Multipliers of Families of Cauchy-Stieltjes Transforms

R. A. Hibschweiler, T. H. MacGregor
1992 Transactions of the American Mathematical Society  
For a > 0 let ^ denote the class of functions defined for \z\ < 1 by integrating 1/(1 -xz)a against a complex measure on |jc| = 1 . A function g holomorphic in |z| < 1 is a multiplier of !?a if / e&a implies gf £ ^a . The class of all such multipliers is denoted by J!a . Various properties of Jta are studied in this paper. For example, it is proven that a < ß implies ¿#a C Jtß , and also that Jta C H°° . Examples are given of bounded functions which are not multipliers. A new proof is given of
more » ... theorem of Vinogradov which asserts that if /' is in the Hardy class Hl , then / € Jlx . Also the theorem is improved to f'eH1 implies / 6 Jta , for all a > 0 . Finally, let a > 0 and let / be holomorphic in \z\ < 1. It is known that / is bounded if and only if its Cesàro sums are uniformly bounded in |z| < 1 . This result is generalized using suitable polynomials defined for a > 0.
doi:10.2307/2154014 fatcat:c5a6yupsnjggndg7o3vng36mma