Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Global $ L^p$ continuity of Fourier integral operators


Authors: Sandro Coriasco and Michael Ruzhansky
Journal: Trans. Amer. Math. Soc. 366 (2014), 2575-2596
MSC (2010): Primary 35S30; Secondary 42B30, 46E30, 47B34
DOI: https://doi.org/10.1090/S0002-9947-2014-05911-4
Published electronically: January 13, 2014
MathSciNet review: 3165647
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish global $ L^p(\mathbb{R}^{n})$-regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $ L^p(\mathbb{R}^{n})$, $ 1<p<\infty $, as well as to be bounded from Hardy space $ H^1(\mathbb{R}^{n})$ to $ L^1(\mathbb{R}^{n})$. This extends local $ L^p$-regularity properties of Fourier integral operators, as well as results of global $ L^2(\mathbb{R}^{n})$ boundedness, to the global setting of $ L^p(\mathbb{R}^{n})$. Global boundedness in weighted Sobolev spaces $ W^{\sigma ,p}_s(\mathbb{R}^{n})$ is also established, and applications to hyperbolic partial differential equations are given.


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

  • [1] Kenji Asada, On the $ L^{2}$ boundedness of Fourier integral operators in $ {\bf R}^{n}$, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 5, 249-253. MR 624035 (84c:47053)
  • [2] Kenji Asada and Daisuke Fujiwara, On some oscillatory integral transformations in $ L^{2}({\bf R}^{n})$, Japan. J. Math. (N.S.) 4 (1978), no. 2, 299-361. MR 528863 (80d:47076)
  • [3] M. Beals, $ L^p$ boundedness of Fourier integrals. Mem. Amer. Math. Soc. 264 (1982).
  • [4] A. Boulkhemair, Estimations $ L^2$ précisées pour des intégrales oscillantes, Comm. Partial Differential Equations 22 (1997), no. 1-2, 165-184 (French). MR 1434142 (98m:42014), https://doi.org/10.1080/03605309708821259
  • [5] Alberto-P. Calderón and Rémi Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374-378. MR 0284872 (44 #2096)
  • [6] A. G. Childs, On the $ L^{2}$-boundedness of pseudo-differential operators, Proc. Amer. Math. Soc. 61 (1976), no. 2, 252-254 (1977). MR 0442755 (56 #1135)
  • [7] 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 (81b:47061)
  • [8] Elena Cordero, Fabio Nicola, and Luigi Rodino, Boundedness of Fourier integral operators on $ {\mathcal {F}}L^p$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6049-6071. MR 2529924 (2010j:47066), https://doi.org/10.1090/S0002-9947-09-04848-X
  • [9] Elena Cordero, Fabio Nicola, and Luigi Rodino, On the global boundedness of Fourier integral operators, Ann. Global Anal. Geom. 38 (2010), no. 4, 373-398. MR 2733369 (2012d:35434), https://doi.org/10.1007/s10455-010-9219-z
  • [10] H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal. 18 (1975), 115-131. MR 0377599 (51 #13770)
  • [11] H. O. Cordes, The technique of pseudodifferential operators, London Mathematical Society Lecture Note Series, vol. 202, Cambridge University Press, Cambridge, 1995. MR 1314815 (96b:35001)
  • [12] S. Coriasco, Fourier integral operators in SG classes. I. Composition theorems and action on SG Sobolev spaces, Rend. Sem. Mat. Univ. Politec. Torino 57 (1999), no. 4, 249-302 (2002). MR 1972487 (2004g:47069)
  • [13] Sandro Coriasco, Fourier integral operators in SG classes. II. Application to SG hyperbolic Cauchy problems, Ann. Univ. Ferrara Sez. VII (N.S.) 44 (1998), 81-122 (1999) (English, with English and Italian summaries). MR 1744134 (2001m:47106)
  • [14] S. Coriasco and M. Ruzhansky, On the boundedness of Fourier integral operators on $ L^p\textup {(}\mathbb{R}^n$), C. R. Math. Acad. Sci. Paris 348 (2010), no. 15-16, 847-851. MR 2677978
  • [15] J. J. Duistermaat, Fourier integral operators, Progress in Mathematics, vol. 130, Birkhäuser Boston Inc., Boston, MA, 1996. MR 1362544 (96m:58245)
  • [16] G. I. Èskin, Degenerate elliptic pseudodifferential equations of principal type, Mat. Sb. (N.S.) 82(124) (1970), 585-628 (Russian). MR 0510219 (58 #23202)
  • [17] Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79-183. MR 0388463 (52 #9299)
  • [18] Lars Hörmander, $ L^{2}$ estimates for Fourier integral operators with complex phase, Ark. Mat. 21 (1983), no. 2, 283-307. MR 727350 (85h:47058), https://doi.org/10.1007/BF02384316
  • [19] L. Hörmander, The analysis of linear partial differential operators. Vols. III-IV, Springer-Verlag, New York, Berlin, 1985.
  • [20] Hitoshi 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 1 (1976), no. 1, 1-44. MR 0397482 (53 #1341)
  • [21] Hitoshi Kumano-go, Pseudodifferential operators, MIT Press, Cambridge, Mass., 1981. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase. MR 666870 (84c:35113)
  • [22] Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343-362. MR 542885 (81j:42021), https://doi.org/10.2307/1998180
  • [23] Anders Melin and Johannes Sjöstrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), Springer, Berlin, 1975, pp. 120-223. Lecture Notes in Math., Vol. 459. MR 0431289 (55 #4290)
  • [24] Akihiko Miyachi, On some estimates for the wave equation in $ L^{p}$ and $ H^{p}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 331-354. MR 586454 (83g:35060)
  • [25] Juan C. Peral, $ L^{p}$ estimates for the wave equation, J. Funct. Anal. 36 (1980), no. 1, 114-145. MR 568979 (81k:35089), https://doi.org/10.1016/0022-1236(80)90110-X
  • [26] Michael Ruzhansky, On the sharpness of Seeger-Sogge-Stein orders, Hokkaido Math. J. 28 (1999), no. 2, 357-362. MR 1698428 (2000e:35247)
  • [27] M. V. Ruzhanskiĭ, Singularities of affine fibrations in the theory of regularity of Fourier integral operators, Uspekhi Mat. Nauk 55 (2000), no. 1(331), 99-170 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 1, 93-161. MR 1751819 (2001i:35317), https://doi.org/10.1070/rm2000v055n01ABEH000250
  • [28] M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tract, vol. 131, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2001. MR 1837457 (2002f:58048)
  • [29] Michael Ruzhansky and Mitsuru Sugimoto, Global $ L^2$-boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations 31 (2006), no. 4-6, 547-569. MR 2233032 (2007e:35310), https://doi.org/10.1080/03605300500455958
  • [30] Michael Ruzhansky and Mitsuru Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann. 335 (2006), no. 3, 645-673. MR 2221126 (2007e:35237), https://doi.org/10.1007/s00208-006-0757-4
  • [31] Michael Ruzhansky and Mitsuru Sugimoto, Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations, Pseudo-differential operators and related topics, Oper. Theory Adv. Appl., vol. 164, Birkhäuser, Basel, 2006, pp. 65-78. MR 2243967 (2007d:35299), https://doi.org/10.1007/3-7643-7514-0_5
  • [32] Michael Ruzhansky and Mitsuru Sugimoto, Weighted Sobolev $ L^2$ estimates for a class of Fourier integral operators, Math. Nachr. 284 (2011), no. 13, 1715-1738. MR 2832678 (2012h:45012), https://doi.org/10.1002/mana.200910080
  • [33] Andreas Seeger, Christopher D. Sogge, and Elias M. Stein, Regularity properties of Fourier integral operators, Ann. of Math. (2) 134 (1991), no. 2, 231-251. MR 1127475 (92g:35252), https://doi.org/10.2307/2944346
  • [34] Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579 (94c:35178)
  • [35] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
  • [36] Mitsuru Sugimoto, $ L^p$-boundedness of pseudodifferential operators satisfying Besov estimates. I, J. Math. Soc. Japan 40 (1988), no. 1, 105-122. MR 917398 (89g:47068), https://doi.org/10.2969/jmsj/04010105
  • [37] Mitsuru Sugimoto, On some $ L^p$-estimates for hyperbolic equations, Ark. Mat. 30 (1992), no. 1, 149-163. MR 1171100 (94f:35079), https://doi.org/10.1007/BF02384867
  • [38] Terence Tao, The weak-type $ (1,1)$ of Fourier integral operators of order $ -(n-1)/2$, J. Aust. Math. Soc. 76 (2004), no. 1, 1-21. MR 2029306 (2004j:42008), https://doi.org/10.1017/S1446788700008661

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35S30, 42B30, 46E30, 47B34

Retrieve articles in all journals with MSC (2010): 35S30, 42B30, 46E30, 47B34


Additional Information

Sandro Coriasco
Affiliation: Dipartimento di Matematica “G. Peano”, Università di Torino, V. C. Alberto, n. 10, Torino I-10126, Italy
Email: sandro.coriasco@unito.it

Michael Ruzhansky
Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
Email: m.ruzhansky@imperial.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2014-05911-4
Keywords: Fourier integral operators, global $L^p(\mathbb{R}^{n})$ boundedness
Received by editor(s): December 9, 2010
Received by editor(s) in revised form: July 9, 2012
Published electronically: January 13, 2014
Additional Notes: The second author was supported in part by the EPSRC grants EP/E062873/1 and EP/G007233/1.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society