With Mixed Homogeneous Nature Of The Oscillatory Singular Integral Operator

The purpose of this paper is to consider the Oscillatory Singular Integralwith mixed homogeneity T(f)(x)=p.v. integral from n=Rⁿ e^iP(x,y)K(x-y)f(y)dy,where mixed homogeneity is in the sense that there are positive numbers a₁,…, a_nsuch that K(λ^a₁x₁,…,λ^a_nx_n)=λ^{-sum from i=1 to n a_i}K(x) and P(x,y) is a real polynomial onRⁿ×Rⁿ. Fulvio Ricci and E. M. Stein proved that the operator T as above isbounded on L^p(Rⁿ)(1＜p＜∞) when K(x) is a standard Calderón-Zygmundkernel on Rⁿ([1]). Our major work is to show that the Oscillatory SingularIntegral Operator with mixed homogeneity T is bounded on L^p(Rⁿ)(1＜p＜∞).Our main result is the following theorem:THEOREM 1.The above operator which initially defined for smooth f with compact support canbe extended to be a bounded operator on L^p(Rⁿ) to itself, with 1＜p＜∞, whereK(x) satisfies(1) K(x)∈C¹(Rⁿ/â–³)(2) K(λ^a₁x₁,…,λ^a_nx_n)=λ^{-sum from i=1 to n a_i}K(x), a_i＞(?)|(i=1,2,…,n)(3) integral from n=∑_n K(x)J(φ₁,…,φ_n-1)dσ=0,The bound of this operator(which of course depends on K, p, and n)can be takento depend only on the total degree d of P, and is otherwise independent of thecoefficients of P.