Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Amar, Eric; Hartmann, Andreas

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 1871-1903.

Nous étudions des relations entre l’uniforme minimalité, l’inconditionnalité et l’interpolation pour des familles de noyaux reproduisants dans des espaces invariants par l’adjoint du shift. Cette classe d’espaces contient en particulier les espaces de Paley-Wiener pour lesquels il est connu que l’uniforme minimalité n’entraîne en général pas l’inconditionalité. Par conséquent, et contrairement à la situation dans les espaces de Hardy habituels (et dans d’autres échelles d’espaces), il semble nécessaire de changer la taille de l’espace afin de déduire l’inconditionnalité (ou l’interpolation) de l’uniforme minimalité. Un tel changement de la taille de l’espace peut être opéré de deux façons différentes : en diminuant l’exposant d’intégration, ou en “augmentant” la fonction définissante de l’espace (ce qui revient à augmenter le type dans le cas des espaces de Paley-Wiener). Les inégalités de Khinchin jouent un rôle central dans les preuves de nos résultats principaux.

We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and of other scales of spaces), changing the size of the space seems necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two directions: lowering the power of integration, or “increasing” the defining inner function (e.g. increasing the type in the case of Paley-Wiener space). Khinchin’s inequalities play a substantial role in the proofs of our main results.

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