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Global and local regularity of Fourier integral operators on weighted and unweighted spaces
About this Title
David Dos Santos Ferreira, Université Paris 13, Cnrs, umr 7539 Laga, 99 avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France and Wolfgang Staubach, Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 229, Number 1074
ISBNs: 978-0-8218-9119-3 (print); 978-1-4704-1528-0 (online)
DOI: https://doi.org/10.1090/memo/1074
Published electronically: September 24, 2013
Keywords: Fourier integral operators,
Weighted estimates,
BMO commutators
MSC: Primary 35S30, 42B99
Table of Contents
Chapters
- Introduction
- 1. Prolegomena
- 2. Global Boundedness of Fourier Integral Operators
- 3. Global and Local Weighted $L^p$ Boundedness of Fourier Integral Operators
- 4. Applications in Harmonic Analysis and Partial Differential Equations
Abstract
We investigate the global continuity on $L^p$ spaces with $p\in [1,\infty ]$ of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context we also prove the optimal global $L^2$ boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hörmander class amplitudes i.e. those in $S^{m} _{\varrho , \delta }$ with $\varrho , \delta \in [0,1]$. We also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on weighted $L^{p}$ spaces, $L_{w}^p$ with $1< p < \infty$ and $w\in A_{p},$ (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition. These results are shown to be optimal for operators with amplitudes in classical Hörmander classes and can also be given a geometrically invariant formulation. The weighted results are in turn applied to prove, for the first time, weighted and unweighted estimates for the commutators of Fourier integral operators with functions of bounded mean oscillation BMO, estimates on weighted Triebel-Lizorkin spaces, and finally global unweighted and local weighted estimates for the solutions of the Cauchy problem for $m$-th and second order hyperbolic partial differential equations on $\mathbf {R}^n .$ The global estimates in this context, when the Sobolev spaces are $L^2$ based, are the best possible.- Josefina Álvarez, Richard J. Bagby, Douglas S. Kurtz, and Carlos Pérez, Weighted estimates for commutators of linear operators, Studia Math. 104 (1993), no. 2, 195–209. MR 1211818, DOI 10.4064/sm-104-2-195-209
- Kenji Asada and Daisuke Fujiwara, On some oscillatory integral transformations in $L^{2}(\textbf {R}^{n})$, Japan. J. Math. (N.S.) 4 (1978), no. 2, 299–361. MR 528863, DOI 10.4099/math1924.4.299
- R. Michael Beals, $L^{p}$ boundedness of Fourier integral operators, Mem. Amer. Math. Soc. 38 (1982), no. 264, viii+57. MR 660216, DOI 10.1090/memo/0264
- Richard Beals, Spatially inhomogeneous pseudodifferential operators. II, Comm. Pure Appl. Math. 27 (1974), 161–205. MR 467397, DOI 10.1002/cpa.3160270204
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- Alberto-P. Calderón and Rémi Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374–378. MR 284872, DOI 10.2969/jmsj/02320374
- Sagun Chanillo, Remarks on commutators of pseudo-differential operators, Multidimensional complex analysis and partial differential equations (São Carlos, 1995) Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 33–37. MR 1447213, DOI 10.1090/conm/205/02651
- 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, DOI 10.1007/s10455-010-9219-z
- Elena Cordero, Fabio Nicola, and Luigi Rodino, Boundedness of Fourier integral operators on ${\scr F}L^p$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6049–6071. MR 2529924, DOI 10.1090/S0002-9947-09-04848-X
- Sandro Coriasco and Michael Ruzhansky, On the boundedness of Fourier integral operators on $L^p(\Bbb R^n)$, C. R. Math. Acad. Sci. Paris 348 (2010), no. 15-16, 847–851 (English, with English and French summaries). MR 2677978, DOI 10.1016/j.crma.2010.07.025
- Johannes J. Duistermaat, Fourier integral operators, Birkäuser 1995.
- G. I. Èskin, Degenerate elliptic pseudodifferential equations of principal type, Mat. Sb. (N.S.) 82(124) (1970), 585–628 (Russian). MR 0510219
- Daisuke Fujiwara, A global version of Eskin’s theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 2, 327–339. MR 467400
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
- Allan Greenleaf and Gunther Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal. 89 (1990), no. 1, 202–232. MR 1040963, DOI 10.1016/0022-1236(90)90011-9
- Miguel de Guzmán, A change-of-variables formula without continuity, Amer. Math. Monthly 87 (1980), no. 9, 736–739. MR 602833, DOI 10.2307/2321865
- Lars Hörmander, Pseudo-differential operators and hypoelliptic equations, Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 138–183. MR 0383152
- Lars Hörmander, The analysis of linear partial differential operators I. Distribution theory and Fourier analysis, Springer Verlag 1985.
- Lars Hörmander, The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. MR 781537
- Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
- Lars Hörmander, On the $L^{2}$ continuity of pseudo-differential operators, Comm. Pure Appl. Math. 24 (1971), 529–535. MR 281060, DOI 10.1002/cpa.3160240406
- Jean-Lin Journé, Calderón-Zygmund operators, pseudodifferential operators and the Cauchy integral of Calderón, Lecture Notes in Mathematics, vol. 994, Springer-Verlag, Berlin, 1983. MR 706075
- Carlos E. Kenig and Wolfgang Staubach, $\Psi$-pseudodifferential operators and estimates for maximal oscillatory integrals, Studia Math. 183 (2007), no. 3, 249–258. MR 2357989, DOI 10.4064/sm183-3-3
- Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362. MR 542885, DOI 10.1090/S0002-9947-1979-0542885-8
- Nicholas Miller, Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Amer. Math. Soc. 269 (1982), no. 1, 91–109. MR 637030, DOI 10.1090/S0002-9947-1982-0637030-4
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Wave front sets, local smoothing and Bourgain’s circular maximal theorem, Ann. of Math. (2) 136 (1992), no. 1, 207–218. MR 1173929, DOI 10.2307/2946549
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. MR 1168960, DOI 10.1090/S0894-0347-1993-1168960-6
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- Fabio Nicola, Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations, Studia Math. 198 (2010), no. 3, 207–219. MR 2650986, DOI 10.4064/sm198-3-1
- Luigi Rodino, On the boundedness of pseudo differential operators in the class $L^{m}_{\rho }{}_{,1}.$, Proc. Amer. Math. Soc. 58 (1976), 211–215. MR 410480, DOI 10.1090/S0002-9939-1976-0410480-X
- Michael Ruzhansky, On local and global regularity of Fourier integral operators, New developments in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 189, Birkhäuser, Basel, 2009, pp. 185–200. MR 2509098, DOI 10.1007/978-3-7643-8969-7_{9}
- 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, DOI 10.1007/3-7643-7514-0_{5}
- J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR 0433481
- 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, DOI 10.2307/2944346
- Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579
- Christopher D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), no. 2, 349–376. MR 1098614, DOI 10.1007/BF01245080
- 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
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
- Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
- Kôzô Yabuta, Calderón-Zygmund operators and pseudodifferential operators, Comm. Partial Differential Equations 10 (1985), no. 9, 1005–1022. MR 806253, DOI 10.1080/03605308508820398