乘积空间上极大奇异积分算子的L^p有界性

L^p-boundedness of maximal singular integral operators on product domains

用旋转法结合Fourier估计以及Littlewood-Paley理论给出了乘积空间上带粗糙核的极大奇异积分算子的Ω(x′,y′)dx′=0, y′∈Sm-1,∫Sm-1Lp有界性.证明了对于Ω∈Lq(Sn-1×Sm-1),其中q>1,∫Sn-1Ω(x′,y′)dy′=0, x′∈Sn-1,且b,h∈L∞(R1+),则积域上极大奇异积分算子∫∫|u|>ε1T*(f)=supb(|u|)h(|v|)Ω(u′,v′)|v|n|v|mf(x-u,y-v)dudvε1>0,ε2>0|v|>ε2为Lp(Rn×Rm)有界,其中1<p<∞.从而改进了以往的结果.

By rotation method, Fourier transform estimates and LittlewoodPaley theory, the Lp boundedness of maximal singular integral operators with rough kernels on product domains is got. It is proved that if Ω∈Lq(Sn-1×Sm-1),q>1,∫Sn-1Ω(x′,y′)dx′=0,y′∈Sm-1,∫Sm-1Ω(x′,y′)dy′=0,x′∈Sn-1,and b,h∈L∞(R1+), then the maximal singular integral operatorT*(f)=supε1>0,ε2>0∫∫|u|>ε1|v|>ε2b(|u|)h(|v|)Ω(u′,v′)|v|n|v|mf(x-u,y-v)dudvis Lp bounded for 1<p<+∞. 