Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators

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March 2015, 14(2): 627-636. doi: 10.3934/cpaa.2015.14.627

Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators

1.	Department of Mathematics, Hunan University, Changsha 410082, China

Received January 2014 Revised July 2014 Published December 2014

Abstract
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In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the pseudo-differential operator is obtained.

Keywords: BMO space, Toeplitz type operator, pseudo-differential operator, Orlicz space..

Mathematics Subject Classification: Primary: 42B20; Secondary: 42B2.

Citation: Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure & Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627

References:

[1]	S. Chanillo, A note on commutators,, \emph{Indiana Univ. Math. J.}, 31 (1982), 7. doi: 10.1512/iumj.1982.31.31002.
[2]	S. Chanillo and A. Torchinsky, Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators,, \emph{Ark. for Mat.}, 24 (1986), 1. doi: 10.1007/BF02384387.
[3]	R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels,, \emph{Ast\'erisque}, 57 (1978).
[4]	R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables,, \emph{Ann. of Math.}, 103 (1976), 611.
[5]	C. Fefferman, $L^p$ bounds for pseudo-differential operators,, \emph{Israel J. Math.}, 14 (1973), 413.
[6]	J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, North-Holland Math., (1985).
[7]	S. Janson, Mean oscillation and commutators of singular integral operators,, \emph{Ark. for Mat.}, 16 (1978), 263. doi: 10.1007/BF02386000.
[8]	S. Janson and J. Peetre, Paracommutators boundedness and Schatten-von Neumann properties,, \emph{Tran. Amer. Math. Soc.}, 305 (1988), 467. doi: 10.2307/2000875.
[9]	S. Janson and J. Peetre, Higher order commutators of singular integral operators,, Interpolation spaces and allied topics in analysis, (1070), 125. doi: 10.1007/BFb0099097.
[10]	L. Z. Liu, Sharp and weighted boundedness for multilinear operators associated with pseudo-differential operators on Morrey space,, \emph{J. of Contemporary Math. Analysis}, 45 (2010), 136. doi: 10.3103/S1068362310030039.
[11]	L. Z. Liu, Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to pseudo-differential operator,, \emph{J. of Pseudo-Differential Operators and Applications}, 3 (2012), 329. doi: 10.1007/s11868-012-0060-y.
[12]	N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols,, \emph{Trans. Amer. Math. Soc.}, 269 (1982), 91. doi: 10.2307/1998595.
[13]	M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1. doi: 10.1512/iumj.1995.44.1976.
[14]	C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators,, \emph{Michigan Math. J.}, 49 (2001), 23. doi: 10.1307/mmj/1008719033.
[15]	C. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators,, \emph{J. London Math. Soc.}, 65 (2002), 672. doi: 10.1112/S0024610702003174.
[16]	M. Saidani, A. Lahmar-Benbernou and S. Gala, Pseudo-differential operators and commutators in multiplier spaces,, \emph{African Diaspora J. of Math.}, 6 (2008), 31.
[17]	E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals,, Princeton Univ. Press, (1993).
[18]	M. E. Taylor, Pseudo-differential Operators and Nonlinear PDE,, Birkhauser, (1991).

show all references

References:

[1]	S. Chanillo, A note on commutators,, \emph{Indiana Univ. Math. J.}, 31 (1982), 7. doi: 10.1512/iumj.1982.31.31002.
[2]	S. Chanillo and A. Torchinsky, Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators,, \emph{Ark. for Mat.}, 24 (1986), 1. doi: 10.1007/BF02384387.
[3]	R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels,, \emph{Ast\'erisque}, 57 (1978).
[4]	R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables,, \emph{Ann. of Math.}, 103 (1976), 611.
[5]	C. Fefferman, $L^p$ bounds for pseudo-differential operators,, \emph{Israel J. Math.}, 14 (1973), 413.
[6]	J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,, North-Holland Math., (1985).
[7]	S. Janson, Mean oscillation and commutators of singular integral operators,, \emph{Ark. for Mat.}, 16 (1978), 263. doi: 10.1007/BF02386000.
[8]	S. Janson and J. Peetre, Paracommutators boundedness and Schatten-von Neumann properties,, \emph{Tran. Amer. Math. Soc.}, 305 (1988), 467. doi: 10.2307/2000875.
[9]	S. Janson and J. Peetre, Higher order commutators of singular integral operators,, Interpolation spaces and allied topics in analysis, (1070), 125. doi: 10.1007/BFb0099097.
[10]	L. Z. Liu, Sharp and weighted boundedness for multilinear operators associated with pseudo-differential operators on Morrey space,, \emph{J. of Contemporary Math. Analysis}, 45 (2010), 136. doi: 10.3103/S1068362310030039.
[11]	L. Z. Liu, Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to pseudo-differential operator,, \emph{J. of Pseudo-Differential Operators and Applications}, 3 (2012), 329. doi: 10.1007/s11868-012-0060-y.
[12]	N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols,, \emph{Trans. Amer. Math. Soc.}, 269 (1982), 91. doi: 10.2307/1998595.
[13]	M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1. doi: 10.1512/iumj.1995.44.1976.
[14]	C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators,, \emph{Michigan Math. J.}, 49 (2001), 23. doi: 10.1307/mmj/1008719033.
[15]	C. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators,, \emph{J. London Math. Soc.}, 65 (2002), 672. doi: 10.1112/S0024610702003174.
[16]	M. Saidani, A. Lahmar-Benbernou and S. Gala, Pseudo-differential operators and commutators in multiplier spaces,, \emph{African Diaspora J. of Math.}, 6 (2008), 31.
[17]	E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals,, Princeton Univ. Press, (1993).
[18]	M. E. Taylor, Pseudo-differential Operators and Nonlinear PDE,, Birkhauser, (1991).

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