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Multipliers of Families of Cauchy-Stieltjes Transforms

1992
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Transactions of the American Mathematical Society
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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

doi:10.2307/2154014
fatcat:c5a6yupsnjggndg7o3vng36mma