| Title:
|
Estimates for the commutator of bilinear Fourier multiplier (English) |
| Author:
|
Hu, Guoen |
| Author:
|
Yi, Wentan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
1113-1134 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $b_1, b_2 \in {\rm BMO}(\mathbb {R}^n)$ and $T_{\sigma }$ be a bilinear Fourier multiplier operator with associated multiplier $\sigma $ satisfying the Sobolev regularity that $\sup _{\kappa \in \mathbb {Z}} \|\sigma _{\kappa }\| _{W^{s_1,s_2}(\mathbb {R}^{2n})}<\infty $ for some $s_1,s_2\in (n/2,n]$. In this paper, the behavior on $L^{p_1}(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_1,p_2\in (1,\infty ))$, on $H^1(\mathbb {R}^n)\times L^{p_2}(\mathbb {R}^n)$ $(p_2\in [2,\infty ))$, and on $H^1(\mathbb {R}^n)\times H^1(\mathbb {R}^n)$, is considered for the commutator $T_{{\sigma }, \vec {b}} $ defined by $$ \begin {aligned} T_{\sigma ,\vec {b}} (f_1,f_2) (x)=&b_1(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(b_1f_1, f_2)(x) &+ b_2(x)T_{\sigma }(f_1, f_2)(x)-T_{\sigma }(f_1, b_2f_2)(x) . \end {aligned} $$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those of the bilinear Fourier multiplier operator which were established by Miyachi and Tomita. (English) |
| Keyword:
|
bilinear Fourier multiplier operator |
| Keyword:
|
commutator |
| Keyword:
|
Hardy space |
| MSC:
|
42B15 |
| idZBL:
|
Zbl 06373964 |
| idMR:
|
MR3165517 |
| DOI:
|
10.1007/s10587-013-0074-5 |
| . |
| Date available:
|
2014-01-28T14:24:16Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143619 |
| . |
| Reference:
|
[1] Anh, B. T., Duong, X. T.: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers.Bull. Sci. Math. 137 (2013), 63-75. Zbl 1266.42019, MR 3007100, 10.1016/j.bulsci.2012.04.001 |
| Reference:
|
[2] Christ, M.: Weak type (1,1) bounds for rough operators.Ann. Math. (2) 128 (1998), 19-42. MR 0951506 |
| Reference:
|
[3] Coifman, R. R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals.Trans. Am. Math. Soc. 212 (1975), 315-331. Zbl 0324.44005, MR 0380244, 10.1090/S0002-9947-1975-0380244-8 |
| Reference:
|
[4] Coifman, R. R., Meyer, Y.: Nonlinear harmonic analysis, operator theory and PDE.Beijing Lectures in Harmonic Analysis. (Summer School in Analysis, Beijing, The People's Republic of China, September 1984). Annals of Mathematics Studies 112 Princeton University Press Princeton (1986), 3-45. MR 0864370 |
| Reference:
|
[5] Fujita, M., Tomita, N.: Weighted norm inequalities for multilinear Fourier multipliers.Trans. Am. Math. Soc. 364 (2012), 6335-6353. Zbl 1275.42015, MR 2958938, 10.1090/S0002-9947-2012-05700-X |
| Reference:
|
[6] García-Cuerva, J., Harboure, E., Segovia, C., Torrea, J. L.: Weighted norm inequalities for commutators of strongly singular integrals.Indiana Univ. Math. J. 40 (1991), 1397-1420. Zbl 0765.42012, MR 1142721, 10.1512/iumj.1991.40.40063 |
| Reference:
|
[7] Grafakos, L., Miyachi, A., Tomita, N.: On multilinear Fourier multipliers of limited smoothness.Can. J. Math. 65 (2013), 299-330. Zbl 1275.42016, MR 3028565, 10.4153/CJM-2012-025-9 |
| Reference:
|
[8] Grafakos, L., Si, Z.: The Hörmander multiplier theorem for multilinear operators.J. Reine Angew. Math. 668 (2012), 133-147. Zbl 1254.42017, MR 2948874 |
| Reference:
|
[9] Grafakos, L., Torres, R. H.: Multilinear Calderón-Zygmund theory.Adv. Math. 165 (2002), 124-164. Zbl 1032.42020, MR 1880324, 10.1006/aima.2001.2028 |
| Reference:
|
[10] Hu, G., Lin, C.-C.: Weighted norm inequalities for multilinear singular integral operators and applications.arXiv: 1208.6346. |
| Reference:
|
[11] Kenig, C. E., Stein, E. M.: Multilinear estimates and fractional integration.Math. Res. Lett. 6 (1999), 1-15. Zbl 0952.42005, MR 1682725, 10.4310/MRL.1999.v6.n1.a1 |
| Reference:
|
[12] Kurtz, D. S., Wheeden, R. L.: Results on weighted norm inequalities for multipliers.Trans. Am. Math. Soc. 255 (1979), 343-362. Zbl 0427.42004, MR 0542885, 10.1090/S0002-9947-1979-0542885-8 |
| Reference:
|
[13] 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 (2009), 1222-1264. Zbl 1160.42009, MR 2483720, 10.1016/j.aim.2008.10.014 |
| Reference:
|
[14] Meda, S., Sjögren, P., Vallarino, M.: On the $H^1-L^1$ boundedness of operators.Proc. Am. Math. Soc. 136 (2008), 2921-2931. Zbl 1273.42021, MR 2399059, 10.1090/S0002-9939-08-09365-9 |
| Reference:
|
[15] Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers.Rev. Mat. Iberoam. 29 495-530 (2013). Zbl 1275.42017, MR 3047426, 10.4171/RMI/728 |
| Reference:
|
[16] Pérez, C., Torres, R. H.: Sharp maximal function estimates for multilinear singular integrals.Harmonic Analysis at Mount Holyoke. Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, Mount Holyoke College, South Hadley, MA, USA, June 25--July 5, 2001 W. Beckner et al. American Mathematical Society Providence Contemp. Math. 320 (2003), 323-331. Zbl 1045.42011, MR 1979948, 10.1090/conm/320/05615 |
| Reference:
|
[17] Tomita, N.: A Hörmander type multiplier theorem for multilinear operators.J. Funct. Anal. 259 (2010), 2028-2044. Zbl 1201.42005, MR 2671120, 10.1016/j.jfa.2010.06.010 |
| . |
