On the boundedness of Fourier integral operators on $ {L}^{p}({\mathbb{R}}^{n})$

Sandro Coriasco; Michael Ruzhansky

doi:10.1016/j.crma.2010.07.025

Mathematical Analysis/Harmonic Analysis

On the boundedness of Fourier integral operators on

L^{p} (R^{n})

[Sur la continuité des opérateurs intégraux de Fourier sur

L^{p} (R^{n})

]

Présenté par : Jean-Michel Bony

Sandro Coriasco ¹ ; Michael Ruzhansky ²

¹ Dipartimento di Matematica, Università di Torino, V. C. Alberto, n. 10, Torino, Italy
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 847-851.

Résumé
Abstract

Dans cette Note nous présentons des estimations globales pour les opérateurs intégraux de Fourier dans les espaces $L^{p} (R^{n})$ . Les questions d'intérêt sont les conditions des décroissance pour les amplitudes. Les résultats sont présentés sous des conditions différentes sur la fonction de phase et l'amplitude.

The aim of this Note is to present global $L^{p}$ boundedness results for Fourier integral operators in $R^{n}$ . The main question is what are the decay conditions on the amplitudes for the operators to be bounded on $L^{p} (R^{n})$ . Results under different sets of assumptions on phase functions and amplitudes are presented.

Reçu le : 2010-03-29
Accepté le : 2010-07-26
Publié le : 2010-08-01

DOI : 10.1016/j.crma.2010.07.025

Affiliations des auteurs :

Sandro Coriasco ¹ ; Michael Ruzhansky ²

¹ Dipartimento di Matematica, Università di Torino, V. C. Alberto, n. 10, Torino, Italy
² Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

@article{CRMATH_2010__348_15-16_847_0,
     author = {Sandro Coriasco and Michael Ruzhansky},
     title = {On the boundedness of {Fourier} integral operators on $ {L}^{p}({\mathbb{R}}^{n})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {847--851},
     publisher = {Elsevier},
     volume = {348},
     number = {15-16},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.025},
     language = {en},
}

TY  - JOUR
AU  - Sandro Coriasco
AU  - Michael Ruzhansky
TI  - On the boundedness of Fourier integral operators on $ {L}^{p}({\mathbb{R}}^{n})$
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 847
EP  - 851
VL  - 348
IS  - 15-16
PB  - Elsevier
DO  - 10.1016/j.crma.2010.07.025
LA  - en
ID  - CRMATH_2010__348_15-16_847_0
ER  -

%0 Journal Article
%A Sandro Coriasco
%A Michael Ruzhansky
%T On the boundedness of Fourier integral operators on $ {L}^{p}({\mathbb{R}}^{n})$
%J Comptes Rendus. Mathématique
%D 2010
%P 847-851
%V 348
%N 15-16
%I Elsevier
%R 10.1016/j.crma.2010.07.025
%G en
%F CRMATH_2010__348_15-16_847_0

Sandro Coriasco; Michael Ruzhansky. On the boundedness of Fourier integral operators on $ {L}^{p}({\mathbb{R}}^{n})$. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 847-851. doi : 10.1016/j.crma.2010.07.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.025/

Bibliographie
Cité par

[1] K. Asada; D. Fujiwara On some oscillatory integral transformations in $L^{2} (R^{n})$ , Japan. J. Math. (N.S.), Volume 4 (1978), pp. 299-361

[2] M. Beals $L^{p}$ boundedness of Fourier integrals, Mem. Amer. Math. Soc., Volume 264 (1982)

[3] A. Boulkhemair Estimations $L^{2}$ precisees pour des intégrales oscillantes, Comm. Partial Differential Equations, Volume 22 (1997), pp. 165-184

[4] A.P. Calderón; R. Vaillancourt On the boundedness of pseudo-differential operators, J. Math. Soc. Japan, Volume 23 (1971), pp. 374-378

[5] A.G. Childs On the $L^{2}$ -boundedness of pseudo-differential operators, Proc. Amer. Math. Soc., Volume 61 (1976), pp. 252-254

[6] R.R. Coifman; Y. Meyer Au-delà des opérateurs pseudo-différentiels, Astérisque, Volume 57 (1978)

[7] E. Cordero; F. Nicola; L. Rodino Boundedness of Fourier integral operators on $F L^{p}$ spaces, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 6049-6071

[8] H.O. Cordes On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., Volume 18 (1975), pp. 115-131

[9] S. Coriasco Fourier integral operators in SG classes I: composition theorems and action on SG Sobolev spaces, Rend. Sem. Mat. Univ. Pol. Torino, Volume 57 (1999), pp. 249-302

[10] G.I. Eskin Degenerate elliptic pseudo-differential operators of principal type, Math. USSR Sbornik, Volume 11 (1970), pp. 539-585

[11] L. Hörmander Fourier integral operators. I, Acta Math., Volume 127 (1971), pp. 79-183

[12] H. Kumano-go A calculus of Fourier integral operators on $R^{n}$ and the fundamental solution for an operator of hyperbolic type, Comm. Partial Differential Equations, Volume 1 (1976), pp. 1-44

[13] A. Miyachi On some estimates for the wave operator in $L^{p}$ and $H^{p}$ , J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 27 (1980), pp. 331-354

[14] J. Peral $L^{p}$ estimates for the wave equation, J. Funct. Anal., Volume 36 (1980), pp. 114-145

[15] M.V. Ruzhansky Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys, Volume 55 (2000), pp. 99-170

[16] M. Ruzhansky Regularity Theory of Fourier Integral Operators with Complex Phases and Singularities of Affine Fibrations, CWI Tract, vol. 131, Math. Centrum, CWI, Amsterdam, 2001

[17] M. Ruzhansky; M. Sugimoto Global $L^{2}$ boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations, Volume 31 (2006), pp. 547-569

[18] M. Ruzhansky; M. Sugimoto A smoothing property of Schrödinger equations in the critical case, Math. Ann., Volume 335 (2006), pp. 645-673

[19] M. Ruzhansky; M. Sugimoto Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations, Oper. Theory Adv. Appl., Volume 164 (2006), pp. 65-78

[20] M. Ruzhansky; M. Sugimoto Weighted Sobolev $L^{2}$ estimates for a class of Fourier integral operators (Math. Nachr., in press) | arXiv

[21] A. Seeger; C.D. Sogge; E.M. Stein Regularity properties of Fourier integral operators, Ann. of Math., Volume 134 (1991), pp. 231-251

[22] M. Sugimoto $L^{2}$ -boundedness of pseudo-differential operators satisfying Besov estimates I, J. Math. Soc. Japan, Volume 40 (1988), pp. 105-122

[23] T. Tao The weak-type $(1, 1)$ of Fourier integral operators of order $- (n - 1) / 2$ , J. Aust. Math. Soc., Volume 76 (2004), pp. 1-21