With ¦Ì-calder¨®n-zygmund Nuclear Oscillatory Integral

The oscillatory integral operator is defined bywhere P(x, y) is a real polynomial on R~n x R~n, and K(x — y) is a standard Calderon-Zygmund kernel. First, in the translation invariant case, this operator is partly in connection with singular integrals on the lower-dimensional varieties. Also it is connected with the Heisenberg group in relation to twisted convolution (and generalization of this to other nilpotent groups). Third, it can be as the model operator occurring in the theory of the singular Radon transforms and their application to the study of the (?)-Neumann problem.F.Ricci and E.M.Stein showed that T is bounded on L~p(R~n) (1 < p < ∞). Furthermore, they obtained a similar theorem for the above operator when the conditions imposed on the kernel K is loosen in a certain extent.As the continuer of the work of F.Ricci E.M.Stem, we define a μ-Calderou-Zygmund kernel (Chapter 1) and show that the oscillatory singular integral operator T with this kind of kernel is bounded on L~p(R~n) (1 < p < ∞), improving the previously known result in the sense that the scope of μ, can be extendable. On the base of that, we obtain a weighted result: T is also bounded on L_ω~p(R~n), (ω∈ A_p, 1 < p < ∞).In this process, the major obstacle lies in the fact that the local part of T may not be converge. As three important tools, some inequalities for polynomials (Chapter 2), some estimates of Van Der Corput type (Chapter 3), and the general multi-dimentional case of Young's inequality (Chapter 4) are presented in detail. Finally, Theorem 1 and Theorem 2 (Chapter 5, Chapter 6) as the two major results of this paper are proved.