The Growth of Derivatives of Multipliers of Fractional Cauchy Transforms

In this paper we prove a number of sharp results on the permissible growth of derivatives of multipliers of fractional Cauchy transforms. For example, we prove that if fgM1 then there exists a positive constant C such that p 1 1 H < f 9 reiu .< du FC5 f 5 M1 1yr 1qlog 1r 1yr yp .. for rgw0, 1.. This result is proved to be sharp