沿高阶单调曲线的Hilbert变换和极大算子的L^p估计

L^p Estimates for Hilbert Transform and Maximal Operators Associated with Highly Monotone Curves

设γ:[-1,1]→R^n是R^n中的曲线,沿曲线的γ的Hilbert变换是如下定义的主值积分: Hf(x)=P.V.integral -1 to 1 f(x-γ(t))dt/t,相应的极大算子定义为: Mf(x)=sup 1/h| integral O to h f(x-γ(t))dt|. 对高阶单调曲线本文证明了相应的算子M和H都是L^p(R^n)有界的,从而改进了Nestlerode的结果。

Let γ:( -1>1]→Rn be the curve in Rn. The Hilbert transform associated with γ is defined by. and the maximal operator M associated with γ is defined by For highly monotone curves,both M and H are shown to be bounded on Lp(Rn),1<p<∞.