Singular Integrals Of Vector Values

The main aim of this paper is to use the space decomposition theorem and singular integral theory proposed by A.P.Calderon to extend the known L^p(Rⁿ)from L^p(Rⁿ)and translational commutative bounded linear operators to the Banach space values of L² and L¹.Assuming H is a separable Hilbert space,the characterization of the multipli-er(?L²(Rⁿ,H),L²(Rⁿ,H))?is first yielded by introducing the tempered distribution of the H value.The precise equation of the multiplier norm is then obtained.Furthermore,by virtue of vector or operator measures,the convolution integral expressions of the multiplier operators of the three types of Banach space values,(?L¹(Rⁿ,H),L¹(Rⁿ,H))?,(?L¹(Rⁿ,H),L¹(Rⁿ))?and(?L¹(Rⁿ),L¹(Rⁿ,H))?are obtained,and the basic relational ex-pression of the multiplier operator norm and the total variation measure of the vector value are also obtained.To characterize the multiplier(?L¹,L¹)?,the representational the-orems of bounded linear operators from C₀(?,H)to H,C₀(?,H)to C,and C₀(?)to H are derived.This is a profound extension of the famous Riesz representation theorem and?here is a locally compact Hausdorff space.