A famous theorem of Hardy asserts that if f ∊ H1, then the sequence
of Fourier coefficients satisfies
. For this reason we say that the sequence (1, 1/2, 1/3, …) belongs to the multiplier class (H1, l1). In this paper, we investigate the multiplier classes (Hp, l1) for 1 ≧ p ≧ ∞. Our observations are based on the fact that a sequence (λ(0), λ(l), …) belongs to (Hp, l1) independent of the arguments of its terms. We also show that (Hp, l1) may be thought of as the conjugate space of a certain Banach space.
1. Preliminaries.Lp denotes the space of complex-valued Lebesgue measurable functions f defined on the circle |z| = 1 such that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00041468/resource/name/S0008414X00041468_eqn1.gif?pub-status=live)
is finite.