In this paper, the weighted boundedness for some multilinear operators generated by the pseudo-differential operators and the weighted Lipschitz functions are obtained.
Citation: Dazhao Chen. Weighted boundedness of multilinear pseudo-differential operators[J]. AIMS Mathematics, 2021, 6(11): 12698-12712. doi: 10.3934/math.2021732
In this paper, the weighted boundedness for some multilinear operators generated by the pseudo-differential operators and the weighted Lipschitz functions are obtained.
| [1] | S. Bloom, A commutator theorem and weighted $BMO$, Trans. Amer. Math. Soc., 292 (1985), 103–122. |
| [2] | S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7–16. |
| [3] | S. Chanillo, A. Torchinsky, Sharp function and weighted $L^p$ estimates for a class of pseudo-differential operators, Ark. Math., 24 (1986), 1–25. |
| [4] | W. G. Chen, Besov estimates for a class of multilinear singular integrals, Acta Math. Sinica, 16 (2000), 613–626. |
| [5] | J. Cohen, A sharp estimate for a multilinear singular integral on $R^n$, Indiana Univ. Math. J., 30 (1981), 693–702. |
| [6] | J. Cohen, J. Gosselin, On multilinear singular integral operators on $R^n$, Studia Math., 72 (1982), 199–223. |
| [7] | J. Cohen, J. Gosselin, A BMO estimate for multilinear singular integral operators, Illinois J. Math., 30 (1986), 445–465. |
| [8] | R. Coifman, Y. Meyer, Wavelets, Calderón-Zygmund and multilinear operators, Cambridge Studies in Advanced Math., Vol. 48, Cambridge University Press, Cambridge, 1997. |
| [9] | R. R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103 (1976), 611–635. |
| [10] | Y. Ding, S. Z. Lu, Weighted boundedness for a class rough multilinear operators, Acta Math. Sinica, 17 (2001), 517–526. |
| [11] | C. Fefferman, $L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413–417. |
| [12] | J. Garcia-Cuerva, Weighted $H^p$ spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1979. |
| [13] | J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math., Vol. 116, Amsterdam, 1985. |
| [14] | B. Hu, J. Gu, Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz spaces, J. Math. Anal. Appl., 340 (2008), 598–605. |
| [15] | L. Z. Liu, Sharp and weighted boundedness for multilinear operators associated with pseudo-differential operators on Morrey space, J. Contemp. Math. Anal., 45 (2010), 136–150. |
| [16] | L. Z. Liu, Boundedness for multilinear operators of pseudo-differential operators for the extreme cases, J. Math. Inequal., 4 (2010), 217–232. |
| [17] | L. Z. Liu, Sharp maximal function inequalities and boundedness for Toeplitz type operator associated to pseudo-differential operator, J. Pseudo-Differ. Oper., 4 (2013), 91–112. |
| [18] | N. Miller, Weighted Sobolev spaces and pseudo-differential operators with smooth symbols, Trans. Amer. Math. Soc., 269 (1982), 91–109. |
| [19] | M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44 (1995), 1–17. |
| [20] | C. Pérez, R. Trujillo-Gonzalez, Sharp weighted estimates for vector-valued singular integral operators and commutators, Tohoku Math. J., 55 (2003), 109–129. |
| [21] | C. Pérez, R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65 (2002), 672–692. |
| [22] | M. Saidani, A. Lahmar-Benbernou, S. Gala, Pseudo-differential operators and commutators in multiplier spaces, African Diaspora J. of Math., 6 (2008), 31–53. |
| [23] | E. M. Stein, Harmonic analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993. |
Dazhao Chen. Weighted boundedness of multilinear pseudo-differential operators[J]. AIMS Mathematics, 2021, 6(11): 12698-12712. doi: 10.3934/math.2021732
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