References
[BP]E. Bombieri, V. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337–357.
[Bo1]J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math. 67:3 (1989), 291-311.
[Bo2]J. Bourgain, Exponential sums and nonlinear Schrödinger equations, GAFA 3 (1993), 157–178.
[Bo3]J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I: Schrödinger Equations, GAFA 3 (1993), 107–156.
[CW]T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation inH s, Nonlinear Analysis, Theory Methods and Applications 14:10 (1990), 807–836.
[GiV1]J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, H. Poincaré Analyse Non Linéaire 2 (1985), 309–327.
[GiV2]J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Klein-Gorobon equation, Math. Z 189 (1985), 487–505.
[Gr]E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag (1985).
[K1]T. Kato, StrongL p-solutions of the Navier-Stokes equations in ℝm with applications to weak solutions, Math. Z 187 (1984), 471–480.
[K2]T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Théorique 46 (1987), 113–129.
[K3]T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Math. Suppl. Studies, Studies in Applied Math. 8 (1983), 93–128.
[KePoVe1]C. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. AMS 4 (1991), 323–347.
[KePoVe2]C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction princple, preprint.
[KePoVe3]C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolov spaces of negative indices, to appear in Duke Math. J.
[KruF]S. Kruzhkov, A. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sbornik 48 (1984), 93–138.
[L]P. Lax, Periodic solutions of the KDV equations, Comm. Pure and Applied Math. 26, 141–188 (1975).
[LeRSp]J. Lebowitz, H. Rose, E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statistical Physics 50:3/4 (1988), 657–687.
[MTr]H. P. McKean, E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143–226.
[MiGKr]R. Miura, M. Gardner, K. Kruskal, Korteweg-de Vries equation and generalizations II. Existence of conservation laws and constant of motion. J. Math. Physics 9:8 (1968), 1204–1209.
[PosTr]J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press 37 (1987).
[Sj]A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579.
[St]R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.
[T]P. Tomas, A restriction for the Fourier transform, Bull. AMS 81 (1975), 477–478.
[Tr]E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 325–341.
[Ts]Y. Tsutsumi,L 2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj 30 (1987), 115–125.
[Vi]I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers, Intersciences, NY (1954).
[ZS]V. Zakharov, A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34:1, 62–69 (1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geometric and Functional Analysis 3, 209–262 (1993). https://doi.org/10.1007/BF01895688
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01895688