Pointwise multipliers in Hardy-Orlicz spaces, and interpolation

Abstract

We study multipliers of Hardy-Orlicz spaces ${\mathcal H}_{\Phi}$ which are strictly contained between $\bigcup_{p>0}H^p$ and so-called "big" Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces $H^p$, the multipliers reduce to $H^{\infty}$. For Hardy-Orlicz spaces ${\mathcal H}_{\Phi}$ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to $\Phi$. In general it cannot be expected to obtain a characterization of the multiplier algebra in terms of Hardy-Orlicz spaces since these are in general not algebras. Nevertheless, some examples show that we are not very far from such a characterization. In certain situations we see how the multiplier algebra grows in a sense from $H^{\infty}$ to big Hardy-Orlicz spaces when we go from classical $H^p$ spaces to big Hardy-Orlicz spaces. However, the multiplier algebras are not always ordered as their underlying Hardy-Orlicz spaces. Such an ordering holds in certain situations, but examples show that there are large Hardy-Orlicz spaces for which the multipliers reduce to $H^{\infty}$ so that the multipliers do in general not conserve the ordering of the underlying Hardy-Orlicz spaces. We apply some of the multiplier results to construct Hardy-Orlicz spaces close to $\bigcup_{p>0}H^p$ and for which the free interpolating sequences are no longer characterized by the Carleson condition which is well known to characterize free interpolating sequences in $H^p$, $p>0$.