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

Yıl 2019, Cilt: 48 Sayı: 3, 657 - 668, 15.06.2019

Dazhao Chen

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

Cited By: 1

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• 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.