Recent progress in smoothing estimates for evolution equations

Abstract

This paper is a survey article of results and arguments from authors' papers (Ruzhansky and Sugimoto in Proc. Lond. Math. Soc. 105:393-423, 2012; Ruzhansky and Sugimoto in Smoothing properties of non-dispersive equations; Ruzhansky and Sugimoto in Smoothing properties of inhomogeneous equations via canonical transforms), and describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators a(Dx) of order m satisfying the dispersiveness condition ∇a(ξ) ≠ 0, the smoothing estimate is established, while it is known to fail for general non-dispersive operators. Especially, time-global smoothing estimates for the operator a(Dx) with lower order terms are the benefit of our new method. For the case when the dispersiveness breaks, we suggest a form which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(Dx). It does continue to hold for a variety of non-dispersive operators a(Dx), where ∇a(ξ) may become zero on some set. It is remarkable that our method allows us to carry out a global microlo-cal reduction of equations to the translation invariance property of the Lebesgue measure.