Weighted boundedness of Hardy and P\'{o}lya-Knopp type operators with variable limits on product space

TL;DR

This paper establishes conditions for the boundedness of Hardy and Pólya-Knopp type operators with variable limits on product spaces, providing explicit bounds and extending previous results in weighted inequalities.

Contribution

It introduces new sufficient conditions for the boundedness of these operators with variable limits on product spaces, including explicit bounds and applications.

Findings

Derived explicit bounds for Hardy operators with variable limits.

Characterized sufficient conditions for weighted boundedness.

Extended results to Pólya-Knopp type operators with specific weights.

Abstract

We characterize the sufficient conditions which three weight functions $u$ and $v_{1}, v_{2}$ satisfy ensure the boundedness of the Hardy operator with variable limits on product space. The corresponding bound is explicitly worked out. Moreover, as application, we can obtain an explicit scale of bound for the P\'{o}lya-Knopp type operator with certain weights.