AN INTEGRAL OPERATOR ON H v AND HARDY'S INEQUALITY ALEXANDRUALEMAN AND JOSEPH A. CIMA 1 Introduction One of the first integral operators defined on the Hardy spaces of the unit disc was the primitive of a function in the Hardy space. Hardy and Littlewood showed that for 0 < p < 1 taking the primitive of an H v function f yields a function in Hq with q = p/(1 -p). This result is sharp. Among the other integral operators defined and investigated on Hv since then is the Ces/tro operator (cf. [5], [7]), which became the paradigm for an integral operator considered by Aleman and Siskakis in [1]. This operator T9 depends on an analytic symbol g and is defined as follows: iff and g are analytic functions on the unit disc D and z is in D, let /: Tj(z) = f(r An important step in their work is to show that T9 is bounded on H p for p _> 1 if and only if the symbol g is a BMOA function on the disc. For p = 2, this had been proved earlier by Pommerenke [6]; and in their study [3, p. 144 and p. 160], Coifman and Meyer generalized Pommerenke's lemma to a large class of integral operators which do not necessarily act on spaces of analytic functions. In this paper, we continue to investigate the behaviour of these operators on HV spaces for arbitrary p > 0. More precisely, we consider the following general problem: given p, q 9 (0, co), characterize the symbols g for which T9 maps Hv into Hq. Of course, by the closed graph theorem, this happens exactly when To maps HP boundedly into Hq. Our main theorem gives a complete answer to this question. In particular, it extends the result in [1] for all p > 0 and yields an improvement of the Hardy-Littlewood Theorem. Theorem 1. Let p, q > O. Then we have the following. (i) For p > q, T 9 maps g p into Hq if and only if g 9 H s, where 1/s = 1/q - 1/p. 157 JOURNAL D'ANALYSE MATHI~MATIQUE, VOI. 85 (2001)