The boundedness of bilinear Fourier integral operators on $L^2\times L^2$

Document Type : Research Paper

Authors

Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran1, Ahmed Ben Bella. B.P. 1524 El M'naouar, Oran, Algeria

Abstract

In this paper, the regularity of bilinear Fourier integral operators on $L^2\times L^2$ are determined in the framework of Besov spaces. Our result improves the $L^2\times L^2\rightarrow L^1$ boundedness of those operators with symbols in the bilinear Hormander classes.

Keywords

[1] H. Abels, Pseudodifferential and singular integral operators. An introduction with applications, de Gruyter, Berlin,
2012.
[2] OF. Aid and A. Senoussaoui, The boundedness of h-admissible Fourier integral operators on Bessel potential
spaces, Turk. J. Math. 43 (2019), no. 5, 2125 – 2141.
[3] O.F. Aid and A. Senoussaoui, Hs
-Boundeness of a class of a Fourier integral operators, Math. Slovaca 71 (2021),
no. 4, 889–902.
[4] K. Asada and D. Fujiwara, On some oscillatory transformations in L
2
(R
n), Japanese J. Math. 4 (1978), no. 2,
299–361.
[5] A. B´enyi and T. Oh, ´ On a class of bilinear pseudodifferential operators, J. Funct. Spaces Appl. 2013 (2013), 1–5.
[6] A. B´enyi and K.A. Okoudjou, ´ Bilinear pseudodifferential operators on modulation spaces, J. Fourier Anal. Appl.
10 (2004), 301–313.
[7] A. B´enyi and K.A. Okoudjou, ´ Modulation spaces estimates for multilinear pseudodifferential operators, Stud.
Math. 172 (2006), 169–180.
[8] A. B´enyi, K. Groechenig, C. Heil and K. Okoudjou, ´ Modulation spaces and a class of bounded multilinear pseudodifferential operators, J. Oper. Theory 54 (2005), 389–401.
[9] A. B´enyi, D. Maldonado, V. Naibo and R.H. Torres, ´ On the H¨ormander classes of bilinear pseudodifferential
operators, Integr. Equ. Oper. Theory 67 (2010), 341–364.
[10] M. Cappiello and J. Toft, Pseudo-differential operators in a Gelfand–Shilov setting, Math. Nachr. 290 (2017),
738–755.
[11] R.R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer.
Math. Soc. 212 (1975), 315–331.[12] R.R. Coifman and Y. Meyer, Commutateurs d’int´egrales singuli`eres et op´erateurs multilin´eaires, Ann. Inst. Fourier
(Grenoble) 28 (1978), 177–202.
[13] R.R. Coifman and Y. Meyer, Au del`a des op´erateurs pseudo-diff´erentiels, Ast´erisque, No. 57, Soci´et´e
Math´ematique de France, 1979.
[14] E. Cordero and KA. Okoudjou, Multilinear localization operators, J. Math. Anal. Appl. 325 (2007), 1103–1116.
[15] L. Grafakos and R.H. Torres, Multilinear Calder´on–Zygmund theory, Adv. Math. 165 (2002), 124–164.
[16] L. Grafakos, Modern fourier analysis, Third edition, Springer, New York, 2014.
[17] N. Hamada, N. Shida and N. Tomita, On the ranges of bilinear pseudo-differential operators of S0,0-type on
L
2 × L
2
, J. Funct. Anal. 280 (2021), no. 3, 108826.
[18] T. Kato, Bilinear pseudo-differential operators with exotic class symbols of limited smoothness, J. Fourier Anal.
Appl. 27 (2021), 1–56.
[19] K. Koezuka and N. Tomita, Bilinear pseudo-differential operators with symbols in BSm
1,1 on Triebel–Lizorkin
spaces, J. Fourier Anal. Appl. 24 (2018) 309–319.
[20] S. Molahajloo, K.A. Okoudjou and G.E. Pfander, Boundedness of multilinear pseudodifferential operators on
modulation spaces, J. Fourier Anal. Appl. 22 (2016), 1381–1415.
[21] S. Rodr´─▒guez-L´opez, D. Rule and W. Staubach, Global boundedness of a class of multilinear Fourier integral
operators, Forum Math. Sigma 9 (2021), 1–45.
Volume 13, Issue 2
July 2022
Pages 1565-1575
  • Receive Date: 07 October 2021
  • Revise Date: 22 April 2022
  • Accept Date: 03 May 2022