We characterize the optimal non-absolute domain for the Cesàro operator minus the identity (C−I), in the sequence space ℓp(ℕ), 1≤p≤∞, and compare the results obtained with the case of C, showing the different behavior in both cases. We also address this question for the Copson operator C∗ 

For a large class of operators acting between weighted l(infinity) 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 Cesaro 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 l(infinity). As an application, best constants are given for inequalities relating the weighted l(infinity) norms of the Cesaro and Copson operators both for general weights and for power weights.

We characterize the optimal non-absolute domain for the Hardy operator minus the identity, in the Lebesgue space Lp(0,∞), 1≤p≤∞. We also address a similar question for the dual Hardy operator minus the identity.

Some new inequalities of Hardy-type in Banach function space settings are proved anddiscussed. In particular, these results generalize and unify several classical Hardy-type inequalities. Some results are new also in the classical situation.

Some new inequalities of Carleson- and Hardy- type in Banach function space settings are proved and discussed. In particular, these theorems both generalize and unify several classical inequalities e.g. those by Hardy, Polya-Knopp, Carleman, Hardy-Knopp and Carleson. As applications some new inequalities are pointed out.

We present factorizations of weighted Lebesgue, Cesàro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy’s integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.

We obtain a new variational characterization of the Sobolev space $W_p^1(\Omega)$ (where $\Omega\subseteq\R^n$ and $p>n$). This is a generalization of a classical result of F. Riesz. We also consider some related results.

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert C*-modules. In particular, we pay special attention to an operator Wielandt inequality.

The generalized functionals of Merentes type generate a scale of spaces connecting the class of functions of bounded second p -variation with the Sobolev space of functions with p-integrable second derivative. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of order 2−1/p . These results extend the results in Lind (Math Inequal Appl 16:2139, 2013) for functions of bounded variation but are not consequence of the last.