EMBEDDING DERIVATIVES OF WEIGHTED HARDY-SPACES INTO LEBESGUE SPACES

Let 1 less-than-or-equal-to p < infinity and let w be a non-negative function defined on the unit circle T which satisfies the A(p) condition of Muckenhoupt. The weighted Hardy space H(p)(w) consists of those functions f in the classical Hardy space H-1 whose boundary values belong to L(p)(w). Recently McPhail (Studia Math. 96, 1990) has characterized those positive Borel measures mu on the, unit disc DELTA for which H(p)(w) is continuously contained in L(p)(dmu). In this paper we study the question of finding necessary and sufficient conditions on a positive Borel measure It on A for the differentiation operator D defined by Df = f' to map HP(w) continuously into L(p)(dmu). We prove that a necessary condition is that there exists a positive constant C such that mu(S(I)) less-than-or-equal-to C\I\p integral-I w(e(itheta))dtheta, for every interval I subset-of T, (A) where for any interval I subset-of T, S(I)={z=re(itheta):e(itheta) is-an-element-of I, 0 < 1 - r less-than-or-equal-to min(1,\I\)}. We prove that this condition is also sufficient in some cases, namely for 2 less-than-or-equal-to p < infinity and w(e(itheta)) = \theta\alpha, (\theta\ less-than-or-equal-to pi), -1 < alpha < p-1, but not in general. In the general case we prove the sufficiency of a condition which is slightly stronger than (A).