1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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). $\endgroup$ Feb 18, 2015 at 10:14

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.