Banach envelopes of vector valued \(H^p\) spaces. (English) Zbl 1030.46022
Let \(X\) be a quasi-Banach space and \(\|\cdot \|_X: X\to [0,\infty[\) a \(q\)-norm on \(X\) which generates its topology. For \(0<p<1\) and \(d \in\mathbb{N}\) denote by \(H^p(\mathbb{D}^d,X)\) the Hardy space on the polydisk \(\mathbb{D}^d\) with values in \(X\) and define \(B_p(\mathbb{D}^d,X)\) as the space of all analytic functions \(F:\mathbb{D}^d\to X\) for which
\[
\|F\|_{B_p(\mathbb{D}^d,X)}: = \int_{\mathbb{D}^d} \bigl\|F(z)\bigr \|_X\prod^d_{j=1} \bigl(1-|z_j |^2 \bigr)^{{1\over p}-2} dA_d(z)
\]
is finite, where \(A_d\) denotes the normalized Lebesgue measure on \(\mathbb{D}^d\). \(B_p(\mathbb{D}^d,X)\) is endowed with the quasi-norm \(\|\;\|_{B_p (\mathbb{D}^d,X)}\). The main result of the paper under review is to show that the Banach envelope \(\widetilde{H^p (\mathbb{D}^d,X)}\) of \(H^p(\mathbb{D}^d,X)\) is equal to \(B_p(\mathbb{D}^d, \widetilde X)\), where \(\widetilde X\) denotes the Banach envelope of \(X\).
Reviewer: Reinhold Meise (Düsseldorf)
MSC:
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |
| 32A35 | \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables |
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