向量值函数的加权模不等式

Weighted Norm Inequalities of Vector-Valued Functions

基于H-L极大算子在加权向量值函数空间的推广,证明了权函数υ(x)≥0,存在一个与υ(x)有关的权函数ω(x)且ω(x)<∞,a.e.x∈Rn,使得向量值的H-L极大算子M从Llpq(Rn,ωdx)空间到Lp(Rn,υdx)空间是有界的,当且仅当∫Rnυ(x)(1+|x|n)-pdx<∞成立.利用双倍性质、H lder’s不等式等证明了其充分性;利用特征函数构造出向量函数证明了其必要性.

To generalize H-L maximal function to vector-valued weighted space, it is proved that for a weighted function v（x） ≥0, the necessary and sufficient conditions are obtained for ∫R^nv（x）（1＋｜x｜^n）^-pdx〈∞, such that the vector-valued H-L maximal operator is hounded from Llqp （R^n, ωdx ） to L^p（R^n, vdx ） for some ω（x） that is related to v（x） and ω（x）〈∞, a.e.x∈R^n. Based on the double property, Holder＇s inequality et al, the sufficiency condition of the theorem are proved. Employing the eigenfunction, the vector-valued functions are set up, and conditions of necessity of the theorem are completed.