Oscillatory integrals and regularity of dispersive equations. (English) Zbl 0738.35022

The paper under review deals with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations of the type \[ \partial_ tu-iP(D)u=0,\quad x\in\mathbb{R}^ n,\;\;t\in\mathbb{R}, \] where \(D=(1/i)(\partial_ 1,\ldots,\partial_ n)\), \(\partial_ j=\partial/\partial x_ j\), and \(P(D)\) is defined via its real symbol, i.e., \[ P(D)f(x)=\int\exp(ix\xi)P(\xi)\hat f(\xi) d\xi. \] The authors study also some applications of these smoothing properties to nonlinear problems and their link with restriction theorem for the Fourier transform and positive convergence results.
Reviewer: R.Manthey (Jena)


35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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