We study the maximal super-singular integral operator T*Ω,α,β(f)(x,y)=sup ∈1〉0,∈2〉0|∫|u|〉ε1,|v|〉ε2 b1(|u|)b2(|u|)Ω(u',u')/|u|^n+α|u|^m+β-f(x-u,y-u)dudu|defined on all f ...We study the maximal super-singular integral operator T*Ω,α,β(f)(x,y)=sup ∈1〉0,∈2〉0|∫|u|〉ε1,|v|〉ε2 b1(|u|)b2(|u|)Ω(u',u')/|u|^n+α|u|^m+β-f(x-u,y-u)dudu|defined on all f ∈ S(R^n ×R^m), where 0 ≤ α,β〈∞, b1 b2 ∈ L∞(R+1 ),Ω satisfies certain cancellation conditions and Ω∈L1(S^n-1×S^m-1)in the case α,β〉0;Ω∈L(log+L)(S^n-1×S^m-1)in the case αβ=0 and α+β 〉0. It is proved that, for 1〈p〈∞.T*Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p(R^n×R^m)to the Lebesgue space L^p(R^n×R^m).展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 10871173, 10931001)
文摘We study the maximal super-singular integral operator T*Ω,α,β(f)(x,y)=sup ∈1〉0,∈2〉0|∫|u|〉ε1,|v|〉ε2 b1(|u|)b2(|u|)Ω(u',u')/|u|^n+α|u|^m+β-f(x-u,y-u)dudu|defined on all f ∈ S(R^n ×R^m), where 0 ≤ α,β〈∞, b1 b2 ∈ L∞(R+1 ),Ω satisfies certain cancellation conditions and Ω∈L1(S^n-1×S^m-1)in the case α,β〉0;Ω∈L(log+L)(S^n-1×S^m-1)in the case αβ=0 and α+β 〉0. It is proved that, for 1〈p〈∞.T*Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p(R^n×R^m)to the Lebesgue space L^p(R^n×R^m).