Volume 27 • Issue 2 • PP: 542-550 • 2026
Generalized Marcinkiewicz Operators Aassociated to Twisted Surfaces
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This paper is concerned with studying the mapping properties of generalized Marcinkiewicz integral operators along twisted surfaces. Under certain conditions on thes esurfaces we_obtain certain Lp estimates for these operators provided that the kernel functions are in Lq Sn−1 × Sm−1 . By an extrapolation argument, we prove that these operators are bounded on Lp(Rn × Rm) for 1 < p < ∞ under very weak conditions on the kernel functions. Our results extend and improve many previously known results.
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References
[1] Y. Ding, L2-boundedness of Marcinkiewicz integral with rough kernel, Hokk. Math. J. 27 (1998) 105–115.
[2] Y. Choi, Marcinkiewicz integrals with rough homogeneous kernel of degree zero in product domains, J. Math. Anal. Appl. 261 (2001) 53–60.
[3] J. Chen, D. Fan and Y. Ying, The method of rotation and Marcinkiewicz integrals on product domains, Stud. Math. 153 (2002) 41–58.
[4] H. Al-Qassem, A. Al-Salman, L. Cheng, and Y. Pan, Marcinkiewicz integrals on product spaces, Studia Math. 167 (2005) 227– 234.
[5] H. Al-Qassem, Rough Marcinkiewicz integral operators on product spaces, Collec. Math. 36 (2005) 275–297.
[6] Y. Jiang and S. Lu, A class of singular integral operators with rough kernel on product domains, Hokkaido Math. J. 25 (1995) 1–7. [7] J. Chen, D. Fan and Y. Ying, Rough Marcinkiewicz integrals with L(log L)2 kernels, Adv. Math. 30 (2001) 179–181.
[8] D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997) 799–839.
[9] W. Kim, S. Wainger, J. Wright, and S. Ziesler, Singular Integrals and Maximal Functions Associated to Surfaces of Revolution, Bull. Lond. Math. Soc. 28 ( 1996) 291–296.
[10] J. Chen, Y. Ding and D. Fan, Lp boundedness of the rough Marcinkiewicz integral on product domains, Chin. J. Contemp. Math. 21 (2000) 47–54.
[11] H. Wu and J. Xu, Rough Marcinkiewicz integrals associated to surfaces of revolution on product domains, Acta Math. Sci. 29 (2009) 294–304.
[12] G. Hu, S. Lu and D. Yan, Lp(Rm × Rn) boundedness of Marcinkiewicz integral on product spaces, Sci China Ser A 46 (2003) 75–82.
[13] F. Liu and H. Wu, Rough Marcinkiewicz Integrals with Mixed Homogeneity on Product Spaces, Acta Math. Sci. 29 (2013) 231–244.
[14] F. Liu and H. Wu, On the L2 boundedness for the multiple Littlewood Paley functions with rough kernels, J. Math. Anal. Appl. 410 (2014) 403–410.
[15] M. Ali and O. Al-Mohammed, Boundedness of a class of rough maximal functions, J. Ineq. Appl. 9 (2018) art. 305.
[16] Q. Xue and J. Yu, Marcinkiewicz Integrals Associated with Schrodinger Operator on Campanato Type Spaces, Front. Math. 20 (2025) 1201–1237.
[17] H. Wu and L. Wu, An extension of Marcinkiewicz integrals with rough kernels, Acta Math. Sci. 45 (2025) 789–808.
[18] H. Al-Qassem and M. Ali, On the functions of Marcinkiewicz integrals along surfaces of revolution on product domains via extrapolation, Symmetry 15 (2023) art. 1814.
[19] M. Ali and H. Al-Qassem, H. On certain estimates for parabolic Marcinkiewicz integrals related to surfaces of revolution on product spaces and extrapolation, Axioms 12 (2023) 1–16.
[20] M. Ali and A. Al-Senjlawi, Boundedness of Marcinkiewicz integrals on product spaces and extrapolation, Inter. J. Pure Appl. Math. 97 (2014) 49-66.
[21] B. Al-Azriyah and A. Al-Salman, Singular and Marcinkiewicz integral operators on product domains, Commun. Korean Math. Soci. 38 (2023) 401–430.
[22] A. Al-Salman, Rough Marcinkiewicz integrals on product spaces, Inter. Math. Forum 2 (2007) 1119–1128.
[23] A. Al-Salman, Marcinkiewicz integral operators along twisted surfaces, Commun. Pure Appl. Anal. 12 (2022) 159–181.
[24] D. Fan and H.Wu, On the generalized Marcinkiewicz integral operators with rough kernels, Canada Math. Bull. 54 (2011) 100–112.
[25] H. Al-Qassem, L. Cheng and Y. Pan, Generalized Littlewood-Paley functions on product spaces, Turk. J. Math. 45 (2021) 319–345.
[26] M. Ali and H. Al-Qassem, A Class of rough generalized Marcinkiewicz integrals on product domains, Symmetry, 15 (2023) art. 823.
[27] M. Ali and H. Al-Qassem, Estimates for generalized parabolic Marcinkiewicz integrals with rough kernels on product domains, Axioms 12 (2023) 1–10.
[28] M. Ali and H. Al-Qassem, On rough generalized Marcinkiewicz integrals along surfaces of revolution on product spaces, AIMS Math. 9 (2024) 4816–4829.
[29] S. Yano, Notes on Fourier analysis. XXIX. An extrapolation theorem, J. Math. Soc. Japan 3 (1951) 296–305.
[30] S. Sato, Estimates for singular integrals and extrapolation, Studia Math. 192 (2009) 219–233.
[31] J. Duoandikoetxea and J. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent Math. 84 (1986) 541–561.2Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan
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