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Multipliers of fractional Cauchy transforms and smoothness conditions


Published:19980601
Printed: Jun 1998
Donghan Luo
Thomas MacGregor
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Abstract
This paper studies conditions on an analytic function that imply it
belongs to ${\cal M}_\alpha$, the set of multipliers of the family of
functions given by $f(z) = \int_{\zeta=1} {1 \over
(1\overline\zeta z)^\alpha} \,d\mu (\zeta)$ $(z<1)$ where $\mu$ is a
complex Borel measure on the unit circle and $\alpha >0$. There are
two main theorems. The first asserts that if $0<\alpha<1$ and
$\sup_{\zeta=1} \int^1_0 f'(r\zeta) (1r)^{\alpha1} \,dr<\infty$
then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha
\leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {f(e^{i(t+s)}) 
2f(e^{it}) + f(e^{i(ts)}) \over s^{2\alpha}} \, ds < \infty$ then
$f \in {\cal M}_\alpha$. The conditions in these theorems are shown
to relate to a number of smoothness conditions on the unit circle for
a function analytic in the open unit disk and continuous in its closure.
MSC Classifications: 
30E20, 30D50 show english descriptions
Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part
30E20  Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 30D50  Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part
