Study of the boundedness of one-sided operators and the well-posedness of KdV type dispersion equations

Abstract

Many important issues in Mathematics and Physics can often attribute to the boundedness of operators on some function spaces. The study for the boundedness of singular integral is one of the important components in Harmonic analysis. A variety of methods and techniques in Harmonic analysis have been widely used in the study of partial differential equations. By the space-time estimates for oscillatory type integral operators on Lebesgue spaces or Morrey type spaces and semigroup theory, one can obtain the well-posedness for the Cauchy problem of nonlinear dispersion equations in the low order Sobolev spaces. We give the definitions of one kind of one-sided oscillatory singular integral operators and study some classical boundedness of these operators. Inspired by the classical results for the characterizations via commutators, we introduce the idea of characterizations via commutators on Morrey spaces and obtain characterizations via commutators with symbol belonging to two kinds of function spaces on these spaces. The results obtained above offer some new tools to the study of partial differential equations. Finally, combining energy method and Number theory, we establish the well-posedness of the Cauchy problem for a class of KdV type dispersion equations.