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# Multipliers of fractional Cauchy transforms and smoothness conditions

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 $\sup_{|\zeta|=1} \int^1_0 |f'(r\zeta)| (1-r)^{\alpha-1} \,dr<\infty$ then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha \leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {|f(e^{i(t+s)}) - 2f(e^{it}) + f(e^{i(t-s)})| \over s^{2-\alpha}} \, ds < \infty$ 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.