Proceedings of the American Mathematical Society

Author(s): A. Alexandrou Himonas; Gerard Misiolek
Journal: Proc. Amer. Math. Soc. 130 (2002), 3043-3050.
MSC (1991): Primary 42B15; Secondary 35G25
Posted: March 14, 2002
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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.

[B1]: J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 1: Schrodinger equation, Geom. Funct. Anal. 3 (1993). MR 95d:35160a
[B2]: J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part 2: KdV equation, Geom. Funct. Anal. 3 (1993). MR 95d:35160b
[B3]: J. Bourgain, Nonlinear Schrödinger equations, in Hyperbolic Equations and Frequency Interactions, IAS/Park City, AMS (1999). MR 2000c:35216
[FG]: Y. Fang and M. Grillakis, Existence and uniqueness for Boussinessq type equations on a circle, Comm. Partial Differential Equations 21 (1996). MR 97d:35190
[G]: J. Ginibre, Le probleme de Cauchy pour des edp semi-lineaires periodiques en variables d'espace [d'apres Bourgain], Seminaire Bourbaki, 237 (1996). MR 98e:35154
[GV]: J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995). MR 97a:46047
[HM1]: A. A. Himonas and G. Misio $\l$ ek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations 23 (1998). MR 99b:35176
[HM2]: A. A. Himonas and G. Misio $\l$ ek, A'priori estimates for Schrödinger type multipliers, to appear in Illinois J. Math.
[KPV1]: C. Kenig, G. Ponce and L. Vega, Higher order nonlinear dispersive equations, Proceedings A.M.S. 122 (1994). MR 94k:35073
[KPV2]: C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991). MR 92d:35081
[KPV3]: C. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteveg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993). MR 94g:35196
[P]: G. Ponce, On nonlinear dispersive equations, Proceedings of the ICM Berlin 1998, Doc. Math. (1998). MR 99k:35087
[ST]: J.C. Saut and N. Tzvetkov, The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000). MR 2001d:35175
[SO]: C. Sogge, Lectures on Nonlinear Wave Equations, International Press 1995. MR 2000g:35153
[St]: E. Stein, Harmonic analysis, Princeton University Press 1993. MR 95c:42002
[SW]: E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press 1971. MR 46:4102
[STR]: R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977). MR 58:23577
[Z]: A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189-209. MR 52:8788

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A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: alex.a.himonas.1@nd.edu

Gerard Misiolek
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

DOI: 10.1090/S0002-9939-02-06439-0
PII: S 0002-9939(02)06439-0
Keywords: Fourier transform, multiplier inequalities, interpolation
Received by editor(s): May 29, 2001
Posted: 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 of article: Copyright 2002, American Mathematical Society

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