End-point Norm Estimates for Ces\`aro and Copson Operators

TL;DR

This paper provides exact formulas for the norms of Cesàro and Copson operators on weighted inity spaces, including restrictions to cones of sequences, with applications to inequalities and best constants.

Contribution

It introduces explicit formulas for operator norms on weighted inity spaces and their cones, specifically for Cesro and Copson operators, including special cases with power weights.

Findings

Exact formulas for operator norms and restrictions to cones.

Simplified formulas for power-weighted inity spaces.

Determination of best constants in related inequalities.

Abstract

For a large class of operators acting between weighted $\ell^\infty$ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Ces\`aro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted $\ell^\infty$. As an application, best constants are given for inequalities relating the weighted $\ell^\infty$ norms of the Ces\`aro and Copson operators both for general weights and for power weights.