A Survey of Some Norm Inequalities

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