Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Weighted norm inequalities for multilinear Fourier multipliers with critical Besov regularity


Authors: Mai Fujita and Naohito Tomita
Journal: Proc. Amer. Math. Soc. 146 (2018), 555-569
MSC (2010): Primary 42B15, 42B25, 42B35
DOI: https://doi.org/10.1090/proc/13680
Published electronically: October 23, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, weighted norm inequalities for multilinear Fourier multipliers with Besov regularity are discussed. As a result, we obtain a limiting case of Hörmander type multiplier theorem for multilinear operators.


References [Enhancements On Off] (What's this?)

  • [1] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
  • [2] Bui Huy Qui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), no. 3, 581-605. MR 676560
  • [3] The Anh Bui and Xuan Thinh Duong, Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers, Bull. Sci. Math. 137 (2013), no. 1, 63-75. MR 3007100, https://doi.org/10.1016/j.bulsci.2012.04.001
  • [4] Ronald R. Coifman and Yves Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (French). With an English summary. MR 518170
  • [5] Javier Duoandikoetxea, Fourier analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001. MR 1800316
  • [6] Mai Fujita and Naohito Tomita, Weighted norm inequalities for multilinear Fourier multipliers, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6335-6353. MR 2958938, https://doi.org/10.1090/S0002-9947-2012-05700-X
  • [7] Mai Fujita and Naohito Tomita, A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity, J. Math. Anal. Appl. 409 (2014), no. 2, 630-636. MR 3103182, https://doi.org/10.1016/j.jmaa.2013.07.041
  • [8] Loukas Grafakos and Hanh Van Nguyen, Multilinear Fourier multipliers with minimal Sobolev regularity, I, Colloq. Math. 144 (2016), no. 1, 1-30. MR 3485878, https://doi.org/10.4064/cm6771-10-2015
  • [9] Loukas Grafakos, Akihiko Miyachi, and Naohito Tomita, On multilinear Fourier multipliers of limited smoothness, Canad. J. Math. 65 (2013), no. 2, 299-330. MR 3028565, https://doi.org/10.4153/CJM-2012-025-9
  • [10] Loukas Grafakos and Hanh Van Nguyen, Multilinear Fourier multipliers with minimal Sobolev regularity, I, Colloq. Math. 144 (2016), no. 1, 1-30. MR 3485878, https://doi.org/10.4064/cm6771-10-2015
  • [11] Loukas Grafakos and Zengyan Si, The Hörmander multiplier theorem for multilinear operators, J. Reine Angew. Math. 668 (2012), 133-147. MR 2948874
  • [12] Loukas Grafakos and Rodolfo H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 51 (2002), no. 5, 1261-1276. MR 1947875, https://doi.org/10.1512/iumj.2002.51.2114
  • [13] Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $ L^{p}$spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235-254. MR 561835, https://doi.org/10.2307/1998156
  • [14] Andrei K. Lerner, Sheldy Ombrosi, Carlos Pérez, Rodolfo H. Torres, and Rodrigo Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. MR 2483720, https://doi.org/10.1016/j.aim.2008.10.014
  • [15] Kangwei Li and Wenchang Sun, Weighted estimates for multilinear Fourier multipliers, Forum Math. 27 (2015), no. 2, 1101-1116. MR 3334095, https://doi.org/10.1515/forum-2012-0128
  • [16] Wenjuan Li, Qingying Xue, and Kôzô Yabuta, Weighted version of Carleson measure and multilinear Fourier multiplier, Forum Math. 27 (2015), no. 2, 787-805. MR 3334082, https://doi.org/10.1515/forum-2012-0083
  • [17] Akihiko Miyachi and Naohito Tomita, Minimal smoothness conditions for bilinear Fourier multipliers, Rev. Mat. Iberoam. 29 (2013), no. 2, 495-530. MR 3047426, https://doi.org/10.4171/RMI/728
  • [18] Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319
  • [19] Andreas Seeger, Estimates near $ L^1$ for Fourier multipliers and maximal functions, Arch. Math. (Basel) 53 (1989), no. 2, 188-193. MR 1004277, https://doi.org/10.1007/BF01198570
  • [20] Mitsuru Sugimoto, Pseudo-differential operators on Besov spaces, Tsukuba J. Math. 12 (1988), no. 1, 43-63. MR 949902
  • [21] Naohito Tomita, A Hörmander type multiplier theorem for multilinear operators, J. Funct. Anal. 259 (2010), no. 8, 2028-2044. MR 2671120, https://doi.org/10.1016/j.jfa.2010.06.010
  • [22] Naohito Tomita, A remark on multilinear Fourier multipliers satisfying Besov estimates, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, pp. 111-121. MR 3050809
  • [23] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42B15, 42B25, 42B35

Retrieve articles in all journals with MSC (2010): 42B15, 42B25, 42B35


Additional Information

Mai Fujita
Affiliation: Faculty of Integrated Media, Wakkanai Hokusei Gakuen University, Wakkanai, Hokkaido 097-0013, Japan
Email: fujita@wakhok.ac.jp

Naohito Tomita
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: tomita@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/proc/13680
Keywords: Multilinear Fourier multipliers, $A_p$-weights, Besov spaces
Received by editor(s): September 9, 2016
Published electronically: October 23, 2017
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society