Multipliers of families of Cauchy-Stieltjes transforms

Abstract:

For $\alpha > 0$ let ${\mathcal {F}_\alpha }$ denote the class of functions defined for $|z| < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $|x|= 1$. A function $g$ holomorphic in $|z| < 1$ is a multiplier of ${\mathcal {F}_\alpha }$ if $f \in {\mathcal {F}_\alpha }$ implies $gf \in {\mathcal {F}_\alpha }$. The class of all such multipliers is denoted by ${\mathcal {M}_\alpha }$. Various properties of ${\mathcal {M}_\alpha }$ are studied in this paper. For example, it is proven that $\alpha < \beta$ implies ${\mathcal {M}_\alpha } \subset {\mathcal {M}_\beta }$, and also that ${\mathcal {M}_\alpha } \subset {H^\infty }$. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if $f’$ is in the Hardy class ${H^1}$, then $f \in {\mathcal {M}_1}$. Also the theorem is improved to $f’ \in {H^1}$ implies $f \in {\mathcal {M}_\alpha }$, for all $\alpha > 0$. Finally, let $\alpha > 0$ and let $f$ be holomorphic in $|z| < 1$. It is known that $f$ is bounded if and only if its Cesàro sums are uniformly bounded in $|z| \leq 1$. This result is generalized using suitable polynomials defined for $\alpha > 0$.