The Boundedness Of Commutators With Hilbert Transforms Along Special Curves

In this paper, we study the boundedness of commutators generated by Hilbert trans-forms along cubic parabola (t,t~3), multilinear commutator generated by Hilbert trans-forms along parabola (t,t2) with BMO and Lipschitz function.In chapter 1, we simply introduce the background and the development in the bound-edness of commutators with Hilbert transforms along curves.In chapter 2, we discuss the commutators of Hilbert transforms along cubic parabola(t,t~3) with a BMO function which is associated with cubic parabola. We prove that thecommutators and maximal operators are bounded on L~p(R~2) spaces.In chapter 3, we study the boundedness of multilinear commutators of Hilbert trans-forms along parabola (t,t2). We prove that the commutators and maximal operators arebounded on L~p(R~2) spaces, if .In chapter 4, we consider the boundedness of commutator of Hilbert transforms alongparabola (t,t~2) with a Lipschitz function which is associated with parabola.We obtain thatif then the commutator is a bounded operator from L~p(R~2) to L2(R2) where .