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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

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