Weighted Norm Inequalities of Vector-Valued Functions

To generalize H-L maximal function to vector-valued weighted space, it is proved that for a weighted function υ（x）≥0, the necessary and sufficient conditions are obtained for ∫_Rⁿυ（x）(1+（|x|ⁿ）^-pdx<)∞, such that the vector-valued H-L maximal operator is bounded from L^<sup>p_<sub>l^q（Rⁿ, ωdx） to L^p（Rⁿ, υdx） for some ω（x） that is related to υ（x） and ω（x）<∞, a.e.x∈Rⁿ.Based on the double property, Hlder'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.