Banach Journal of Mathematical Analysis

Carleson measures on circular domains and canonical embeddings of Hardy spaces into function lattices

Paweł Mleczko and Michał Rzeczkowski

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Abstract

We study general variants of spaces of holomorphic functions on circular domains on the complex plane. We define Hardy-type spaces generated by Banach function lattices, for which we prove the Carleson theorem. We also analyze canonical embeddings of such spaces into appropriate function lattices. Finally, we study composition operators on Hardy-type spaces on circular domains and characterize order-boundedness of such maps.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 4 (2019), 864-883.

Dates
Received: 25 September 2018
Accepted: 28 February 2019
First available in Project Euclid: 9 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1570608166

Digital Object Identifier
doi:10.1215/17358787-2019-0013

Mathematical Reviews number (MathSciNet)
MR4016901

Zentralblatt MATH identifier
07118766

Subjects
Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
Hardy spaces rearrangement-invariant spaces composition operators Carleson measures

Citation

Mleczko, Paweł; Rzeczkowski, Michał. Carleson measures on circular domains and canonical embeddings of Hardy spaces into function lattices. Banach J. Math. Anal. 13 (2019), no. 4, 864--883. doi:10.1215/17358787-2019-0013. https://projecteuclid.org/euclid.bjma/1570608166


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References

  • [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988.
  • [2] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • [3] I. Chalendar and J. R. Partington, Approximation problems and representations of Hardy spaces in circular domains, Studia Math. 136 (1999), no. 3, 255–269.
  • [4] I. Chalendar and J. R. Partington, Interpolation between Hardy spaces on circular domains with applications to approximation, Arch. Math. (Basel) 78 (2002), no. 3, 223–232.
  • [5] J. B. Conway, Functions of One Complex Variable, II, Grad. Texts in Math. 159, Springer, New York, 1995.
  • [6] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995.
  • [7] P. L. Duren, Theory of $H^{p}$ Spaces, Pure Appl. Math. 38, Academic Press, New York, 1970.
  • [8] S. D. Fisher, Function Theory on Planar Domains: A Second Course in Complex Analysis, Wiley, New York, 1983.
  • [9] J. B. Garnett and D. E. Marshall, Harmonic Measure, New Math. Monogr. 2, Cambridge Univ. Press, Cambridge, 2005.
  • [10] S. G. Kreĭn, Y. I. Petunīn, and E. M. Semënov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, 1982.
  • [11] P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez-Piazza, Composition operators on Hardy–Orlicz spaces, Mem. Amer. Math. Soc. 207 (2010), no. 974.
  • [12] M. Lengfield, Duals and envelopes of some Hardy–Lorentz spaces, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1401–1409.
  • [13] M. Mastyło and P. Mleczko, Absolutely summing multipliers on abstract Hardy spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 6, 883–902.
  • [14] M. Mastyło and L. Rodríguez-Piazza, Carleson measures and embeddings of abstract Hardy spaces into function lattices, J. Funct. Anal. 268 (2015), no. 4, 902–928.
  • [15] P. Mleczko and R. Szwedek, Interpolation of Hardy spaces on circular domains, Math. Nachr. 290 (2017), no. 14–15, 2322–2333.
  • [16] Z. Qiu, Carleson measures on circular domains, Houston J. Math. 31 (2005), no. 4, 1199–1206.
  • [17] B.-Z. A. Rubshtein, M. A. Muratov, G. Y. Grabarnik, and Y. S. Pashkova, Foundations of Symmetric Spaces of Measurable Functions, Dev. Math. 45, Springer, Cham, 2016.
  • [18] W. Rudin, Analytic functions of class $H_{p}$, Trans. Amer. Math. Soc. 78 (1955), 46–66.
  • [19] M. Rzeczkowski, Composition operators on Hardy–Orlicz spaces on planar domains, Ann. Acad. Sci. Fenn. Math. 42 (2017), no. 2, 593–609.
  • [20] M. Rzeczkowski, Classical properties of composition operators on Hardy–Orlicz spaces on planar domains, J. Aust. Math. Soc. 107 (2018), no. 2, 256–271.
  • [21] D. Sarason, The $H^{p}$ spaces of an annulus, Mem. Amer. Math. Soc. 56 (1965), 1–78.
  • [22] Q. Xu, Notes on interpolation of Hardy spaces, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 875–889.