The norm of hardy-type oscillation operators in the discrete and continuous settings

Recently, several authors have considered the problem of determining optimal norm inequalities for Hardy-type operators. In this work, we continue this study and obtain sharp bounds for the norm of the difference of the Cesàro operator with the identity when it acts on the cone of nonnegative sequences in ℓ p . This operator controls the oscillation of the sequence with respect to its average. Also, we get optimal constants for the difference of the Copson operator with the identity when restricted to the cones of nonnegative or decreasing sequences in ℓ p , as well as best constants for the adjoint of the classical Hardy operator minus identity when restricted to the cones of nonnegative or decreasing functions in L p (ℝ + ).