沿近似齐次曲线的Hilbert变换和极大算子的有界性

THE L^P—BOUNDEDNESS OF HILBERT'S TRANSFORM AND MAXIMAL OPERATOR ALONG THE APPROXIMATELY HOMOGENEOUS CURVES

设γ:[-1.1]→R^n 是 R^n 中的曲线。沿曲线γ的 Hilbert 变换是如下定义的主值积分Hf(x)=P.V.∫_(-1)~1f(x—γ(t))(dt/t)相应的极大算子 M 定义为Mf(x)=■(1/h)|∫_0~h f(x—γ(t))dt|对近似齐次曲线γ,我们证得 H 和 M 都在 L^p(R^n)上有界,p>1。从而改进了 E.M.Stein.Wainger 和 D.A.Weinberg 在文[1]、[2]中的结果。

Let γ:[—1,1]→R^n be a curve in R^n.The Hilbert's transform H along the curve γ is defined by: Hf(x)=P.V∫_(-1)~1 f(x-γ(t))(dt/t) and the maximal operator M is defined by: Mf(x)=■(1/h)|∫_0~h f[x-γ(t)]dt| For the approximately homogeneous curve γ,the author proves that both H and M are bounded on L^P (R^n),p>1.