Construction Conditions and Applications of a Hilbert-Type Multiple Integral Inequality Involving Multivariable Upper Limit Functions and Higher-Order Partial Derivatives

Abstract

Hilbert-type inequalities derive from the classical Hilbert inequality, and their theoretical work has key applications not only in operator theory but also in various analytic disciplines. In this paper, we have achieved the parametric conditions required for the construction of such inequalities, as well as the expressions for the optimal constant factors. Through the utilization of the construction theorem for Hilbert-type multiple integral inequalities with homogeneous kernels, our investigation centers on a Hilbert-type multiple integral inequality that involves multivariable upper limit functions and higher-order partial derivatives. Furthermore, we apply these results to discuss the boundedness and operator norms of integral operators with identical kernels.

Keywords

• multivariable upper limit functions
• higher-order partial derivatives
• Hilbert-type multiple integral inequality
• gamma function
• bounded integral operator
• operator norm