BMO- and VMO-spaces of slice hyperholomorphic functions
Corresponding Author
Jonathan Gantner
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy
Correspondence
Jonathan Gantner, Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy.
Email: [email protected]
Search for more papers by this authorJ. Oscar González-Cervantes
Departamento de Matemáticas, E.S.F.M. del I.P.N., 07338 Mexico, D.F., Mexico
Search for more papers by this authorTim Janssens
University of Antwerp, Department of Mathematics, Middelheimlaan, 1, 2020 Antwerp, Belgium
Search for more papers by this authorCorresponding Author
Jonathan Gantner
Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy
Correspondence
Jonathan Gantner, Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy.
Email: [email protected]
Search for more papers by this authorJ. Oscar González-Cervantes
Departamento de Matemáticas, E.S.F.M. del I.P.N., 07338 Mexico, D.F., Mexico
Search for more papers by this authorTim Janssens
University of Antwerp, Department of Mathematics, Middelheimlaan, 1, 2020 Antwerp, Belgium
Search for more papers by this authorAbstract
In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO- and VMO-spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space . The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.
References
- 1S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, International Series of Monographs on Physics Volume 88, Oxford University Press, New York, 1995.
- 2D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, American Mathematical Society, Providence, 2001. Translated from the 1998 French original by Stephen S. Wilson, Panoramas et Synthèses.
- 3D. Alpay, F. Colombo, and D. P. Kimsey, The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum, J. Math. Phys. 57(2) (2016), 023503, 27 pp.
- 4D. Alpay, F. Colombo, D. P. Kimsey, and I. Sabadini, The spectral theorem for unitary operators based on the S-spectrum, Milan J. Math. 84(1) (2016), 41–61.
- 5D. Alpay, F. Colombo, I. Lewkowicz, and I. Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, Milan J. Math. 83(1) (2015), 91–144.
- 6D. Alpay, F. Colombo, and I. Sabadini, Pontryagin–de Branges–Rovnyak spaces of slice hyperholomorphic functions, J. Anal. Math. 121 (2013), 87–125.
- 7D. Alpay, F. Colombo, and I. Sabadini, Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72(2) (2012), 253–289.
- 8D. Alpay, F. Colombo, and I. Sabadini, Slice Hyperholomorphic Schur Analysis, Operator Theory: Advances and Applications, Volume 256, Springer, Basel, 2017.
- 9D. Alpay, F. Colombo, I. Sabadini, and G. Salomon, The Fock space in the slice hyperholomorphic setting, In: Hypercomplex Analysis: New Perspectives and Applications, Trends in Mathematics, Birkhäuser, Basel, 2014, pp. 43–59.
- 10L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
10.1093/oso/9780198502456.001.0001 Google Scholar
- 11N. Arcozzi and G. Sarfatti, From Hankel operators to Carleson measures in a quaternionic variable, preprint, arXiv:1407.8479 [math.CV].
- 12L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559.
- 13C. M. P. Castillo Villalba, F. Colombo, and J. Gantner, Bloch, Besov and Dirichlet spaces of slice hyperholomorphic functions, Complex Anal. Oper. Theory 9(2) (2015), 479–517.
- 14F. Colombo, J. O. González-Cervantes, M. E. Luna-Elizarraras, I. Sabadini, and M. Shapiro, On Two Approaches to the Bergman Theory for Slice Regular Functions, In: Advances in Hypercomplex Analysis, Springer INdAM Series Volume 1, Springer, Milan, 2013, pp. 39–54.
10.1007/978-88-470-2445-8_3 Google Scholar
- 15F. Colombo, J. O. González-Cervantes, and I. Sabadini, Further properties of the Bergman spaces of slice regular functions, Adv. Geom. 15(4) (2015), 469–484.
- 16F. Colombo, J. O. González-Cervantes, and I. Sabadini, On slice biregular functions and isomorphisms of Bergman spaces, Complex Var. Elliptic Equ. 58(7–8) (2013), 1355–1372.
- 17F. Colombo, J. O. González-Cervantes, and I. Sabadini, The C-property for slice regular functions and applications to the Bergman space, Complex Var. Elliptic Equ. 58(10) (2013), 1355–1372.
- 18F. Colombo, I. Sabadini, and D. C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254(8) (2008), 2255–2274.
- 19F. Colombo, I. Sabadini, and D. C. Struppa, Entire Slice Regular Functions, Volume of SpringerBriefs in Mathematics, Springer International Publishing, 2016, 10.1007/978-3-319-49265-0.
10.1007/978-3-319-49265-0 Google Scholar
- 20F. Colombo, I. Sabadini, and D. C. Struppa, Noncommutative Functional Calculus, Theory and Applications of Slice Hyperholomorphic Functions, Progress in Mathematics Volume 289, Birkhäuser, Basel, 2011.
- 21F. Colombo, I. Sabadini, and D. C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009), 385–403.
- 22N. Danikas, Some Banach spaces of analytic functions, In: Function Spaces and Complex Analysis R. Aulaskari and I. Laine (Eds.), Univ. Joensuu Dept. Math. Rep. Ser. Volume 2, Univ. Joensuu, Joensuu, 1999, pp. 9–35.
- 23R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 4(1) (1932), 9–20.
10.1007/BF01202702 Google Scholar
- 24R. Fueter, Die Funktionentheorie der Differentialgleichungen
and
mit vier reellen Variablen, Comment. Meth. Helv. 7(1) (1934), 307–330.
10.1007/BF01292723 Google Scholar - 25G. Gentili, C. Stoppato, and D. C. Stuppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Berlin-Heidelberg, 2013.
10.1007/978-3-642-33871-7 Google Scholar
- 26G. Gentili and D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris 342(10) (2006), 741–744.
- 27R. Ghiloni, V. Moretti, and A. Perotti, Continuous slice functional calculus in quatenrionic Hilbert spaces, Rev. Math. Phys. 25(4) (2013), 1350006, 83 pp.
- 28D. Girela, Analytic functions of bounded mean oscillation, In: Complex Function Spaces (Mekrijärvi, 1999). Vol. 4. Univ. Joensuu Dept. Math. Rep. Ser. Univ. Joensuu, Joensuu, 2001, 61–170.
- 29I. Sabadini and A. Saracco, Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball, to appear in: J. London Math. Soc. (2), https://doi.org/10.1112/jlms.12035.
- 30G. Sarfatti, Elements of Function Theory in the Unit Ball of Quaternions, PhD Thesis, Università degli Studi di Firenze, 2013.
- 31C. Stoppato, Regular Moebius transformations of the space of quaternions, Ann. Global Anal. Geom. 39(4) (2011), 387–401.
- 32K. Zhu, Operator Theory in Function Spaces, Second Edition, Mathematical Surveys and Monographs Volume 138, American Mathematical Society, Providence, 2007.
10.1090/surv/138 Google Scholar