A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications

Journal of inequalities and applications

article

Yong Hong¹  Jianquan Liao²  Bicheng Yang²  Qiang Chen³ 

[1] Department of Mathematics, Guangdong Baiyun University;Department of Mathematics, Guangdong University of Education;Department of Computer Science, Guangdong University of Education

关键词: Generalized homogeneous kernel;    Hilbert-type multiple integral inequality;    Necessary and sufficient condition;    The best constant factor;    Bounded operator;    Operator norm;   

DOI  :  10.1186/s13660-020-02401-0

学科分类：电力

来源: SpringerOpen

【 摘 要 】

Let $x=(x_{1},x_{2},\ldots,x_{n})$, and let $K(u(x),v(y))$ satisfy $u(rx)=ru(x)$, $v(ry)=rv(y)$, $K(ru,v)=r^{\lambda\lambda_{1}}K(u, r^{-\frac{\lambda_{1}}{\lambda_{2}}}v)$, and $K(u,rv)=r^{\lambda\lambda_{2}}K(r^{-\frac{\lambda_{2}}{\lambda_{1}}}u, v)$. In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel $K(u(x),v(y))$ and discuss its applications in the theory of operators.

【 授权许可】

CC BY