## Duals to weighted spaces of analytic functions.(Russian, English)Zbl 0976.46011

Sib. Mat. Zh. 42, No. 1, 3-17 (2001); translation in Sib. Math. J. 42, No. 1, 1-14 (2001).
Let $$D$$ be a bounded convex domain in $$\mathbb C^p$$, $$0\in D$$, and let $$\{u_n(z)\}$$ be a decreasing sequence of convex functions defined on $$D$$ such that $$u_n(z)\to\infty$$ as $$z\to\partial D$$, $$z\in D$$. Denote by $$H$$ the space of analytic functions on $$D$$ satisfying the condition $|f(z)|e^{-|u_n(z)|}\to 0\quad \text{ for all }n = 1,2,\dots$ as $$\operatorname{dist}(z,\partial D)\to 0$$. The space $$H$$ is endowed with the topology of inductive limit of $$B$$-spaces. The following theorem by Epifanov describes the image of the dual space $$H^*$$ under the Laplace transformation:
Theorem (Epifanov). Let $$v_n(\lambda) = \sup_{z\in D}(\operatorname{Re} \langle\lambda,z\rangle - u_n(z))$$, $$\lambda\in\mathbb C^p$$, $$n = 1,2,\dots$$, and let $$P$$ be the space of entire functions $$F(\lambda)$$ with the topology of inductive limit such that, for some $$n = n(F)$$, $\|F\|_n = \sup_{\lambda\in\mathbb C^p}|F(\lambda)|e^{-v_n(\lambda)} < \infty.$ If the condition $v_{n+1}(\lambda) \geq v_n(\lambda) + \ln(1 + |\lambda|) + c_n$ holds, then the Laplace transformation $$L$$ determines a topology isomorphism from the strong dual space $$H^*$$ onto the space $$P$$ and, conversely, the operator $$L$$ maps isomorphically $$P^*$$ onto $$H$$.
The aim of the article is to prove the Epifanov theorem under the assumption that the weighted functions $$v_n\in C^2(\mathbb C^p)$$ satisfy the so-called “accuracy” growth condition. The proof of this theorem is based on using the Hörmander theorem.

### MSC:

 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 30D55 $$H^p$$-classes (MSC2000) 30D20 Entire functions of one complex variable (general theory) 30H05 Spaces of bounded analytic functions of one complex variable 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
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