非双倍测度下多线性奇异积分算子及其交换子在Morrey空间上的有界性

Boundedness of multilinear singular integrals and their commutators on Morrey spaces for non-doubling measures

非齐型空间上多线性奇异积分算子的有界性问题,自20世纪末由Tolsa等人提出后,广为人们所关注。设μ是非双倍测度,借助RBMO函数的等价刻划,以及多线性奇异积分算子的核满足的条件,证明如果多线性奇异积分算子T从L1(μ)×…×L1(μ)到Lm1,∞(μ)有界,则T以及它与有界平均振荡函数生成的交换子是从Mpq11(μ)×…×Mpqmm(μ)到Mpq(μ)有界的算子。

Many mathematicians have paid their attentions to the boundedness problem of multilinear singular integral operators since it was put forward at the end of last century.Assume that μ is a non-doubling measure.By the equivalent characterization of RBMO functions and the properties of the kernel of multilinear singular integral operators,it is proven that T and its commutators with RBMO functions are bounded form Mq1^p1（μ）×…×Mqm^pm（μ） to Mq^p（μ） provided that the multilinear singular operator T is bounded from L^1（μ）×…×L^1（μ） to L^1/m,∞（μ）.