logo CRAS
Comptes Rendus. Mathématique
Functional analysis
Periodic Fourier integral operators in L p -spaces
Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 547-553.

In this note we give sufficient conditions for the L p boundedness of periodic Fourier integral operators. We also refer to them as Fourier series operators (FSOs). The main tool will be the notion of full symbol and the periodic analysis on the torus introduced by Ruzhansky and Turunen [34].

Dans cette note nous présentons les conditions suffisantes pour la continuité des opérateurs intégraux de Fourier périodique qui sont appelés aussi séries des opérateurs de Fourier. Le principal outil est la notion des opérateurs intégraux de Fourier et l’analyse discrête notamment l’analyse périodique dans le tore introduite par Ruzhansky et Turunen [34].

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.194
Duván Cardona 1; Rekia Messiouene 2; Abderrahmane Senoussaoui 2

1 Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, Ghent, Belgium
2 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Ben Bella. B.P. 1524 El M’naouar, Oran, Algeria
@article{CRMATH_2021__359_5_547_0,
     author = {Duv\'an Cardona and Rekia Messiouene and Abderrahmane Senoussaoui},
     title = {Periodic {Fourier} integral operators in $L^p$-spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {547--553},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {5},
     year = {2021},
     doi = {10.5802/crmath.194},
     language = {en},
}
TY  - JOUR
TI  - Periodic Fourier integral operators in $L^p$-spaces
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 547
EP  - 553
VL  - 359
IS  - 5
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.194
DO  - 10.5802/crmath.194
LA  - en
ID  - CRMATH_2021__359_5_547_0
ER  - 
%0 Journal Article
%T Periodic Fourier integral operators in $L^p$-spaces
%J Comptes Rendus. Mathématique
%D 2021
%P 547-553
%V 359
%N 5
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.194
%R 10.5802/crmath.194
%G en
%F CRMATH_2021__359_5_547_0
Duván Cardona; Rekia Messiouene; Abderrahmane Senoussaoui. Periodic Fourier integral operators in $L^p$-spaces. Comptes Rendus. Mathématique, Volume 359 (2021) no. 5, pp. 547-553. doi : 10.5802/crmath.194. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.194/

[1] Mikhail S. Agranovich Spectral properties of elliptic pseudo-differential operators on a closed curve, Funct. Anal. Appl., Volume 13 (1971), pp. 279-281 | Article

[2] Kenji Asada; Daisuke Fujiwara On some oscillatory integral transformations in L 2 ( n ), Jpn. J. Math., Volume 4 (1978) no. 2, pp. 299-361 | Article | MR: 528863 | Zbl: 0402.44008

[3] Duván Cardona Estimativos L 2 para una clase de operadores pseudodiferenciales definidos en el toro, Rev. Integr., Volume 31 (2013) no. 2, pp. 147-152 | Zbl: 1295.47049

[4] Duván Cardona Hölder estimates for pseudo-differential operators on 𝕋 1 , J. Pseudo-Differ. Oper. Appl., Volume 5 (2014) no. 4, pp. 517-525 | Article | Zbl: 1328.58023

[5] Duván Cardona Weak type (1, 1) bounds for a class of periodic pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., Volume 5 (2014) no. 4, pp. 507-515 | Article | MR: 3274884 | Zbl: 1321.58018

[6] Duván Cardona Hölder–Besov boundedness for periodic pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., Volume 8 (2016) no. 1, pp. 13-34 | Article | MR: 3610250 | Zbl: 1376.43003

[7] Duván Cardona On the boundedness of periodic pseudo-differential operators, Monatsh. Math., Volume 185 (2018) no. 2, pp. 189-206 | Article | MR: 3748333 | Zbl: 1397.58013

[8] Duván Cardona Pseudo-differential operators in Hölder spaces revisited. Weyl-Hörmander calculus and Ruzhansky-Turunen classes, Mediterr. J. Math., Volume 16 (2019) no. 6, 148, 17 pages | MR: 4023822 | Zbl: 1427.35383

[9] Duván Cardona; Vishvesh Kumar Multilinear analysis for discrete and periodic pseudo-differential operators in Lp spaces, Rev. Integr., Volume 36 (2018) no. 2, pp. 151-164 | Zbl: 1426.58007

[10] Duván Cardona; Vishvesh Kumar L p -boundedness and L p -nuclearity of multilinear pseudo-differential operators on n and the torus 𝕋 n , J. Fourier Anal. Appl., Volume 25 (2019) no. 6, pp. 2973-3017 | Article | MR: 4029168 | Zbl: 1429.58036

[11] Duván Cardona; Vishvesh Kumar The nuclear trace of periodic vector-valued pseudo-differential operators with applications to index theory (2019) (https://arxiv.org/abs/1901.10010, to appear in Math. Nachr.)

[12] Duván Cardona; Rekia Messiouene; Abderrahmane Senoussaoui L p -bounds for periodic Fourier integral operators (2019) (https://arxiv.org/abs/1807.09892)

[13] Duván Cardona; Michael Ruzhansky Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups (2021) (https://arxiv.org/abs/2008.09651)

[14] Sandro Coriasco; Michael Ruzhansky On the boundedness of Fourier integral operators on L p ( n ), C. R. Math. Acad. Sci. Paris, Volume 348 (2010) no. 15-16, pp. 847-851 | Article | MR: 2677978 | Zbl: 1197.35340

[15] Sandro Coriasco; Michael Ruzhansky Global L p continuity of Fourier integral operators, Trans. Am. Math. Soc., Volume 366 (2014) no. 5, pp. 2575-2596 | Article | MR: 3165647 | Zbl: 1301.35231

[16] Julio Delgado L p bounds for pseudo-differential operators on the torus, Pseudo-differential operators, generalized functions and asymptotics (Operator Theory: Advances and Applications), Volume 231, Birkhäuser/Springer, 2013, pp. 103-116 | Article | MR: 3075935 | Zbl: 1283.58019

[17] Julio Delgado; Michael Ruzhansky L p -bounds for pseudo-differential operators on compact Lie groups, J. Inst. Math. Jussieu, Volume 18 (2019) no. 3, pp. 531-559 | Article | MR: 3936641 | Zbl: 1419.35263

[18] Johannes J. Duistermaat; Lars V. Hörmander Fourier integral operators. II, Acta Math., Volume 128 (1972) no. 3-4, pp. 183-269 | Article | MR: 388464 | Zbl: 0232.47055

[19] G. I. Éskin Degenerate elliptic pseudodifferential equations of principal type, Mat. Sb., Volume 82 (1970) no. 124, pp. 585-628 | MR: 510219 | Zbl: 0239.35047

[20] Daisuke Fujiwara Construction of the fundamental solution for the Schrödinger equations, Proc. Japan Acad., Volume 55 (1979) no. 1, pp. 10-14 | Zbl: 0421.35018

[21] Lars V. Hörmander Fourier integral operators. I, Acta Math., Volume 127 (1971) no. 1-2, pp. 79-183 | Article | MR: 388463

[22] Lars V. Hörmander The analysis of linear partial differential operators. III: Pseudo-differential operators, Grundlehren der Mathematischen Wissenschaften, 274, Springer, 1985 | Zbl: 0601.35001

[23] Hitoshi Kumano-go A calculus of Fourier integral operators on n and the fundamental solution for an operator of hyperbolic type, Commun. Partial Differ. Equations, Volume 1 (1976) no. 1, pp. 1-44 | Article | MR: 397482 | Zbl: 0331.42012

[24] William McLean Local and Global description of periodic pseudo-differential operators, Math. Nachr., Volume 150 (1991), pp. 151-161 | Article | MR: 1109651 | Zbl: 0729.35149

[25] Akihiko Miyachi On some estimates for the wave equation in L p and H p , J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1998), pp. 331-354 | Zbl: 0437.35042

[26] Shahla Molahajloo; Man W. Wong Pseudo-differential operators on 𝕊 1 , New developments in pseudo-differential operators (Operator Theory: Advances and Applications), Volume 189, Birkhäuser, 2008, pp. 297-306 | Article | Zbl: 1210.47073

[27] Shahla Molahajloo; Man W. Wong Ellipticity, Fredholmness and spectral invariance of pseudo-differential operators on 𝕊 1 , J. Pseudo-Differ. Oper. Appl., Volume 1 (2010) no. 2, pp. 183-205 | Article | MR: 2679899 | Zbl: 1222.47075

[28] Juan C. Peral L p -estimates for the wave equation, J. Funct. Anal., Volume 36 (1980), pp. 114-145 | Article | MR: 568979 | Zbl: 0442.35017

[29] Michael Ruzhansky Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tracts, 131, Centrum voor Wiskunde en Informatica, 2001 | MR: 1837457 | Zbl: 0974.35138

[30] Michael Ruzhansky; Mitsuru Sugimoto Global L 2 -boundedness theorems for a class of Fourier integral operators, Commun. Partial Differ. Equations, Volume 31 (2006) no. 4-6, pp. 547-569 | Article | MR: 2233032 | Zbl: 1106.35158

[31] Michael Ruzhansky; Mitsuru Sugimoto A smoothing property of Schrödinger equations in the critical case, Math. Ann., Volume 335 (2006) no. 3, pp. 645-673 | Article | Zbl: 1109.35096

[32] Michael Ruzhansky; Mitsuru Sugimoto Weighted Sobolev L 2 estimates for a class of Fourier integral operators, Math. Nachr., Volume 284 (2011) no. 13, pp. 1715-1738 | Article | MR: 2832678 | Zbl: 1234.35334

[33] Michael Ruzhansky; Mitsuru Sugimoto A local-to-global boundedness argument and Fourier integral operators, J. Math. Anal. Appl., Volume 473 (2019) no. 2, pp. 892-904 | Article | MR: 3912857 | Zbl: 07052455

[34] Michael Ruzhansky; Ville Turunen Pseudo-differential operators and symmetries: Background analysis and advanced topics, Pseudo-Differential Operators. Theory and Applications, 2, Birkhäuser, 2010 | Zbl: 1193.35261

[35] Michael Ruzhansky; Ville Turunen Quantization of pseudo-sifferential operators on the torus, J. Fourier Anal. Appl., Volume 16 (2010) no. 6, pp. 943-982 | Article | Zbl: 1252.58013

[36] Michael Ruzhansky; Jens Wirth Dispersive type estimates for Fourier integrals and applications to hyperbolic systems, Discrete Contin. Dyn. Syst., Volume 2011 (2011), pp. 1263-1270 | Article | MR: 3012928 | Zbl: 1306.35068

[37] Michael Ruzhansky; Jens Wirth L p Fourier multipliers on compact Lie groups, Math. Z., Volume 280 (2015) no. 3-4, pp. 621-642 | Article | MR: 3369343 | Zbl: 1331.43002

[38] Andreas Seeger; Christopher D. Sogge; Elias M. Stein Regularity properties of Fourier integral operators, Ann. Math., Volume 134 (1991) no. 2, pp. 231-251 | Article | MR: 1127475 | Zbl: 0754.58037

[39] Elias M. Stein Harmonic Analysis, Princeton Mathematical Series, 43, Princeton University Press, 1993 | Zbl: 0821.42001

[40] Terence Tao The weak-type (1,1) of Fourier integral operators of order -(n-1)/2, J. Aust. Math. Soc., Volume 76 (2004) no. 1, pp. 1-21 | MR: 2029306 | Zbl: 1059.42013

[41] Ville Turunen; Gennadi Vainikko On symbol analysis of periodic pseudodifferential operators, Z. Anal. Anwend., Volume 17 (1998) no. 1, pp. 9-22 | Article | MR: 1616044 | Zbl: 0899.47039

Cited by Sources: