Weighted estimates for maximal bilinear rough singular integrals via sparse dominations

Zhidan Wang1, Qingying Xue1, Xinchen Duan1
1School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, People’s Republic of China

Tóm tắt

Let $$x=(x_1,x_2)$$ with $$x_1,x_2 \in \mathbb {R}^n$$ and let $$K(x)={\Omega \big ({x}/{|x|}\big )}{\big |x\big |^{-2n}}$$ , where $$\Omega \in L^{\infty }(\mathbb {S}^{2n-1})$$ and satisfies $$\int _{\mathbb {S}^{2n-1}}\Omega =0$$ . We show that the maximal truncated bilinear singular integrals with rough kernel $$K(x_1,x_2)$$ satisfy a sparse bound by (p, p, p)-averages for all $$p>1$$ . As consequences, we obtain some quantitative weighted estimates for these rough singular integrals. A pointwise sparse domination for commutators of bilinear rough singular integrals were also established, which can be used to establish some weighted inqualities.

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Tài liệu tham khảo

Barron, A.: Weighted estimates for rough bilinear singular integrals via sparse domination. New York J. Math. 23, 779–811 (2017) Buriánková, E., Honzík, P.: Rough maximal bilinear singular integrals. Collect. Math. 70(3), 431–446 (2019) Calderon, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952) Calderon, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78(2), 289–309 (1956) Cao, M., Yabuta, K.: The multilinear littlewood-paley operators with minimal regularity conditions. J. Fourier Anal. Appl. 25(3), 1203–1247 (2019) Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975) Conde-Alonso, J.M., Culiuc, A., Di Plinio, F., Ou, Y.: A sparse domination principle for rough singular integrals. Anal. PDE 10(5), 1255–1284 (2017) Culiuc, A., Di Plinio, F., Ou, Y.: Domination of multilinear singular integrals by positive sparse forms. J. Lond. Math. Soc. 98(2), 369–392 (2018) Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via fourier transform estimates. Invent. Math. 84(3), 541–561 (1986) Grafakos, L., He, D., Honzík, P.: Rough bilinear singular integrals. Adv. Math. 326, 54–78 (2018) Grafakos, L., He, D., Slavíková, L.: \({L}^2 \times {L}^2 \rightarrow {L}^1 \) boundedness criteria. Math. Ann., to appear Grafakos, L., Stefanov, A.: L p bounds for singular integrals and maximal singular integrals with rough kernels. Indiana University Math. J. 455–469 (1998) Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana U. Mathe. J. 51(5), 1261–1276 (2002) Grafakos, L., Wang, Z., Xue, Q.: Sparse domination and weighted estimates for rough bilinear singular integrals. Preprint arXiv:2009.02456 Hytönen, T., Pérez, C., Rela, E.: Sharp reverse holder property for \(A_{\infty }\) weights on spaces of homogeneous type. J. Funct. Anal. 263(12), 3883–3899 (2012). 12 Lerner, A.K.: On pointwise estimates involving sparse operators. New York J. Math. 22, 341–349 (2016) 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) Li, K.: Sparse domination theorem for multilinear singular integral operators with \({L}^{r}\) -Hörmander condition. Michigan Math. J. 67(2), 253–265 (2018). 05 Li, K., Martell, J.M., Ombrosi, S.: Extrapolation for multilinear muckenhoupt classes and applications to the bilinear hilbert transform. Preprint arXiv:1802.03338v2 Plinio, F.D., Hytönen, T.P., Li, K.: Sparse bounds for maximal rough singular integrals via the fourier transform. Preprint arXiv:1706.09064, (2017) Rivera-Rios, I.P.: Improved \(A_1- A_{\infty }\) and related estimates for commutators of rough singular integrals. Proc. Edinburgh Math. Soc. 61(4), 1069–1086 (2018) Seeger, A.: Singular integral operators with rough convolution kernels. J. Am. Math. Soc. 9(1), 95–105 (1996) Tao, T.: The weak-type (1,1) of LlogL homogeneous convolution operator. Indiana Univ. Math. J. 48(4), 1547–1584 (1999)