A priori estimates for higher order multipliers on a circle
HTML articles powered by AMS MathViewer
- by A. Alexandrou Himonas and Gerard Misiołek PDF
- Proc. Amer. Math. Soc. 130 (2002), 3043-3050 Request permission
Abstract:
We present an elementary proof of an a priori estimate of Bourgain for a general class of multipliers on a circle using an extension of methods developed in our previous work. The main tool is a suitable version of a counting argument of Zygmund for unbounded regions.References
- J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
- J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
- Jean Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (Park City, UT, 1995) IAS/Park City Math. Ser., vol. 5, Amer. Math. Soc., Providence, RI, 1999, pp. 3–157. MR 1662829, DOI 10.1090/coll/046
- Yung-Fu Fang and Manoussos G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm. Partial Differential Equations 21 (1996), no. 7-8, 1253–1277. MR 1399198, DOI 10.1080/03605309608821225
- Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Astérisque 237 (1996), Exp. No. 796, 4, 163–187 (French, with French summary). Séminaire Bourbaki, Vol. 1994/95. MR 1423623
- J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50–68. MR 1351643, DOI 10.1006/jfan.1995.1119
- A. Alexandrou Himonas and Gerard Misiołek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations 23 (1998), no. 1-2, 123–139. MR 1608504, DOI 10.1080/03605309808821340
- A. A. Himonas and G. Misiołek, A’priori estimates for Schrödinger type multipliers, to appear in Illinois J. Math.
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc. 122 (1994), no. 1, 157–166. MR 1195480, DOI 10.1090/S0002-9939-1994-1195480-8
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), no. 1, 1–21. MR 1230283, DOI 10.1215/S0012-7094-93-07101-3
- Gustavo Ponce, On nonlinear dispersive equations, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 67–76. MR 1648141
- J. C. Saut and N. Tzvetkov, The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. (9) 79 (2000), no. 4, 307–338 (English, with English and French summaries). MR 1753060, DOI 10.1016/S0021-7824(00)00156-2
- Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
- 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 Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. MR 387950, DOI 10.4064/sm-50-2-189-201
Additional Information
- A. Alexandrou Himonas
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 86060
- Email: alex.a.himonas.1@nd.edu
- Gerard Misiołek
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556 – and – Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Cambridge, CB3 9EW, United Kingdom
- Email: misiolek.1@nd.edu
- Received by editor(s): May 29, 2001
- Published electronically: March 14, 2002
- Additional Notes: Both authors were partially supported by the NSF under grant number DMS-9970857 and by the Faculty Research Program of the University of Notre Dame.
The second author was also supported by the Isaac Newton Institute for Mathematical Sciences of the University of Cambridge. - Communicated by: Mei-Chi Shaw
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3043-3050
- MSC (1991): Primary 42B15; Secondary 35G25
- DOI: https://doi.org/10.1090/S0002-9939-02-06439-0
- MathSciNet review: 1908929