Multilinear square functions and multiple weights
DOI:
https://doi.org/10.7146/math.scand.a-105504Abstract
In this paper we prove weighted estimates for a class of smooth multilinear square functions with respect to multilinear $A_{\vec P}$ weights. In particular, we establish weighted estimates for the smooth multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions.
References
Benea, C. and Bernicot, F., A bilinear Rubio de Francia inequality for arbitrary squares, Forum Math. Sigma 4 (2016), e26, 34 pp. https://doi.org/10.1017/fms.2016.21
Bernicot, F., $L^p$ estimates for non-smooth bilinear Littlewood-Paley square functions on ℝ, Math. Ann. 351 (2011), no. 1, 1–49. https://doi.org/10.1007/s00208-010-0588-1
Bernicot, F. and Shrivastava, S., Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators, Indiana Univ. Math. J. 60 (2011), no. 1, 233–268. https://doi.org/10.1512/iumj.2011.60.4527
Culiuc, A., Plinio, F. D., and Ou, Y., Domination of multilinear singular integrals by positive sparse forms, preprint arxiv:1603.05317 [math.CA], 03 2016.
Diestel, G., Some remarks on bilinear Littlewood-Paley theory, J. Math. Anal. Appl. 307 (2005), no. 1, 102–119. https://doi.org/10.1016/j.jmaa.2005.01.014
Grafakos, L., He, S., and Xue, Q., Certain multi(sub)linear square functions, Potential Anal. 45 (2016), no. 1, 55–64. https://doi.org/10.1007/s11118-016-9534-5
Grafakos, L. and Torres, R. H., Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. https://doi.org/10.1006/aima.2001.2028
Lacey, M. and Thiele, C., $L^p$ estimates on the bilinear Hilbert transform for $2 < p < infty $, Ann. of Math. (2) 146 (1997), no. 3, 693–724. https://doi.org/10.2307/2952458
Lacey, M. and Thiele, C., On Calderón's conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. https://doi.org/10.2307/120971
Lacey, M. T., On bilinear Littlewood-Paley square functions, Publ. Mat. 40 (1996), no. 2, 387–396. https://doi.org/10.5565/PUBLMAT_40296_10
Lerner, A. K. and Nazarov, F., Intuitive dyadic calculus: The basics, Expositiones Mathematicae (2018), to appear. https://doi.org/https://doi.org/10.1016/j.exmath.2018.01.001
Lerner, A. K., Ombrosi, S., Pérez, C., Torres, R. H., and Trujillo-González, R., New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222–1264. https://doi.org/10.1016/j.aim.2008.10.014
Li, K., Moen, K., and Sun, W., The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators, J. Fourier Anal. Appl. 20 (2014), no. 4, 751–765. https://doi.org/10.1007/s00041-014-9326-5
Mohanty, P. and Shrivastava, S., A note on the bilinear Littlewood-Paley square function, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2095–2098. https://doi.org/10.1090/S0002-9939-10-10233-0
Ratnakumar, P. K. and Shrivastava, S., A remark on bilinear Littlewood-Paley square functions, Monatsh. Math. 176 (2015), no. 4, 615–622. https://doi.org/10.1007/s00605-014-0668-5
Rubio de Francia, J. L., Estimates for some square functions of Littlewood-Paley type, Publ. Sec. Mat. Univ. Autònoma Barcelona 27 (1983), no. 2, 81–108.
Rubio de Francia, J. L., A Littlewood-Paley inequality for arbitrary intervals, Rev. Mat. Iberoamericana 1 (1985), no. 2, 1–14. https://doi.org/10.4171/RMI/7
