Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of slow growth, which is defined in the following way. We endow $M$ with some metric compatible with is topology, and for each positive integer $k$ consider the Banach space $\mathcal{A}^{-k}(\Omega,X)$ of holomorphic functions $f$ on $\Omega$ with values in the Banach space $X$ for which there is a constant $C>0$ such that for $z\in \Omega$, we have

$$ \Vert{f(z)}\Vert_X\leq \frac{C}{{\rm dist}(z,\partial\Omega)^k}.$$

Then $\mathcal{A}^{-k}(\Omega, X)$ is a Banach space with the norm

$$ \Vert{f}\Vert_{\mathcal{A}^{-\infty}}=\sup_{z\in \Omega}\left\{\Vert{f(z)}\Vert_X\cdot {{\rm dist}(z,\partial\Omega)^k}\right\} $$

We define $\mathcal{A}^{-\infty}(\Omega, X)$ to be the inductive limit of the spaces $\mathcal{A}^{-k}(\Omega,X)$.

There is another way of constructing $X$-valued holomorphic functions of slow growth on $\Omega$. We begin with the space $\mathcal{A}^{-\infty}(\Omega)= \mathcal{A}^{-\infty}(\Omega,\mathbb{C})$ and form its injective tensor product $\mathcal{A}^{-\infty}(\Omega)\hat{\otimes}_\epsilon X$ with the Banach space $X$. My first question is:

(1) Is it true that the space $\mathcal{A}^{-\infty}(\Omega,X)$ and the space $\mathcal{A}^{-\infty}(\Omega) \hat{\otimes}_\epsilon X$ are isomorphic as topological vector spaces?

I suspect that the answer is yes.

(2) Does it help to have $\mathcal{A}^{-\infty}(\Omega)$ nuclear (I know this happens when $\Omega$ has smooth boundary.)

I don't know much functional analysis but I need to understand these spaces for some application to a problem on holomorphic extension of distributions.

  • $\begingroup$ Such questions have been investigated e.g. by K.-D. Bierstedt and R. Meise in the early 1970s. Your case might be contained in Meise's article Räume holomorpher Vektorfunktionen mit Wachstumsbedingungen und topologische Tensorprodukte (Math. Ann. 199 (1972), 293–312). $\endgroup$ – Jochen Wengenroth Feb 18 '15 at 10:14

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