Toeplitz operators on Hardy space \(H^ p(S)\) with \(0<p\leq{}1\). (English) Zbl 0781.47034

Li, Song-Ying

Integral Equations Oper. Theory 15, No. 5, 807-824 (1992).

Let \(\mathbb{B}^ n\) be the unit ball in \(\mathbb{C}^ n\), \(\mathbb{S}\) the boundary of \(\mathbb{B}^ n\), \(L^ p(\mathbb{S})\) the usual Lebesgue spaces

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over \(\mathbb{S}\) with respect to the normalized surface measure, \(H^ p(\mathbb{B}^ n)\) its usual holomorphic subspaces. \(H^ p(\mathbb{S})\) denotes the atomic Hardy space. Let \(P\): \(L^ 2(\mathbb{S}) \to H^ 2(\mathbb{B}^ n)\) be the orthogonal projection

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. For each \(f \in L^ \infty(\mathbb{S})\), \(M_ f:L^ p(\mathbb{S})\to L^ p(\mathbb{S})\) is the multiplication operator and \(T_ f=PM_ f\) is the Toeplitz operator

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. The paper gives a characterization theorem on \(f\) such that the Toeplitz operators \(T_ f\) and \(T_{\bar f}\) are bounded from \(H^ p(\mathbb{S})\) to \(H^ p(\mathbb{B}^ n)\) with \(p\in(0,1]\).

Reviewer: V.S.Rabinovich (Rostov-na-Donu)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A35	\(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Keywords:

atomic Hardy space; multiplication operator; Toeplitz operator

References:

[1]	[BCZ] Berger, C. A., Coburn, L. A. and Zhu, K. H., Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus,Amer. J. Math. 110(1988), 921-952. · Zbl 0657.32001 · doi:10.2307/2374698
[2]	[BGS] Burkholder, D. L., Gundy, R. F. and Silverstein, M. L., A maximal function characterization of the classH p ,Trans. Amer. Math. Sco. 157(1971), 137-153. · Zbl 0223.30048
[3]	[C] Coifman, R. R., A real variable characterization ofH p ,Studia Math., 51(1974), 269-274. · Zbl 0289.46037
[4]	[CRW] Coifman, R. R., Rochberg, R. and Weiss, G., Factorization theorems for Hardy spaces in several variables.Ann. Math. 103 (1976), 611-635. · Zbl 0326.32011 · doi:10.2307/1970954
[5]	[CS] J. Cima and D. Stegenga, Hankel operators onH p ,Analysis at Urbana 1, London Math Soc., Lecture Note Series 137, 133-150.
[6]	[CW] Coifman R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogenes,Lecture Notes in Mathematics 242, Springer-Verlag, Berlin 1971.
[7]	[DRS] Duren, P. L., Romberg, B. W. and Shields, A. L., Linear functional onH p space with 0<p<1.Reine Angew. Math. 238(1969), 32-60. · Zbl 0176.43102
[8]	[FS] Fefferman, C. and Stein, E. M.,H p spaces of several variables,Acta Math. 129(1972), 137-193. · Zbl 0257.46078 · doi:10.1007/BF02392215
[9]	[GL] Garnett, J. B. and Latter, R. H., The atomic decomposition for Hardy space in several complex variablesDuke Math. J. 45(1978), 815-845. · Zbl 0403.32006 · doi:10.1215/S0012-7094-78-04539-8
[10]	[JPS] Janson, S., Peetre, J. and Semmes, S., On the action of Hankel on some function space,Duke Math. J. 51(1984), 937-958. · Zbl 0579.47022 · doi:10.1215/S0012-7094-84-05142-1
[11]	[R] Rudin, W. Function theory in the unit ball inC n , Springer Verlag, 1980. · Zbl 0495.32001
[12]	[S] Stegenga, D. A., Bounded Toeplitz-operator onH 1 and applications of duality betweenH 1 and functions of bounded mean oscillation,Amer. J. Math., 98(1976), 573-589. · Zbl 0335.47018 · doi:10.2307/2373807
[13]	[Z1] Zhu, K., Multiplers of BMO in the Bergman metric with applications to the Toeplitz operators,J. Funct. Anal. 87(1989), 31-50. · Zbl 0705.47025 · doi:10.1016/0022-1236(89)90003-7
[14]	[Z2] Zhu, K., Hankel-Toeplitz type operator onL a 1 (?),Integral Equations and Operator Theory, 13(1990), 285-302. · Zbl 0697.47023 · doi:10.1007/BF01193761