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Let 11' be ii non-negative radial function on R?\(o}. If 0 -=YJ -=-, let fl:, denote the spare of functions f. holomorphic i n the right half plane. such that is uniformly bounded for x =-O. It is a well known fact that if zc(x) = 1, then this condition suffices to insure the existence of boundary values o f f . Indeed, if the w ( x ) are certain power weights then ROONEY [I81 has shown that the uniform boundedness of ,u,,,,(f; x) implies the existence of boundary values. More recently, GARCIA-CUERVA [6] considered weighted HARDY spaces and has shown that if w Trans.

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