Abstract
In this article, we establish a multilinear Cotlar-type inequality for the maximal multilinear singular integrals in Dunkl setting whose kernels possess less regularity conditions compared to the multilinear Calderón–Zygmund kernels in spaces of homogeneous type. As applications, we achieve weighted boundedness of maximal multilinear Dunkl–Calderón–Zygmund singular integrals and pointwise convergence of principal value integrals associated with multilinear Dunkl–Calderón–Zygmund kernels.
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References
Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)
Duong, X.T., Gong, R., Grafakos, L., Li, J., Yan, L.: Maximal operator for multilinear singular integrals with non-smooth kernels. Ind. Univ. Math. J. 58(6), 2517–2541 (2009)
Grafakos, L., Liu, L., Maldonado, D., Yang, D.: Multilinear analysis on metric spaces. Diss. Math. 497, 121 (2014)
Grafakos, L., Liu, L., Yang, D.: Multiple-weighted norm inequalities for maximal multilinear singular integrals with non-smooth kernels. Proc. R. Soc. Edinb. Sect. A 141(4), 755–775 (2011)
Grafakos, L., Torres, R.H.: Discrete decompositions for bilinear operators and almost diagonal conditions. Trans. Am. Math. Soc. 354(3), 1153–1176 (2002)
Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Ind. Univ. Math. J. 51(5), 1261–1276 (2002)
Grafakos, L., Torres, R.H.: Multilinear Calderón–Zygmund theory. Adv. Math. 165(1), 124–164 (2002)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)
Mukherjee, S., Parui, S.: Weighted bilinear multiplier theorems in Dunkl setting via singular integrals. Preprint arXiv:2311.04754 (2023)
Tan, C., Han, Y., Han, Y., Lee, M.-Y., Li, J.: Singular integral operators, T1 theorem, Littlewood–Paley theory and Hardy spaces in Dunkl setting. Preprint arXiv:2204.01886 (2022)
Tan, C., Han, Y., Li, J.: Maximal operator, Cotlar’s inequality and pointwise convergence for singular integral operators in Dunkl setting. J. Geom. Anal. 33(5), 164, 18 (2023)
Acknowledgements
Suman Mukherjee is supported by research fellowship from Department of Atomic Energy (DAE), Government of India.
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Communicated by Ferenc Weisz.
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Mukherjee, S. Cotlar-type inequality and weighted boundedness for maximal multilinear singular integrals in Dunkl setting. Adv. Oper. Theory 9, 38 (2024). https://doi.org/10.1007/s43036-024-00338-5
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DOI: https://doi.org/10.1007/s43036-024-00338-5
Keywords
- Dunkl transform
- Weighted inequalities
- Multilinear Dunkl–Calderón–Zygmund operators
- Multilinear Cotlar-type inequality