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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, Indiana Univ. Math. J., 31 (1982), 7-16. doi: 10.1512/iumj.1982.31.31002.  Google Scholar [2] S. Chanillo and A. Torchinsky, Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators, Ark. for Mat., 24 (1986), 1-25. doi: 10.1007/BF02384387.  Google Scholar [3] R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels, Astérisque, 57 (1978).  Google Scholar [4] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  Google Scholar [5] C. Fefferman, $L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.  Google Scholar [6] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math., 116, Amsterdam, 1985.  Google Scholar [7] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. for Mat., 16 (1978), 263-270. doi: 10.1007/BF02386000.  Google Scholar [8] S. Janson and J. Peetre, Paracommutators boundedness and Schatten-von Neumann properties, Tran. Amer. Math. Soc., 305 (1988), 467-504. doi: 10.2307/2000875.  Google Scholar [9] S. Janson and J. Peetre, Higher order commutators of singular integral operators, Interpolation spaces and allied topics in analysis, Lecture Notes in Math., 1070, Springer, Berlin, 1984, 125-142. doi: 10.1007/BFb0099097.  Google Scholar [10] L. Z. Liu, Sharp and weighted boundedness for multilinear operators associated with pseudo-differential operators on Morrey space, J. of Contemporary Math. Analysis, 45 (2010), 136-150. doi: 10.3103/S1068362310030039.  Google Scholar [11] L. Z. Liu, Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to pseudo-differential operator, J. of Pseudo-Differential Operators and Applications, 3 (2012), 329-350. doi: 10.1007/s11868-012-0060-y.  Google Scholar [12] N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols, Trans. Amer. Math. Soc., 269 (1982), 91-109. doi: 10.2307/1998595.  Google Scholar [13] M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44 (1995), 1-17. doi: 10.1512/iumj.1995.44.1976.  Google Scholar [14] C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49 (2001), 23-37. doi: 10.1307/mmj/1008719033.  Google Scholar [15] C. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65 (2002), 672-692. doi: 10.1112/S0024610702003174.  Google Scholar [16] M. Saidani, A. Lahmar-Benbernou and S. Gala, Pseudo-differential operators and commutators in multiplier spaces, African Diaspora J. of Math., 6 (2008), 31-53.  Google Scholar [17] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993.  Google Scholar [18] M. E. Taylor, Pseudo-differential Operators and Nonlinear PDE, Birkhauser, Boston, 1991. Google Scholar

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##### References:
 [1] S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7-16. doi: 10.1512/iumj.1982.31.31002.  Google Scholar [2] S. Chanillo and A. Torchinsky, Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators, Ark. for Mat., 24 (1986), 1-25. doi: 10.1007/BF02384387.  Google Scholar [3] R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels, Astérisque, 57 (1978).  Google Scholar [4] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  Google Scholar [5] C. Fefferman, $L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.  Google Scholar [6] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math., 116, Amsterdam, 1985.  Google Scholar [7] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. for Mat., 16 (1978), 263-270. doi: 10.1007/BF02386000.  Google Scholar [8] S. Janson and J. Peetre, Paracommutators boundedness and Schatten-von Neumann properties, Tran. Amer. Math. Soc., 305 (1988), 467-504. doi: 10.2307/2000875.  Google Scholar [9] S. Janson and J. Peetre, Higher order commutators of singular integral operators, Interpolation spaces and allied topics in analysis, Lecture Notes in Math., 1070, Springer, Berlin, 1984, 125-142. doi: 10.1007/BFb0099097.  Google Scholar [10] L. Z. Liu, Sharp and weighted boundedness for multilinear operators associated with pseudo-differential operators on Morrey space, J. of Contemporary Math. Analysis, 45 (2010), 136-150. doi: 10.3103/S1068362310030039.  Google Scholar [11] L. Z. Liu, Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to pseudo-differential operator, J. of Pseudo-Differential Operators and Applications, 3 (2012), 329-350. doi: 10.1007/s11868-012-0060-y.  Google Scholar [12] N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols, Trans. Amer. Math. Soc., 269 (1982), 91-109. doi: 10.2307/1998595.  Google Scholar [13] M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44 (1995), 1-17. doi: 10.1512/iumj.1995.44.1976.  Google Scholar [14] C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49 (2001), 23-37. doi: 10.1307/mmj/1008719033.  Google Scholar [15] C. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65 (2002), 672-692. doi: 10.1112/S0024610702003174.  Google Scholar [16] M. Saidani, A. Lahmar-Benbernou and S. Gala, Pseudo-differential operators and commutators in multiplier spaces, African Diaspora J. of Math., 6 (2008), 31-53.  Google Scholar [17] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993.  Google Scholar [18] M. E. Taylor, Pseudo-differential Operators and Nonlinear PDE, Birkhauser, Boston, 1991. Google Scholar
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