Pointwise multipliers in Hardy-Orlicz spaces, and interpolation

TL;DR

This paper investigates the structure of multiplier algebras of Hardy-Orlicz spaces, revealing how they relate to classical Hardy spaces and providing examples that challenge existing characterizations of interpolating sequences.

Contribution

It characterizes multipliers of intermediate Hardy-Orlicz spaces, showing their relation to defining functions and illustrating how multiplier algebras can differ from the underlying spaces.

Findings

Multiplier algebras can grow from $H^{ ext{infty}}$ to big Hardy-Orlicz spaces.

Examples show that multiplier algebras do not always preserve the ordering of Hardy-Orlicz spaces.

Constructed Hardy-Orlicz spaces with non-Carleson interpolating sequences.

Abstract

We study multipliers of Hardy-Orlicz spaces $\mH_{\Phi}$ 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 $\mH_{\Phi}$ 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 $\Phi$. Even if the results do not entirely characterize the multiplier algebra, some examples show that we are not very far from precise conditions. In certain situations we see how the multiplier algebra grows in a sense from $\Hi$ 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 $\Hi$ 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 $H^p$, $p>0$.