Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T10:17:44.048Z Has data issue: false hasContentIssue false
  • English
  • Français

CARLESON INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS

Part of: Spaces and algebras of analytic functions Miscellaneous topics of analysis in the complex domain Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  29 March 2021

Mikael Lindström Open the ORCID record for Mikael Lindström [Opens in a new window] ,
Mieczysław Mastyło Open the ORCID record for Mieczysław Mastyło [Opens in a new window] ,
Paweł Mleczko Open the ORCID record for Paweł Mleczko [Opens in a new window] ,
David Norrbo Open the ORCID record for David Norrbo [Opens in a new window]  and
Michał Rzeczkowski Open the ORCID record for Michał Rzeczkowski [Opens in a new window]
Mikael Lindström
Affiliation:
Department of Mathematics, Åbo Akademi University, FI-20500 Åbo, Finland ([email protected])
Mieczysław Mastyło
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland ([email protected])
Paweł Mleczko
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland ([email protected])
David Norrbo
Affiliation:
Department of Mathematics, Åbo Akademi University, FI-20500 Åbo, Finland ([email protected])
Michał Rzeczkowski
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result for H∞ can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spaces Hp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.

Keywords

Carleson interpolating sequencesinterpolation of operatorsHardy spaces

MSC classification

Primary: 30E05: Moment problems, interpolation problems
Secondary: 46B70: Interpolation between normed linear spaces 30H10: Hardy spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 1915 - 1945
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agler, J. and McCarthy, J. E., Pick Interpolation and Hilbert Function Spaces (American Mathematical Society, Providence, RI, 2002).CrossRefGoogle Scholar
Akilov, G. P. and Kantorovich, L. V., Functional Analysis, 2nd ed. (Pergamon Press, Oxford-Elmsford, NY, 1982).Google Scholar
Amar, É., Extension de fonctions holomorphes et courants, Bull. Sci. Math. 107 (1983), 25–48.Google Scholar
Ambrosie, C. and Müller, V., Invariant subspaces for polynomially bounded operators, J. Funct. Anal. 213 (2004), 321–345.CrossRefGoogle Scholar
Bennett, C. and Sharpley, R., Interpolation of Operators (Academic Press, Boston, 1988).Google Scholar
Bergh, J. and Löfström, J., Interpolation Spaces: An Introduction (Springer-Verlag, Berlin, 1976).CrossRefGoogle Scholar
Brudnyi, Yu. A. and Krugljak, N. Ya., Interpolation Functors and Interpolation Spaces I (North-Holland, Amsterdam, 1991).Google Scholar
Calderón, A. P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.CrossRefGoogle Scholar
Carleson, L., An interpolation problem for analytic functions, Amer. J. Math. 80 (1958), 921–930.CrossRefGoogle Scholar
Clark, D. N., On interpolating sequences and the theory of Hankel and Toeplitz matrices, J. Funct. Anal. 5 (1970), 247–258.CrossRefGoogle Scholar
Duren, P., Theory of ${H}^p$ Spaces (Academic Press, New York, 1970).Google Scholar
Garnett, J. B., Bounded Analytic Functions (Springer, New York, 2007).Google Scholar
Georgiou, T. T. and Smith, M. C., Graphs, causality, and stabilizability: linear, shift-invariant systems on ${L}^2\left[0,\infty \right)$ , Math. Control Signals Systems 6(3) (1993), 195–223.CrossRefGoogle Scholar
Hartmann, A., The generalized Carleson condition in certain spaces of analytic functions, in Proceedings of the 13th International Conference on Banach Algebras, pp. 245–260 (Walter de Gruyter, Berlin, 1998).CrossRefGoogle Scholar
Hartmann, A., Free interpolation in Hardy–Orlicz spaces, Studia Math. 135 (1999), 179–190.CrossRefGoogle Scholar
Hoffman, K., Banach Spaces of Analytic Functions (Prentice Hall, Englewood Cliffs, NJ, 1962).Google Scholar
Jones, P. W., Interpolation between Hardy spaces, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II, pp. 437–451, Wadsworth Math. Ser. (Wadsworth, Belmont, CA, 1983).Google Scholar
Krein, S. G., Petunin, Yu. I. and Semenov, E. M., Interpolation of Linear Operators, Translations of Mathematical Monographs, 54 (American Mathematical Society, Providence, RI, 1982).Google Scholar
Lozanovskiĭ, G. Ya., On some Banach lattices IV , Sibirsk. Mat. Zh. 14 (1973), 140–155 (in Russian). English translation: Sib. Math. J. 14 (1973), 97–108.Google Scholar
Lozanovskiĭ, G. Ya., Transformations of ideal Banach spaces by means of concave functions, in Qualitative and Approximate Methods for the Investigation of Operator Equations, pp. 122–147 (Yaroslavl 1978) (in Russian).Google Scholar
Mastyło, M. and Rodríguez-Piazza, L., Carleson measures and embeddings of abstract Hardy spaces into function lattices, J. Funct. Anal. 268 (2015), 902–928.CrossRefGoogle Scholar
Nakazi, T., Interpolation of weighted ${\ell}^q$ sequences by ${H}^p$ functions, Taiwanese J. Math. 9(3) (2005), 457–467.CrossRefGoogle Scholar
Ovchinnikov, V. I., The method of orbits in interpolation theory, Math. Rep. (Bucur.) 1(2) (1984), i–x, 349–515.Google Scholar
Schuster, A. P. and Seip, K., A Carleson-type condition for interpolation in Bergman spaces, J. Reine Angew. Math. 497 (1998), 223–233.CrossRefGoogle Scholar
Seip, K., Interpolation and Sampling in Spaces of Analytic Functions, (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
Shapiro, H. S. and Shields, A. L., On some interpolation problems for analytic functions, Amer. J. Math. 83(3) (1961), 513–532.CrossRefGoogle Scholar
Sparr, G., Interpolation of weighted spaces, Studia Math. 62 (1978), 229–271.CrossRefGoogle Scholar
Taylor, B. A. and Williams, D. L., Interpolation of ${\ell}^q$ sequences by a ${H}^p$ functions, Proc. Amer. Math. Soc. 34(1) (1972), 181–186.Google Scholar
Xu, Q., Notes on interpolation of Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), 875–889.CrossRefGoogle Scholar