggl(\int_{|z|=R}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j v_j) (z)}v(z)\,d\overline z[j] \wedge dz\biggr)=0.
\end{equation}
\tag{3.7}
$$
Now it follows from (3.6) and (3.7) that
$$
\begin{equation*}
0=\langle w, v \rangle=\sum_{j=1}^n \|\overline \partial_j v\|^2_{L^2({\mathbb C}^n \setminus \overline D)},
\end{equation*}
\notag
$$
that is, $v\in{\mathcal O} ( \widehat{\mathbb C}^n \setminus \overline D) $. Moreover, $v$ belongs to ${\mathcal O} ( \widehat{\mathbb C}^n \setminus D) $ because it is harmonic in a neighbourhood of $\widehat{\mathbb C}^n \setminus D$. For $n>1$ Hartogs’s theorem yields immediately that $v$ extends holomorphically to ${\mathbb C}^n$. In particular, since $v\in{\mathcal O}({\mathbb C}^n)$ vanishes at infinity, it vanishes identically by Liouville’s theorem, so that the map (3.5) is antilinear and injective. 3.3. The surjectivity of the mapWe show that the map (3.5) is surjective. In fact, fix some $f\in ({\mathcal O} (D))^*$. Since ${\mathcal O} (D)$ is a closed subspace of $C(D)$, by the Hahn–Banach theorem there exists a functional $F\in (C(D))^*$ that is equal to $f$ on ${\mathcal O}(D)$. By the classical Riesz duality for $C(D)$, there exists a Radon measure $\mu$ with compact support in $D$ such that
$$
\begin{equation}
f(u)=\int_{K} u (z)\,d\mu (z) \quad\text{for all } u\in{\mathcal O} (D),
\end{equation}
\tag{3.8}
$$
where the compact set $K\Subset D$ contains the support $\operatorname{supp} (\mu)$ of $\mu$: for instance, see [30], § 4.10. As $K\Subset D$, there exists a domain $G$ with smooth boundary such that $K\Subset G\Subset D$. In particular, by the Bochner–Martinelli formula (2.6)
$$
\begin{equation}
(M_{\partial G} u) (z) =\begin{cases} u(z), & z\in G, \\ 0, & z \notin \overline G, \end{cases}
\end{equation}
\tag{3.9}
$$
for each function $u\in{\mathcal O} (D)$. It follows from (2.2), (3.8), (3.9) and Fubini’s theorem that
$$
\begin{equation}
\begin{aligned} \, \notag f(u) &=\int_{K}\biggl(\int_{\partial G}{\mathfrak U}_n (\zeta,z) u(\zeta)\biggr)\,d\mu (z) \\ &=\begin{cases} \displaystyle \int_{\partial G} h (\zeta)u(\zeta) \, d\zeta, & n=1, \\ \displaystyle \int_{\partial G} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j \widehat v) (\zeta)}u(\zeta) \, d\overline \zeta[j] \wedge d\zeta, & n>1, \end{cases} \end{aligned}
\end{equation}
\tag{3.10}
$$
for all $u\in{\mathcal O} (D)$, where
$$
\begin{equation*}
h (\zeta)=\frac{1}{2\pi \iota}\int_{K} \frac{d \mu (z)}{\zeta- z}\quad\text{and} \quad \widehat v(\zeta)=\int_{K} \Phi_{2n} (\zeta,z)\,d\overline \mu (z).
\end{equation*}
\notag
$$
If $n=1$, then $h$ is holomorphic in ${\mathbb C} \setminus K$ and vanishes at infinity by the properties of the Cauchy kernel, so $h\in{\mathcal O} (\widehat{\mathbb C} \setminus D)$ because $K\Subset D$. This yields the classical duality (1.1) (see [1]–[3]) by means of the pairing (2.8). After the substitution $g=\overline h$ we obtain Grothendieck’s classical duality (2.12) (see [15]) by means of the pairing (2.9). For $n>1$, as $\Phi_{2n} (\zeta,z)$ is a two-sided fundamental solution of the Laplace operator, the function $\widehat v $ is harmonic in ${\mathbb C}^n \setminus K$ and vanishes at infinity. In particular, $\widehat v\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$ because $K\Subset D$. However, we cannot ensure that $\widehat v\in \Sigma (\widehat{\mathbb C}^n \setminus D)$. To cope with this fix $u\in{\mathcal O}(D)$ and a sequence $\{u_\nu \} \subset{\mathcal O} (D)\cap H^1 (D)$ approximating $u$ in $C (D)$ (which exists by the hypotheses). Then from (3.10) and Stokes’ formula we obtain
$$
\begin{equation}
\begin{aligned} \, \notag f(u) &=\lim_{\nu \to\infty} f(u_\nu) \\ \notag &=\lim_{\nu \to\infty}\int_{\partial G}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j \widehat v) (\zeta)}u_\nu(\zeta) \, d\overline \zeta[j]\wedge d\zeta \\ &=\lim_{\nu \to\infty}\int_{\partial D} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j \widehat v) (\zeta)}u_\nu(\zeta) \, d\overline \zeta[j]\wedge d\zeta. \end{aligned}
\end{equation}
\tag{3.11}
$$
On the other hand recall a result in [31]. Given a function $w_0\in H^{1/2} (\partial D)$, let ${\mathcal P}_{D} (w_0)$ denote the unique solution $w\in H^1 (D)$ of the (interior with respect to $D$) Dirichlet problem for the Laplace operator:
$$
\begin{equation}
\begin{cases} \Delta_{2n}{\mathcal P}_{D} (w_0)=0 & \text{in } D, \\ {\mathcal P}_{D} (w_0)=w_0 & \text{on } \partial D \end{cases}
\end{equation}
\tag{3.12}
$$
(in fact, ${\mathcal P}_{D} (w_0)$ is the Poisson integral of $w_0$ for $D$). In a similar way let $\widetilde{\mathcal P}_{D} (w_0) $ denote the unique solution of the (exterior) Dirichlet problem for the Laplace operator:
$$
\begin{equation}
\begin{cases} \Delta_{2n} \widetilde{\mathcal P}_{D} (w_0)=0 & \text{in } {\mathbb C}^n \setminus \overline D, \\ \widetilde{\mathcal P}_{D} (w_0)=w_0 & \text{on } \partial D,\ \end{cases}
\end{equation}
\tag{3.13}
$$
that satisfies (1.3) and the relation $\overline \partial \widetilde{\mathcal P}_{D} (w_0)\in L^2 ({\mathbb C}^n \setminus \overline D)$. Lemma 2. The Hermitian form For the proof, see [31] (or see [32], where a more detailed discussion is presented). Then it follows form Stokes’ formula that
$$
\begin{equation*}
-\int_{\partial D} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j \widehat v) (\zeta)}h(\zeta) \, d\overline \zeta[j] \wedge d\zeta=h_D (h,{\mathcal P}_D \widehat v)
\end{equation*}
\notag
$$
for each $h\in H^1 (D)\cap{\mathcal O} (D)$. In particular, (3.11) means that
$$
\begin{equation}
f(u)=- \lim_{\nu \to\infty} h_{D} (u_\nu,{\mathcal P}_D \widehat v)=- \lim_{\nu \to\infty} h_{D} (u_\nu, \Pi_D{\mathcal P}_D \widehat v),
\end{equation}
\tag{3.14}
$$
where $\Pi_D\colon H^1 (D) \to H^1 (D)\cap{\mathcal O} (D)$ is the orthogonal projection with respect to the Hermitian form $ h_{D} (\,\cdot\,{,}\,\cdot\,)$. We show that the function $\widetilde{\mathcal P}_{D}( \Pi_D{\mathcal P}_D w)$ belongs to ${\mathcal H} (\widehat{\mathbb C}^n \setminus D)$, provided that $w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$. To do this fix $w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$. By definition there exists a domain $G\Subset D$ with Lipschitz boundary such that $w\in{\mathcal H}(\widehat{\mathbb C}^n \setminus G)$. As the duality relation is independent of $G$, we select $G$ so that the set $D\setminus G$ has no compact components in $D$ (this can be dome because ${\mathbb C}^n \setminus D$ is connected). We endow the spaces $H^1 (D) $ and $H^1 (G) $ with the inner products $h_{D} (\,\cdot\,{,}\,\cdot\,)$ and $h_{G} (\,\cdot\,{,}\,\cdot\,)$, respectively. Now note that the exterior Dirichlet problem is well posed in the sense of Hadamard, and therefore the Hermitian form
$$
\begin{equation*}
\widetilde h_{D} (W, \widetilde W)=\sum_{j=1}^n \int_{D} \overline{(\overline \partial_j{\mathcal P}_D W)}\, \overline \partial_j{\mathcal P}_D (\widetilde W) \,dx+ \sum_{j=1}^n\int_{{\mathbb C}^n \setminus \overline D} \overline{( \overline \partial_j W)}\, \overline \partial_j \widetilde W \,dx
\end{equation*}
\notag
$$
defines an inner product on the space $\widetilde H^1 ({\mathbb C}^n \setminus \overline D)\cap{\mathcal H}(\widehat{\mathbb C}^n \setminus \overline D)$ of elements of ${\mathcal H} (\widehat{\mathbb C}^n \setminus \overline D)\cap H^1_{\mathrm{loc}} ({\mathbb C}^n \setminus D)$ such that $\overline \partial_j W\in L^2 ({\mathbb C}^n \setminus \overline D)$ for all $1\leqslant j \leqslant n$. Moreover, by construction
$$
\begin{equation}
h_{D} ({\mathcal P}_D W, h)= \widetilde h_{{\mathbb C}^n \setminus \overline D} (W, \widetilde{\mathcal P}_{D} h )
\end{equation}
\tag{3.15}
$$
for all $h\in H^1 (D)\cap{\mathcal H} (D) $ and $W\in \widetilde H^1 ({\mathbb C}^n \setminus\overline D)\cap{\mathcal H} (\widehat{\mathbb C}^n \setminus\overline D)$. In other words $\widetilde{\mathcal P}_{D}$ defines an isomorphism between the Hilbert spaces $H^1 (D)\cap{\mathcal H} (D)$ and ${\widetilde H^1 ({\mathbb C}^n \setminus \overline D)\cap{\mathcal H}(\widehat{\mathbb C}^n \setminus\overline D)}$, and in a similar way
$$
\begin{equation}
h_{G} ({\mathcal P}_G W, h)=\widetilde h_{G} (W, \widetilde{\mathcal P}_{G} h )
\end{equation}
\tag{3.16}
$$
for all $h\in H^1 (G)\cap{\mathcal H} (G)$ and $W\in \widetilde H^1 ({\mathbb C}^n \setminus \overline G)\cap{\mathcal H} (\widehat{\mathbb C}^n \setminus \overline G)$. Now let $Y^1(G)$ denote the closure of $H^1 (D)\cap{\mathcal O}(D)$ in $H^1 (G)\cap{\mathcal O} (G)$. By the Stieltjes–Vitali theorem the embedding $R\colon H^1 (D)\cap{\mathcal H} (D) \to H^1 (G)\cap{\mathcal H} (G)$ is compact. Hence by the Hilbert–Schmidt theorem on the spectrum of a compact selfadjoint operator there exists an orthonormal basis $\{b_m\}_{m\in \mathbb N}$ (with respect to $h_{D} (\,\cdot\,{,}\,\cdot\,)$) in $H^1 (D)\cap{\mathcal H} (D)$ that forms an orthogonal system (with respect to $h_{G} (\,\cdot\,{,}\,\cdot\,)$) in $H^1 (G)\cap{\mathcal H} (G)$ and satisfies Moreover, since the set $D\setminus G$ has no compact components in $D$, by Mergelyan’s theorem $H^1 (D)\cap{\mathcal H}(D) $ is dense in $H^1 (G)\cap{\mathcal H}(G) $, that is, $\{R b_m \}_{m\in \mathbb N}$ is an orthonormal basis in $H^1 (G)\cap{\mathcal H}(G) $; see [33]. In particular, the eigenvalues $\lambda_m \ne 0$ correspond to the subspace $H^1 (D)\cap{\mathcal O}(D)$, the system $\{b_m \}_{\lambda_m\ne 0}$ is an orthonormal basis in $H^1 (D)\cap{\mathcal O}(D) $, and $\{R b_m \}_{\lambda_m\ne 0}$ is an orthogonal basis in $H^1 (G)\cap{\mathcal H}(G) $ $Y^1 (G)$. Then the projection $\Pi_D$ is expressed by the formula
$$
\begin{equation*}
\Pi_D h=\sum_{\lambda_m\ne 0} h_{D} (h, b_m) b_m \quad\text{for all } h\in H^1 (D)\cap {\mathcal H}(D).
\end{equation*}
\notag
$$
However, by (3.15) we can actually treat $\Pi_D{\mathcal P}_D$ as an orthogonal projection from $\widetilde H^1 ({\mathbb R}^n \setminus \overline D)\cap{\mathcal H} (\widehat{\mathbb C}^n \setminus \overline D)$ onto the closed subspace of functions $W$ such that ${\overline \partial P_D W=0}$ in $D$, that is,
$$
\begin{equation*}
\widetilde{\mathcal P}_{D}\Pi_D h=\sum_{\lambda_m\ne 0} h_{D} (h, b_m) \widetilde{\mathcal P}_{D} b_m \quad\text{for all } h\in H^1 (D)\cap {\mathcal H}(D).
\end{equation*}
\notag
$$
We let $\widetilde R$ denote the operator of continuous embedding $\widetilde R\colon {\mathcal H} (\widehat{\mathbb C}^n \setminus \overline G)\cap \widetilde H^1 ({\mathbb C}^n \setminus \overline G) \to{\mathcal H} (\widehat{\mathbb C}^n \setminus \overline D)\cap \widetilde H^1 ({\mathbb C}^n \setminus \overline D)$ and conclude that for each function $w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$ we have
$$
\begin{equation*}
\begin{gathered} \, w=\sum_{m=1}^\infty h_{G} ({\mathcal P}_{G} w, R b_m) \frac{\widetilde{\mathcal P}_{G} Rb_m} {h_G (Rb_m, Rb_m)}, \\ \widetilde R w=\sum_{m=1}^\infty h_{D} ({\mathcal P}_{D} w, b_m) \widetilde{\mathcal P}_{D} b_m \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\widetilde{\mathcal P}_{D} \Pi_D{\mathcal P}_D\widetilde R w=\sum_{\lambda_m\ne 0} h_{D} ({\mathcal P}_{D} \widetilde R w, b_m) \widetilde{\mathcal P}_{D} b_m.
\end{equation}
\tag{3.17}
$$
Of course, by the uniqueness theorem for harmonic functions $w$ is the unique harmonic extension of $\widetilde Rw$ from ${\mathbb C}^n \setminus D$ to ${\mathbb C}^n \setminus G$. Furthermore, if $\lambda_m \ne 0$, then
$$
\begin{equation*}
h_G (Rb_m, Rb_m)=h_D (\Pi_DR^*R\Pi_D b_m, b_m)=\lambda_m,
\end{equation*}
\notag
$$
because $h_D (b_m, b_m)=1$, and therefore the series
$$
\begin{equation}
\sum_{\lambda_m\ne 0} h_{G} ({\mathcal P}_{G} w, R b_m) \frac{\widetilde{\mathcal P}_{G} Rb_m} {h_G (Rb_m, Rb_m)}=\sum_{\lambda_m\ne 0} h_{G} ({\mathcal P}_{G} w, R b_m) \frac{\widetilde{\mathcal P}_{G} Rb_m}{\lambda_m}
\end{equation}
\tag{3.18}
$$
converges in ${\mathcal H} (\widehat{\mathbb C}^n \setminus \overline G)\cap \widetilde H^1 ({\mathbb C}^n \setminus \overline G)$. On the other hand, if $w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$, then by Stokes’ formula
$$
\begin{equation}
\begin{aligned} \, \notag h_{D} ({\mathcal P}_D \widetilde R w, \Pi_D h) &=-\int_{\partial D}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j w) (\zeta)}\Pi_D h(\zeta) \, d\overline \zeta[j]\wedge d\zeta \\ \notag &=-\int_{\partial G}\sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial _j w) (\zeta)}\Pi_D h(\zeta) \, d\overline \zeta[j]\wedge d\zeta \\ &= h_{G} ({\mathcal P}_G w, R \Pi_D h) \end{aligned}
\end{equation}
\tag{3.19}
$$
for all $h\in H^1 (D)\cap{\mathcal H} (D)$. In particular,
$$
\begin{equation}
\widetilde{\mathcal P}_{D} \Pi_D{\mathcal P}_D\widetilde R w=\sum_{\lambda_m\ne 0} h_{G} ({\mathcal P}_G w, R b_m) \widetilde{\mathcal P}_{D} b_m.
\end{equation}
\tag{3.20}
$$
Moreover, since ${\mathcal P}_G \widetilde{\mathcal P}_G=I$, it follows from (3.19) that
$$
\begin{equation*}
h_{D} ({\mathcal P}_D \widetilde R \widetilde{\mathcal P}_G Rb_m, \Pi_D h)= h_{G} (Rb_m, R R\Pi_D h)=\lambda_m h_{D} (b_m, \Pi_D h)
\end{equation*}
\notag
$$
for all $h\in H^1 (D)\cap{\mathcal H} (D)$ for $\lambda_m \ne 0$. Thus, as $H^1 (D)\cap{\mathcal O}(D)$ is densely embedded in $Y^1 (G)$ and $\widetilde{\mathcal P}_{G}$ is a Hilbert space isomorphism between $H^1 (G)\cap{\mathcal H} (G)$ and $\widetilde H^1 ({\mathbb C}^n \setminus \overline G)\cap{\mathcal H} (\widehat{\mathbb C}^n \setminus \overline G )$, it follows from (3.17), (3.18) and (3.20) that
$$
\begin{equation*}
\widetilde{\mathcal P}_{D} \Pi_D{\mathcal P}_D\widetilde R w= \sum_{\lambda_m\ne 0} h_{G} ({\mathcal P}_{G} w, R b_m) \frac{\widetilde{\mathcal P}_{G} Rb_m} {h_G (Rb_m, Rb_m)}.
\end{equation*}
\notag
$$
This means that $\widetilde{\mathcal P}_{D} \Pi_D{\mathcal P}_D w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D)$ and
$$
\begin{equation*}
\overline \partial{\mathcal P}_D \widetilde{\mathcal P}_{D}\Pi_D{\mathcal P}_D w=0 \quad\text{in}\ D \quad\text{for all } w\in{\mathcal H} (\widehat{\mathbb C}^n \setminus D).
\end{equation*}
\notag
$$
Hence $v=\widetilde{\mathcal P}_{D} \Pi_D{\mathcal P}_D \widehat v $ belongs to $ \Sigma (\widehat{\mathbb C}^n \setminus D)$ and, according to (3.11) and (3.14),
$$
\begin{equation*}
f(u)=-\lim_{\nu \to\infty} h_{D} (u_\nu, v)= \int_{\partial G} \sum_{j=1}^n (-1)^{j-1} \overline{(\overline \partial_j v (\zeta))}u(\zeta) \, d\overline \zeta [j]\wedge d\zeta=\langle u, v \rangle,
\end{equation*}
\notag
$$
that is the map (3.5) is surjective. Finally, in follows from the closed graph theorem for $\mathrm{DF}$-spaces (see [30], Ch. 6, or [34], Corollary A.6.4) that the inverse map of (3.5) is also continuous, that is, (3.5) is an (antilinear) topological isomorphism. Theorem 1 is proved. § 4. Various observationsAs mentioned in the introduction, for a bounded domain $D \Subset{\mathbb C}^n$ with real analytic boundary such that the space ${\mathcal O}(\overline D)$ is dense in ${\mathcal O}(D)$, Theorem 1 can be extracted from [9], where the duality (1.2) was established for a quite special duality relation related to (3.1) (cf. also [6]–[8] and see [19] for a duality (1.2) constructed using other pairings in domains with real analytic boundaries). The considerable difference arising in the assumptions on $D$ in this particular case is due to the fact that the proof in [9] uses the Grothendieck duality
$$
\begin{equation*}
({\mathcal H} (D ))^* \cong{\mathcal H} (\widehat{\mathbb C}^{n} \setminus D ),
\end{equation*}
\notag
$$
which follows from Theorem 4 in [15] for each bounded domain with connected complement, and the continuous map
$$
\begin{equation*}
\widetilde{\mathcal P}_{D} \colon {\mathcal O} (\overline D) \to{\mathcal H} (\widehat{\mathbb C}^{n} \setminus D ),
\end{equation*}
\notag
$$
provided by the results in [35] for domains with real analytic boundaries (of course, this latter is impossible for all bounded domain, even for ones with $C^\infty$-smooth boundaries). As follows from [9], Theorem 8.1, the duality relation in [9] induces a topological isomorphism (1.2) if and only if the space ${\mathcal O} (\overline D)$ is dense in ${\mathcal O}(D)$. This observation shows a way to the proof of the corresponding result for the map (3.5). Corollary 2. Let $D\subset{\mathbb C}^n$, $n>1$, be a bounded Lipschitz domain with connected complement. Then the pairing (3.1) induces a topological isomorphism (1.4) if and only if the space $H^1 (D)\cap{\mathcal O}(D)$ is dense in ${\mathcal O} (D)$. Proof. In view of Theorem 1 we must prove that $H^1 (D)\cap{\mathcal O}(D) $ is dense in ${\mathcal O}(D) $ under the assumptions of the corollary. With this aim in mind let $F$ be a continuous linear functional on ${\mathcal O}(D) $ which vanishes on $H^1 (D)\cap{\mathcal O}(D) $. By the Hahn–Banach theorem we will prove our theorem, once we will have shown that $F\equiv 0$. By assumption there exists $v\in \Sigma (\widehat{\mathbb C}^n \setminus D)$ such that $\langle v, u\rangle=0$ for all $u\in H^1 (D)\cap{\mathcal O}(D) $. As follows from Corollary 1, we have ${\mathcal P}_D v\in H^1 (D)\cap{\mathcal O}(D)$, so that ${\langle v,{\mathcal P}_D v\rangle=0}$. Hence the arguments analogous to the ones used in the proof of injectivity in Theorem 1 show that $v=0$ in $D$. Thus, $F=0$, as required.
The proof is complete. 
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\Bibitem{KhoShl24} |
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