Abstract
Let \({\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}\) denote the space of \({\mathcal{L}}\)-valued analytic functions \({\phi}\) for which the Hankel operator \({\Gamma_\phi}\) is \({H^2\left(\mathcal{H}\right)}\)-bounded. Obtaining concrete characterizations of \({\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}\) has proven to be notoriously hard. Let \({D^\alpha}\) denote fractional differentiation. Motivated originally by control theory, we characterize \({H^2\left(\mathcal{H}\right)}\)-boundedness of \({D^\alpha \Gamma_\phi}\), where \({\alpha > 0}\), in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that \({\textrm{BMOA}_{\mathcal{N}\mathcal{P}} \left(\mathcal{L}\right)}\) is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of \({\Gamma_\phi}\). The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aleksandrov A.B., Peller V.V.: Hankel operators and similarity to a contraction. Internat. Math. Res. Notices 6, 263–275 (1996)
Aleman A., Perfekt K.-M.: Hankel forms and embedding theorems in weighted Dirichlet spaces. Int. Math. Res. Not. IMRN 19, 4435–4448 (2012)
Blasco O.: Hardy spaces of vector-valued functions: duality. Trans. Amer. Math. Soc. 308(2), 495–507 (1988)
Blasco O.: Vector-valued analytic functions of bounded mean oscillation and geometry of Banach s-paces. Illinois J. Math. 41(4), 532–558 (1997)
Blasco O., Arregui J.-L.: Multipliers on vector valued Bergman spaces. Canad. J. Math. 54(6), 1165–1186 (2002)
O. Blasco and J.-L. Arregui, Bergman and Bloch spaces of vector-valued functions, Math. Nachr. 261/262 (2003), 3–22.
Blasco O., Pott S.: Embeddings between operator-valued dyadic BMO spaces. Illinois J. Math. 52(3), 799–814 (2008)
F.F. Bonsall, Boundedness of Hankel matrices, J. London Math. Soc. (2) (2)29 (1984), 289–300.
Bourgain J.: On the similarity problem for polynomially bounded operators on Hilbert space. Israel J. Math. 54(2), 227–241 (1986)
J. Bourgain, Vector-valued singular integrals and the \({{H}^{1}}\)-BMO duality, Probability theory and harmonic analysis (Cleveland, Ohio, 1983) (1986), pp. 1–19.
Buckley S.M., Koskela P., Vukotić D.: Fractional integration, differentiation, and weighted Bergman spaces. Math. Proc. Cambridge Philos. Soc. 126(2), 369–385 (1999)
A.V. Bukhvalov and A.A. Danilevich, Boundary properties of analytic and harmonic functions with values in a Banach space, Mat. Zametki (2)31 (1982), 203–214, 317 (Russian).
Carleson L.: An interpolation problem for bounded analytic functions. Amer. J. Math. 80, 921–930 (1958)
Carleson L.: Interpolations by bounded analytic functions and the corona problem. Ann. of Math. 76(2), 547–559 (1962)
Cohn W.S., Verbitsky I.E.: Factorization of tent spaces and Hankel operators. J. Funct. Anal. 175(2), 308–329 (2000)
Davidson K.R., Paulsen V.I.: Polynomially bounded operators. J. Reine Angew. Math. 487, 153–170 (1997)
J. Diestel and J.J. Jr. Uhl, Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis, Mathematical Surveys, No. 15.
Fefferman C.: Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77, 587–588 (1971)
Fefferman C., Stein E.M.: Hp spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Flett T.M.: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 38, 746–765 (1972)
Foguel S.R.: A counterexample to a problem of Sz-Nagy. Proc. Amer. Math. Soc. 15, 788–790 (1964)
J.B. Garnett, Bounded analytic functions, first revised, Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007.
Gillespie T.A., Pott S., Treil S., Volberg A.: Logarithmic growth for weighted Hilbert transforms and vector Hankel operators. J. Operator Theory 52(1), 103–112 (2004)
Haagerup U., Pisier G.: Factorization of analytic functions with values in noncommutative L1-spaces and applications. Canad. J. Math. 41(5), 882–906 (1989)
Halmos P.R.: Ten problems in Hilbert space. Bull. Amer. Math. Soc. 76, 887–933 (1970)
H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000.
Jacob B., Rydhe E., Wynn A.: The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators. J. Evol. Equ. 14(1), 85–120 (2014)
Janson S., Peetre J.: Paracommutators—boundedness and Schatten-von Neumann properties. Trans. Amer. Math. Soc. 305(2), 467–504 (1988)
S.V. Kislyakov, Operators that are (dis)similar to a contraction: Pisier’s counterexample in terms of singular integrals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 247 (1997), no. Issled. po Linein. Oper. i Teor. Funkts. 25, 79–95, 300 (Russian).
Mei T.: Notes on matrix valued paraproducts. Indiana Univ. Math. J. 55(2), 747–760 (2006)
Nazarov F., Pisier G., Treil S., Volberg A.: Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts. J. Reine Angew. Math. 542, 147–171 (2002)
Nazarov F., Treil S., Volberg A.: Counterexample to the infinite-dimensional Carleson embedding theorem. C. R. Acad. Sci. Paris Sér. I Math. 325(4), 383–388 (1997)
Nehari Z.: On bounded bilinear forms. Ann. of Math. 65(2), 153–162 (1957)
N.K. Nikolski, Operators, functions, and systems: an easy reading. Vol. I, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz, Translated from the French by Andreas Hartmann.
Page L.B.: Bounded and compact vectorial Hankel operators. Trans. Amer. Math. Soc. 150, 529–539 (1970)
Paulsen V.I.: Every completely polynomially bounded operator is similar to a contraction. J. Funct. Anal. 55(1), 1–17 (1984)
V.I. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge (2002).
Peller V.V.: Estimates of functions of power bounded operators on Hilbert spaces. J. Operator Theory 7(2), 341–372 (1982)
Peller V.V.: Vectorial Hankel operators, commutators and related operators of the Schatten-von Neu-mann class \({\gamma_p}\). Integral Equations Operator Theory 5(2), 244–272 (1982)
Peller V.V.: Hankel operators and their applications, Springer Monographs in Mathematics. Springer, New York (2003)
Pisier G.: A polynomially bounded operator on Hilbert space which is not similar to a contraction. J. Amer. Math. Soc. 10(2), 351–369 (1997)
M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (1985). Oxford Science Publications.
E. Rydhe, On the characterization of triebel–lizorkin type space of analytic functions, arXiv:1609.09229 (2016), preprint.
E. Rydhe, Two more counterexamples to the infinite-dimensional carleson embedding theorem, arXiv:1608.06728 (2016), preprint.
Sarason D.: Generalized interpolation in \({H^\infty}\). Trans. Amer. Math. Soc. 127, 179–203 (1967)
B. Sz.-Nagy, Completely continuous operators with uniformly bounded iterates, Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 (1959), 89–93 (Hungarian).
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge (1991).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Rydhe, E. Vectorial Hankel operators, Carleson embeddings, and notions of BMOA. Geom. Funct. Anal. 27, 427–451 (2017). https://doi.org/10.1007/s00039-017-0400-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-017-0400-4