Multipliers of families of Cauchy-Stieltjes transforms

Abstract

For α > 0 \alpha > 0 let F α {\mathcal {F}_\alpha } denote the class of functions defined for | z | > 1 |z| > 1 by integrating 1 / ( 1 − x z ) α 1/{(1 - xz)^\alpha } against a complex measure on | x | = 1 |x|= 1 . A function g g holomorphic in | z | > 1 |z| > 1 is a multiplier of F α {\mathcal {F}_\alpha } if f ∈ F α f \in {\mathcal {F}_\alpha } implies g f ∈ F α gf \in {\mathcal {F}_\alpha } . The class of all such multipliers is denoted by M α {\mathcal {M}_\alpha } . Various properties of M α {\mathcal {M}_\alpha } are studied in this paper. For example, it is proven that α > β \alpha > \beta implies M α ⊂ M β {\mathcal {M}_\alpha } \subset {\mathcal {M}_\beta } , and also that M α ⊂ H ∞ {\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 ′ f’ is in the Hardy class H 1 {H^1} , then f ∈ M 1 f \in {\mathcal {M}_1} . Also the theorem is improved to f ′ ∈ H 1 f’ \in {H^1} implies f ∈ M α f \in {\mathcal {M}_\alpha } , for all α > 0 \alpha > 0 . Finally, let α > 0 \alpha > 0 and let f f be holomorphic in | z | > 1 |z| > 1 . It is known that f f is bounded if and only if its Cesàro sums are uniformly bounded in | z | ≤ 1 |z| \leq 1 . This result is generalized using suitable polynomials defined for α > 0 \alpha > 0 .