A Survey of Some Norm Inequalities

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type ‖ A f‖ _𝒳^2 ≤ C ‖ f‖ _𝒳‖ A^2 f‖ _𝒳, f ∈ dom (A^2 ), and recall that under exceedingly stronger hypotheses on the operator A and/or the Banach space 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 𝒳 ) all the way down to 1 (e.g., when A is a symmetric operator on a Hilbert space ℋ ). We also survey some results in connection with an extension of the Hardy–Littlewood inequality involving quadratic forms as initiated by Everitt.