Weighted bilinear multiplier theorems in Dunkl setting via singular integrals

References

[1] B. Amri, J.-P. Anker and M. Sifi, Three results in Dunkl analysis, Colloq. Math. 118 (2010), no. 1, 299–312. 10.4064/cm118-1-16Search in Google Scholar

[2] B. Amri, A. Gasmi and M. Sifi, Linear and bilinear multiplier operators for the Dunkl transform, Mediterr. J. Math. 7 (2010), no. 4, 503–521. 10.1007/s00009-010-0057-9Search in Google Scholar

[3] B. Amri and M. Sifi, Singular integral operators in Dunkl setting, J. Lie Theory 22 (2012), no. 3, 723–739. Search in Google Scholar

[4] T. A. Bui and X. T. Duong, Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers, Bull. Sci. Math. 137 (2013), no. 1, 63–75. 10.1016/j.bulsci.2012.04.001Search in Google Scholar

[5] M. Cao, Q. Xue and K. Yabuta, Weak and strong type estimates for the multilinear pseudo-differential operators, J. Funct. Anal. 278 (2020), no. 10, Article ID 108454. 10.1016/j.jfa.2019.108454Search in Google Scholar

[6] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331. 10.2307/1998628Search in Google Scholar

[7] R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57, Société Mathématique de France, Paris, 1978. Search in Google Scholar

[8] R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E, Beijing Lectures in Harmonic Analysis (Beijing 1984), Ann. of Math. Stud. 112, Princeton University, Princeton (1986), 3–45. 10.1515/9781400882090-002Search in Google Scholar

[9] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971. 10.1007/BFb0058946Search in Google Scholar

[10] D. V. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Oper. Theory Adv. Appl. 215, Birkhäuser/Springer, Basel, 2011. 10.1007/978-3-0348-0072-3Search in Google Scholar

[11] F. Dai and W. Ye, Local restriction theorem and maximal Bochner–Riesz operators for the Dunkl transforms, Trans. Amer. Math. Soc. 371 (2019), no. 1, 641–679. 10.1090/tran/7285Search in Google Scholar

[12] M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162. 10.1007/BF01244305Search in Google Scholar

[13] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. 10.1090/S0002-9947-1989-0951883-8Search in Google Scholar

[14] C. F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa 1991), Contemp. Math. 138, American Mathematical Society, Providence (1992), 123–138. 10.1090/conm/138/1199124Search in Google Scholar

[15] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math. 29, American Mathematical Society, Providence, 2001. Search in Google Scholar

[16] X. T. Duong, R. Gong, L. Grafakos, J. Li and L. Yan, Maximal operator for multilinear singular integrals with non-smooth kernels, Indiana Univ. Math. J. 58 (2009), no. 6, 2517–2541. 10.1512/iumj.2009.58.3803Search in Google Scholar

[17] J. Dziubański and A. Hejna, Hörmander’s multiplier theorem for the Dunkl transform, J. Funct. Anal. 277 (2019), no. 7, 2133–2159. 10.1016/j.jfa.2019.03.002Search in Google Scholar

[18] J. Dziubański and A. Hejna, Remarks on Dunkl translations of non-radial kernels, J. Fourier Anal. Appl. 29 (2023), no. 4, Paper No. 52. 10.1007/s00041-023-10034-2Search in Google Scholar

[19] M. Fujita and N. Tomita, Weighted norm inequalities for multilinear Fourier multipliers, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6335–6353. 10.1090/S0002-9947-2012-05700-XSearch in Google Scholar

[20] L. Grafakos, Classical Fourier Analysis, 3rd ed., Grad. Texts in Math. 249, Springer, New York, 2014. 10.1007/978-1-4939-1194-3Search in Google Scholar

[21] L. Grafakos, L. Liu, D. Maldonado and D. Yang, Multilinear analysis on metric spaces, Dissertationes Math. 497 (2014), 121. 10.4064/dm497-0-1Search in Google Scholar

[22] L. Grafakos, L. Liu and D. Yang, Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand. 104 (2009), no. 2, 296–310. 10.7146/math.scand.a-15099Search in Google Scholar

[23] L. Grafakos, L. Liu and D. Yang, Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 4, 755–775. 10.1017/S0308210509001383Search in Google Scholar

[24] L. Grafakos and R. H. Torres, Multilinear Calderón–Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. 10.1006/aima.2001.2028Search in Google Scholar

[25] G. Hu and C.-C. Lin, Weighted norm inequalities for multilinear singular integral operators and applications, Anal. Appl. (Singap.) 12 (2014), no. 3, 269–291. 10.1142/S0219530514500043Search in Google Scholar

[26] C. E. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1–15. 10.4310/MRL.1999.v6.n1.a1Search in Google Scholar

[27] A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264. 10.1016/j.aim.2008.10.014Search in Google Scholar

[28] K. Li and W. Sun, Weighted estimates for multilinear Fourier multipliers, Forum Math. 27 (2015), no. 2, 1101–1116. 10.1515/forum-2012-0128Search in Google Scholar

[29] S. Mukherjee and S. Parui, Weighted inequalities for multilinear fractional operators in Dunkl setting, J. Pseudo-Differ. Oper. Appl. 13 (2022), no. 3, Paper No. 34. 10.1007/s11868-022-00464-9Search in Google Scholar

[30] C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis. Vol. II, Cambridge Stud. Adv. Math. 138, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139047081Search in Google Scholar

[31] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), no. 3, 445–463. 10.1215/S0012-7094-99-09813-7Search in Google Scholar

[32] M. Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2413–2438. 10.1090/S0002-9947-03-03235-5Search in Google Scholar

[33] M. Rösler, Dunkl operators: Theory and applications, Orthogonal Polynomials and Special Functions, Lecture Notes in Math. 1817, Springer, Berlin (2003), 93–135. 10.1007/3-540-44945-0_3Search in Google Scholar

[34] J. L. Rubio de Francia, Maximal functions and Fourier transforms, Duke Math. J. 53 (1986), no. 2, 395–404. 10.1215/S0012-7094-86-05324-XSearch in Google Scholar

[35] C. Tan, Y. Han, Y. Han, M.-Y. Lee and J. Li, Singular integral operators, T1 theorem, Littlewood–Paley theory and Hardy spaces in Dunkl setting, preprint (2022), https://arxiv.org/abs/2204.01886. Search in Google Scholar

[36] S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55. 10.1007/BF02807401Search in Google Scholar

[37] N. Tomita, A Hörmander type multiplier theorem for multilinear operators, J. Funct. Anal. 259 (2010), no. 8, 2028–2044. 10.1016/j.jfa.2010.06.010Search in Google Scholar

[38] B. Wróbel, Approaching bilinear multipliers via a functional calculus, J. Geom. Anal. 28 (2018), no. 4, 3048–3080. 10.1007/s12220-017-9945-6Search in Google Scholar PubMed PubMed Central