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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) (zU={|z|<1}) with f ' (z)>0 (zU). 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 p . (Note that f itself need not even be bounded!). One such transform is

F(z)=c 0 z (log(z/t) α-1 f(t)dt

for a specific c and any α>1). Such transforms make sense for a wider range of α and the authors ask if their conclusions are valid for such α.


MSC:
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)