A survey of some norm inequalities

TL;DR

This survey reviews classical norm inequalities related to operators on Banach and Hilbert spaces, highlighting the conditions under which the optimal constants are achieved and discussing extensions involving quadratic forms.

Contribution

It compiles and discusses various classical inequalities, their optimal constants, and recent extensions, providing a comprehensive overview of the topic.

Findings

Optimal constants vary from 4 to 1 depending on operator and space conditions.

Stronger hypotheses on operators lead to smaller constants in inequalities.

Extensions involving quadratic forms expand the scope of classical inequalities.

Abstract

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type \[ \|A f\|_{\mathcal{X}}^2 \leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in dom\big(A^2\big), \] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $\mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $\mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $\mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.