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Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$.

My question is how to characterize all such (Radon) measures $\mu$ on $G$, that $\int_{G} f d\mu=0$ for any $f\in H\left(G\right)$.

A little bit on the motivation. I am interested in some internal description of the dual to $H\left(G\right)$. The only description which I know at this moment is the one from Lueking/Rubel. There is only one-dimensional case considered and in this case the dual is shown to be isomorphic to the space of holomorphic functions on $\mathbb{C}^{d}\backslash G$. Since this description relies on the embedding of $G$ into $\mathbb{C}^{d}$, it is not internal.

Possibly the problems are not well-stated, but I believe that they are meaningful.

Thank you.

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  • $\begingroup$ Ciuld you please give a more precise definition of "internal"? Also, I'm not sure I believe the statement you've given of Luecking--Rubel, could you give a reference? $\endgroup$
    – Yemon Choi
    Commented Jul 10, 2014 at 10:22
  • $\begingroup$ Assume $d=1$. Formally we have $ \mathcal H(G)=\ker \overline \partial, $ so that $$ \bigl(\mathcal H(G)\bigr)^\perp=\bigl(\ker \overline \partial\bigr)^\perp=\overline{\text{ran }\partial }. $$ As a result the orthogonal of $\mathcal H(G)$ is the set of Radon measures $\mu$ of the form $$ \mu=\frac{\partial \nu}{\partial z}, $$ where $\nu$ is an hyperfunction in $U$ such that $\frac{\partial \nu}{\partial z}$ is a Radon measure. $\endgroup$
    – Bazin
    Commented Jul 11, 2014 at 15:37
  • $\begingroup$ @Bazin I don't see why the dual of $H(G)$ consists of measures as opposed to more general distributions -- your sketch does not tell me the domain and codomain of the d-bar operator so I cannot make sense of the perp and the closed range $\endgroup$
    – Yemon Choi
    Commented Jul 11, 2014 at 18:39
  • $\begingroup$ Downvoting until the OP clarifies what is meant by "the dual of H(G)" -- I assume it is what @blackburne describes in his or her answer, but this should really be specified in the question, especially given the claim that the dual coincides with a space of holomorphic functions $\endgroup$
    – Yemon Choi
    Commented Jul 11, 2014 at 18:41
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    $\begingroup$ It seemed clear to me that the question refers to the natural topology, that of uniform convergence on compacta. Since $H(G)$ is then a locally convex subspace of $C(G)$ with the same topology, then by Hahn- Banach its dual is the quotient of the space of Radon measures on $G$ with compact support by the annihilator of $H(G)$ and this is, presumably, the motivation for the question $\endgroup$
    – blackburne
    Commented Jul 11, 2014 at 19:30

2 Answers 2

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This is more of an extended comment than an answer. Let me write $M(U)$ for the space of measures on $U$, and $D(U)$ for the space of distributions. If $V\subseteq U$ we can certainly define a restriction map $M(U)\to M(V)$, and I think this makes $M$ a sheaf, or at least it satisfies the sheaf axiom for finite covers. Moreover, if $\mu\in M(U)$ and $f\in C(V)$ has compact support and $f^U\in C(U)$ is the extension by zero then $\langle\mu|_V,f\rangle=\langle\mu,f^U\rangle$; this links the sheaf structure of $M$ to that of $C$, and we can exploit it using partitions of unity. There is a similar story for $D$ and $C^\infty$. I guess that some analogue of this would qualify as an "internal" description of $H(U)^*$. However, the theory would have a very different flavour because the restriction maps $H(U)\to H(V)$ are injective (provided that $U$ is connected), there is no obvious way to define restriction maps for $H(U)^*$, and there is no obvious analogue of the compact support condition, because holomorphic functions of compact support are zero.

As a more direct example for your original question, note that Cauchy's integral theorem says that for every nullhomotopic closed curve in $U$, we have a measure concentrated on that curve which is orthogonal to all holomorphic functions.

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There is a result of this nature in the one dimensional case but your version is inaccurate. The precise form is due to Gottfried Köthe and identifies the strong dual of $H(U)$ with the Silva space of functions which are holomorphic on the complement of $G$ in the Riemann sphere and which vanish at infinity. This appeared in Crelle's Journal in the early $50$'s and is readily accessible in his monograph on topological vector spaces. The case of functions on a compact Riemann surface was studied in detail by his student R. Kultze. The case of higher dimensions is obviously much more subtle and I, too, would be curious to know if serious work has been done on it. Two simple remarks: you will need some conditions on $G$ in higher dimensions, e.g., that it is a domain of holomorphy. Secondly, if you consider the special case of a product $G_1 \times G_2$ of two complex domains, then using Grothendieck's theory of tensor products, the dual is identifiable with a space of holomorphic functions on the product of their complements (in each case, in a copy of the Riemann sphere) so that even this simplest of cases doesn't produce a direct analogy for higher dimensions.

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