关于一类振荡积分的L^p有界性

L^P—BOUNDEDNESS OF OSCILLATORY SINGULAR INTEGRALS

研究带粗糙核的振荡积分算子 Tf(x)=P.V.integral from n=R^n to e^(ip(x,g))Ω(x-y)|x-y|^(-1)f(y)dy,其中 P(xy)是R^n×R^n中的实多项式,Ω(x)是R^n中的零次齐次函数且在单位球面S^(n-1)的积分为零。通过适当的取值分解,证明了Ω∈Llog^+(S^(n-1))时,Tf(x)

In this note, L^p--boundedness (1<p<∞) is proved for oscillatory singular integrals Tf(x)=P. V. ∫R^(R^e^(iP(x,y))Ω(x--y)|x--y|^(-R)f(y)dy, where P(x,y) is a real ploynormal on R^n×R^n,Ω(x) is homogeneous of degree 0, has mean value 0 and Ω(x)∈Llog^+L(S^(n-1)