Bilinear Bochner-Riesz square function and applications. (English) Zbl 1527.42004
Summary: In this paper, we introduce Stein’s square function associated with bilinear Bochner-Riesz means and investigate its \(L^p\)-boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in [K. Jotsaroop and S. Shrivastava, Adv. Math. 395, Article ID 108100, 38 p. (2022; Zbl 1482.42010)]. Another application concerns the \(L^p\)-estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of L. Grafakos et al. [J. Anal. Math. 143, No. 1, 231–251 (2021; Zbl 1470.42029)] in the context of bilinear radial multipliers and provide a dimension-free sufficient condition on the bilinear multipliers for \(L^2\times L^2\rightarrow L^1\)-boundedness of the associated maximal function. The generalised bilinear spherical maximal function is a particular example of such maximal functions.
MSC:
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 42B25 | Maximal functions, Littlewood-Paley theory |
Keywords:
Stein’s square function; Bochner-Riesz means; bilinear multipliers; sparse operators; maximal functionsReferences:
| [1] | Bernicot, F.; Grafakos, L.; Song, L.; Yan, L., The bilinear Bochner-Riesz problem, J. Anal. Math., 127, 179-217 (2015) · Zbl 1327.42013 · doi:10.1007/s11854-015-0028-y |
| [2] | Bergh, J., Löfström, J.: Interpolation Spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin (1976) · Zbl 0344.46071 |
| [3] | Carleson, L.: Some analytic problems related to statistical mechanics. In: Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md., Lecture Notes in Math. 779. Springer, Berlin 1980, 5-45 (1979) · Zbl 0425.60091 |
| [4] | Carbery, A., The boundedness of the maximal Bochner-Riesz operator on \(L^4({\mathbb{R} }^2)\), Duke Math. J., 50, 2, 409-416 (1983) · Zbl 0522.42015 · doi:10.1215/S0012-7094-83-05018-4 |
| [5] | Carbery, A.: Radial Fourier multipliers and associated Maximal functions. Recent Progress in Fourier Analysis (El Escorial, 1983) North-Holland Math. Stud., 111, Notas Mat., 101, North-Holland, Amsterdam (1985) |
| [6] | Carro, MJ; Domingo-Salazar, C., Stein’s square function \(G_\alpha\) and sparse operators, J Geom. Anal., 27, 1624-1635 (2017) · Zbl 1371.42017 · doi:10.1007/s12220-016-9733-8 |
| [7] | Christ, M., On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Am. Math. Soc., 96, 1, 16-20 (1985) · Zbl 0569.42011 · doi:10.1090/S0002-9939-1985-0796439-7 |
| [8] | Christ, M., Zhou, Z.: A class of singular bilinear maximal functions. arXiv:2203.16725v1 |
| [9] | Dahlberg, B.E.J., Kenig, C.E.: A note on the almost everywhere behaviour of solutions to the Schrödinger equation, in Harmonic Analysis (Minneapolis, Minn. Lecture Notes in Math. 908. Springer, Berlin 1982, 205-209 (1981) · Zbl 0519.35022 |
| [10] | Deleaval, L.; Kriegler, C., Dimension free bounds for the vector-valued Hardy-Littlewood Maximal operator, Rev Matematica Iberoamericana, 35, 1, 101-123 (2019) · Zbl 1418.42027 · doi:10.4171/rmi/1050 |
| [11] | Diestel, G.; Grafakos, L., Unboundedness of the ball bilinear multiplier operator, Nagoya Math. J., 185, 4, 583-584 (2007) · Zbl 1131.43003 |
| [12] | Du, X.; Guth, L.; Li, X., A sharp Schrödinger estimate in \({\mathbb{R} }^2\), Ann. Math.(2), 186, 2, 607-640 (2017) · Zbl 1378.42011 · doi:10.4007/annals.2017.186.2.5 |
| [13] | Du, X.; Zhang, R., Sharp \(L^2\) estimates for Schrodinger operator in higher dimensions, Ann. Math.(2), 189, 3, 837-861 (2019) · Zbl 1433.42010 · doi:10.4007/annals.2019.189.3.4 |
| [14] | Fefferman, C.; Stein, EM, Some maximal inequalities, Am. J. Math., 93, 107-115 (1971) · Zbl 0222.26019 · doi:10.2307/2373450 |
| [15] | Grafakos, L., Classical Fourier Analysis (2014), New York: Springer, New York · Zbl 1304.42001 · doi:10.1007/978-1-4939-1194-3 |
| [16] | Grafakos, L., Modern Fourier Analysis (2014), New York: Springer, New York · Zbl 1304.42002 · doi:10.1007/978-1-4939-1230-8 |
| [17] | Grafakos, L.; He, D.; Honźik, P., Maximal operators associated with bilinear multipliers of limited decay, J. Anal. Math., 143, 1, 231-251 (2021) · Zbl 1470.42029 · doi:10.1007/s11854-021-0154-7 |
| [18] | Grafakos, L.; Li, X., The disc as a bilinear multiplier, Am. J. Math., 128, 1, 91-119 (2006) · Zbl 1143.42015 · doi:10.1353/ajm.2006.0006 |
| [19] | Grafakos, L.; Torres, R., Multilinear Calderón-Zygmund theory, Adv. Math., 165, 1, 124-164 (2002) · Zbl 1032.42020 · doi:10.1006/aima.2001.2028 |
| [20] | Jeong, E.; Lee, S., Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators, J. Funct. Anal., 279, 7, 108629 (2020) · Zbl 1445.42006 · doi:10.1016/j.jfa.2020.108629 |
| [21] | Jeong, E.; Lee, S.; Vergas, A., Improved bound for the bilinear Bochner-Riesz operator, Math. Ann., 372, 1-2, 581-609 (2018) · Zbl 1457.42021 · doi:10.1007/s00208-018-1696-6 |
| [22] | Jotsaroop, K., Shrivastava, S.: Maximal estimates for bilinear Bochner-Riesz means. Adv. Math. 395, Paper No. 108100 · Zbl 1482.42010 |
| [23] | Jotsaroop, K.; Shrivastava, S.; Shuin, K., Weighted estimates for bilinear Bochner-Riesz means at the critical index, Poten. Anal., 55, 4, 603-617 (2021) · Zbl 1486.42032 · doi:10.1007/s11118-020-09870-4 |
| [24] | Kaneko, M.; Sunouchi, G., On the Littlewood-Paley and Marcinkiewicz functions in higher dimensions, Tohoku Math. J., 37, 343-365 (1985) · Zbl 0579.42011 · doi:10.2748/tmj/1178228647 |
| [25] | Lee, S., Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. J., 122, 1, 205-232 (2004) · Zbl 1072.42009 · doi:10.1215/S0012-7094-04-12217-1 |
| [26] | Lee, S., Square function estimates for the Bochner-Riesz means, Anal. PDE, 11, 6, 1535-1586 (2018) · Zbl 1414.42010 · doi:10.2140/apde.2018.11.1535 |
| [27] | Lee, S.; Rogers, KM; Seeger, A., Improved bounds for Stein’s square functions, Proc. Lond. Math. Soc., 3, 104, 1198-1234 (2012) · Zbl 1254.42018 · doi:10.1112/plms/pdr067 |
| [28] | Lee, S.; Rogers, KM; Seeger, A.; Fefferman, C., Square functions and maximal operators associated with radial Fourier multipliers, Advances in Analysis: The Legacy of Elias M. Stein, 273-302 (2014), Princeton University Press: Princeton, Princeton University Press · Zbl 1309.42024 · doi:10.1515/9781400848935-013 |
| [29] | Lerner, A.; Nazarov, F., Intuitive Dyadic Calculus, Expo. Math., 37, 3, 225-265 (2019) · Zbl 1440.42062 · doi:10.1016/j.exmath.2018.01.001 |
| [30] | Liu, H.; Wang, M., Boundedness of the bilinear Bochner-Riesz means in the non-Banach triangle case, Proc. Am. Math. Soc., 148, 1121-1130 (2020) · Zbl 1431.42023 · doi:10.1090/proc/14819 |
| [31] | RubiodeFrancia, JL, Maximal functions and Fourier transforms, Duke Math. J., 52, 2, 395-404 (1986) · Zbl 0612.42008 |
| [32] | Seeger, A., On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math., 370, 61-73 (1986) · Zbl 0584.42008 |
| [33] | Sogge, C., Fourier Integrals in Classical Analysis (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0783.35001 · doi:10.1017/CBO9780511530029 |
| [34] | Stein, EM, Localization and summability of multiple Fourier series, Acta Math., 100, 93-147 (1958) · Zbl 0085.28401 · doi:10.1007/BF02559603 |
| [35] | Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton Mathematical Series, No. 32, Princeton University Press, Princeton (1971) · Zbl 0232.42007 |
| [36] | Sunouchi, G., On the Littlewood-Paley function \(g^*\) of multiple Fourier integrals and Hankel multiplier transformations, Tohoku Math. J. (2), 9, 496-511 (1967) · Zbl 0189.11903 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
