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The Hardy space of analytic functions associated with certain one-parameter families of integral operators. (English) Zbl 0774.30008
The authors consider classes of analytic functions $f(z)$ $(z\in U=\{\vert z\vert<1\})$ with $\Re\ f'(z)>0$ $(z\in U)$. They show (Theorem 1 and (2.16)) that the image of $f$ under a variety of integral transforms is continuous on the closed disk, and in particular belongs to all Hardy spaces $H\sb p$. (Note that $f$ itself need not even be bounded!). One such transform is $$F(z)=c\int\sb 0\sp z(\log(z/t)\sp{\alpha-1}f(t)\,dt$$ for a specific $c$ and any $\alpha>1)$. Such transforms make sense for a wider range of $\alpha$ and the authors ask if their conclusions are valid for such $\alpha$.

30D55H (sup p)-classes (MSC2000)
30C45Special classes of univalent and multivalent functions
46E20Hilbert spaces of continuous, differentiable or analytic functions
47B38Operators on function spaces (general)
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