一类奇异积分算子交换子的L^(p)有界性

L^(p) Boundedness for a Class of Commutators of Singular Integral Operators

研究奇异积分算子的交换子Tb,kf (x) =∫Rn(b(x) - b(y) ) k Ω (x - y)| x - y| nf (y) dy的 Lp有界性 ,其中 b(x) =b(| x| )是径向函数且 b(r)∈ BMO(R+ ) ,k是自然数 ,Ω 是 Rn中的零阶齐次函数 ,在单位球面上的积分为零 .在 Ω 具有某种最小可积性条件时 ,证明了 Tb,k及其相应的极大算子是Lp(Rn) (1<p<∞ )上以 C‖ b‖ k BMO(R+ )

The L p boundedness for the commutators of singular integral operators defined byT b,k f(x)=∫ R n (b(x)-b(y)) kΩ(x-y)|x-y| nf(y) d yis considered, where b(x)=b(|x|) is radial and b(r)∈ BMO (R +),k is a positive integer, Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is shown that the operator T b,k and the corresponding maximal operator are bounded on L p(R n) with bounded C‖b‖ k BMO(R +) when Ω satisfies certain minimum size condition.