Abstract
We characterize the symbols of Hankel operators that extend into bounded operators from the Hardy–Orlicz \({\mathcal H^{\Phi_1}(\mathbb B^n)}\) into \({\mathcal H^{\Phi_2}(\mathbb B^n)}\) in the unit ball of \({\mathbb C^n}\) , in the case where the growth functions \({\Phi_1}\) and \({\Phi_2}\) are either concave or convex. The case where the growth functions are both concave has been studied by Bonami and Sehba. We also obtain several weak factorization theorems for functions in \({\mathcal H^{\Phi}(\mathbb B^n)}\) , with concave growth function, in terms of products of Hardy–Orlicz functions with convex growth functions.
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Bonami A., Grellier S.: Hankel operators and weak factorization for Hardy–Orlicz spaces. Colloq. Math. 118(1), 107–132 (2010)
Bonami A., Grellier S., Sehba B.F.: Boundedness of Hankel operators on \({\mathcal H^1(\mathbb B^n)}\) . C. R. Math. Acad. Sci. Paris 344(12), 749–752 (2007)
Bonami A., Madan S.: Balayage of Carleson measures and Hankel operators on generalized Hardy spaces. Math. Nachr. 153, 237–245 (1991)
Bonami A., Sehba B.F.: Hankel operators between Hardy–Orlicz spaces and products of holomorphic functions. Rev. Math. Arg. Vol. 50(2), 187–199 (2009)
Coifman R., Rochberg R., Weiss G.: Factorization for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)
Grellier S., Peloso M.: Decomposition theorems for Hardy spaces on convex domains of finite type. Ill. J. Math. 46(1), 207–232 (2002)
Janson S., Peetre P., Semmes S.: On the action of Hankel and Toeplitz operators on some function spaces. Duke Math. J. 51(4), 937–958 (1987)
Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Composition operators on Hardy-Orlicz spaces. Memoirs of the American Mathematical Society, vol. 207, no. 974, pp. vi+74, ISBN: 978-0-8218-4637-7 (2010)
Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, pp. xii+449, Marcel Dekker, Inc., New York, ISBN: 0-8247-8478-2 (1991)
Sehba, B., Tchoundja, E.: Hankel operators with Lipschitz symbols in the unit ball. Math. Scand. (to appear)
Tolokonnikov V.A.: Hankel and Toeplitz operators in Hardy spaces. Sov. Math. 37, 1359–1364 (1987)
Tolokonnikov V.A., Volberg A.L.: Hankel operators and problems of best approximation of unbounded functions. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Matematicheskogo Instituta im. V.A. Steklova AN SSSR 141, 5–17 (1985)
Viviani B.E.: An atomic decomposition of the predual of BMO(ρ). Rev. Mat. Iberoam. 3(3–4), 401–425 (1987)
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Benoît F. Sehba was partially supported by the ANR project ANR-09-BLAN-0058-01. Edgar Tchoundja was supported by the Centre of Recerca Matemàtica, Barcelona (Spain).
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Sehba, B.F., Tchoundja, E. Hankel Operators on Holomorphic Hardy–Orlicz Spaces. Integr. Equ. Oper. Theory 73, 331–349 (2012). https://doi.org/10.1007/s00020-012-1974-8
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DOI: https://doi.org/10.1007/s00020-012-1974-8