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

摘要: 利用权函数方法，讨论了非齐次核K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂)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.

Abstract: Using the weight function methods, the equivalent parameter conditions for the validity of Hilbert-type integral inequalities with non-homogeneous kernel K(x, y)=φ^λ(x^λ₁y^λ₂)φ′(x^λ₁y^λ₂) 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.