构建一类非齐次核的Hilbert型积分不等式的等价参数条件

Equivalent Parameter Conditions for the Construction of Hilbert-type Integral Inequalities with a Class of Non-homogeneous Kernels

利用权函数方法,讨论了非齐次核K(x,y)=φλ(xλ1yλ2)φ′(xλ1yλ2)的Hilbert型积分不等式成立的等价参数条件及最佳常数因子,得到了构建此类Hilbert型不等式的充分必要条件及最佳常数因子的表达公式;对一些具体的非齐次核,得到了若干具有最佳常数因子的新的Hilbert型不等式;最后,讨论了相应奇异积分算子的有界性及其范数.

Using the weight function methods,the equivalent parameter conditions for the validity of Hilbert-type integral inequalities with non-homogeneous kernel K(x,y)=φλ(xλ1yλ2)φ′(xλ1yλ2)and the best constant factor are discussed.The necessary and sufficient conditions for constructing such Hilbert-type inequalities and the formula for expressing the best constant factor are obtained.Many new Hilbert-type integral inequalities with some specific non-homogeneous kernels and the best constant factors are also obtained.Finally,the norm and boundedness of corresponding singular integral operators are discussed.