Weighted boundedness for Toeplitz type operator associated to singular integral operator with variable Calderón-Zygmund kernel

Year 2019, Volume: 48 Issue: 3, 657 - 668, 15.06.2019

Dazhao Chen

https://doi.org/10.15672/hujms.546986

Cited By: 1

https://izlik.org/JA74GW55ZT

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for the Toeplitz type operator associated to the singular integral operator with variable Calderón- Zygmund kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.

Keywords

Toeplitz type operator , singular integral operator , sharp maximal function , variable Calderón-Zygmund kernel , weighted BMO , weighted Lipschitz function

References

• S. Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc. 292, 103–122, 1985.
• A.P. Calderón and A. Zygmund, On singular integrals with variable kernels, Appl. Anal. 7, 221–238, 1978.
• S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31, 7-16, 1982.
• R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79, 249–254, 1980.
• R.R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103, 611–635, 1976.
• G. Di FaZio and M.A. Ragusa, Commutators and Morrey spaces, Boll. Un. Mat. Ital. 5-A(7), 323–332, 1991.
• G. Di Fazio and M.A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112, 241–256,1993.
• J. Garcia-Cuerva, Weighted Hp spaces, Dissertationes Math. 162, 63 pp., 1979.
• J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, Amsterdam, 1985.
• Y. X. He and Y. S. Wang, Commutators of Marcinkiewicz integrals and weighted BMO, Acta Math. Sinica (Chinese Series), 54, 513–520, 2011.
• B. Hu and J. Gu, Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions, J. Math. Anal. Appl. 340, 598–605, 2008.
• S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Math. 16, 263–270, 1978.
• Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282, 219–231, 2009.
• S. Krantz and S. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications, J. Math. Anal. Appl. 258, 629–641, 2001.
• Y. Lin and S.Z. Lu, Toeplitz type operators associated to strongly singular integral operator, Sci. in China (ser. A Mathematics), 36, 615–630, 2006.
• L.Z. Liu, Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators, Acta Math. Scientia 25(B), 89–94, 2005.
• L.Z. Liu, The continuity for multilinear singular integral operators with variable Calderón-Zygmund kernel on Hardy and Herz spaces, Siberia Elec. Math. Rep. 2, 156–166, 2005.
• L.Z. Liu, Good \lambda estimate for multilinear singular integral operators with variable Calderón-Zygmund kernel, Kragujevac J. Math. 7, 19–30, 2005.
• L.Z. Liu, Weighted estimates of multilinear singular integral operators with variable Calderón-Zygmund kernel for the extreme cases, Vietnam J. Math. 34, 51–61, 2006.
• S.Z. Lu and H.X. Mo, Toeplitz type operators on Lebesgue spaces, Acta Math. Scientia 29(B), 140–150, 2009.
• S.Z. Lu, D.C. Yang, and Z.S. Zhou, Oscillatory singular integral operators with Calderón-Zygmund kernels, Southeast Asian Bull. Math. 23, 457–470, 1999.
• T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, in "Harmonic Analysis", (Sendai, 1990) ICM-90 Satell. Conf. Proc., Springer, Tokyo, 183–189, 1991.
• C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, 126–166, 1983.
• D.K. Palagachev and L.G. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal. 20, 237–263, 2004.
• M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J. 44, 1–17, 1995.
• J. Peetre, On convolution operators leaving Lp,\lambda-spaces invariant, Ann. Mat. Pura. Appl. 72, 295–304, 1966.
• J. Peetre, On the theory of Lp,\lambda-spaces, J. Funct. Anal. 4, 71–87,1969.
• C. Pérez, Endpoint estimate for commutators of singular integral operators, J. Funct. Anal. 128, 163–185,1995.
• C. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc. 65, 672–692, 2002.
• E.M. Stein, Harmonic analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993.
• H. Xu and L.Z. Liu, Weighted boundedness for multilinear singular integral operator with variable Calderón-Zygmund kernel, African Diaspora J. Math. 6, 1–12, 2008.