Pointwise multipliers in Hardy-Orlicz spaces, and interpolation

• Autores: Andreas Hartmann
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 106, Nº 1, 2010, págs. 107-141
• Idioma: inglés
• DOI: 10.7146/math.scand.a-15128
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• Resumen

  • We study multipliers of Hardy-Orlicz spaces HΦ which are strictly contained between ⋃p>0Hp 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 Hp, the multipliers reduce to H∞. For Hardy-Orlicz spaces HΦ 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 Φ. 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∞ to big Hardy-Orlicz spaces when we go from classical Hp 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∞ 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 ⋃p>0Hp 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 Hp, p>0.