带非卷积位相的粗糙核振荡奇异积分算子的加权L^P估计(英文)

Weighted L^P Estimates for Rough Oscillatory Singular Integrals with Nonconvolution Phases

文研究一类位相较多项式更一般的振荡奇异积分算子.在积分核Ω∈Llog^+L(S^(n-1))的条件下,建立了该类算子在加权L^P空间的有界性.

In this paper, we study the weighted Lp estimates （1 〈 p 〈 ∞） for the oscillatory singular integral operator given by Tf（x）=p.v.∫Rn e^iФ（x,y）b（｜x-y｜）Ω（x-y）/｜x-y｜^n f（y）dy.The phase function Ф has the form Ф（x,y）=^l∑k=0 Pk（x）Фk（y-x）,where Pk is a real polynomial on R^n, Фk is a real homogeneous function on Rn and is analytic on S^n-1. Ω is homogeneous of degree zero and ∫S^n-1Ω（x＇）dσ（x＇）=0, b is a bounded variation function on [0, ∞）. We show that if Ω∈Llog^＋L（S^n-1）,then T is bounded on Lpw, provided w satisfies a condition similar to the Ap condition but involves rectangles arising from a covering of a star-shaped set related to Ω.