Parameter Conditions for Hilbert-Type Operator Bounded with Homogeneous Kernels between Sequence Space l and Function Space L

(1.Department of Applied Mathematics,Guangzhou Huashang Col lege,Guangzhou 511300,China;2.Department of Mathematics,Guangdong University of Education,Guangzhou 510303,China)

Abstract:By using the technique of real analysis,the parameter conditions for Hilberttype series operator and integral operatorbounded with homogeneous kernels are discussed.The necessary and sufficient conditions forbounded are obtained,and their norm expressions are established under certain conditions.

Keywords:Hilbert-type series operator;Hilbert-type integral operator;Mixed Hilberttype inequality;Bounded operator;Operator norm;Homogeneous kernel

§1.Introduction and preliminary knowledge

In this paper,under the condition that the kernelK(n,x)is a homogeneous function of orderλ,what conditions are satisfied by the parametersp,q,α,β,γ1,γ2andλsuch thatT1andT2are bounded operators?It is further proved that such parameter conditions are necessary and sufficient conditions forT1andT2to be bounded.The expressions of the operator norms ofT1andT2are also obtained under certain conditions.

According to the basic theory of Hilbert-type inequality,we know that operator inequalities

§2.Parameter conditions under which mixed Hilbert-type inequality holds with homogeneous kernels

§3.Conditions for Hilbert-type series and integral operators bounded with homogeneous kernels and operator norms