A Survey of Some Norm Inequalities

Abstract

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type

$$\begin{aligned} \Vert A f\Vert _{{{\mathcal {X}}}}^2 \le C \Vert f\Vert _{{{\mathcal {X}}}} \big \Vert A^2 f\big \Vert _{{{\mathcal {X}}}}, \quad f \in {{\,\mathrm{dom}\,}}\big (A^2\big ), \end{aligned}$$

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.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

We are indebted to Man Kam Kwong and Lance Littlejohn for helpful discussions.

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Correspondence to Fritz Gesztesy.

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Dedicated with great pleasure to Henk de Snoo on the happy occasion of his 75th birthday.

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This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.

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Gesztesy, F., Nichols, R. & Stanfill, J. A Survey of Some Norm Inequalities. Complex Anal. Oper. Theory 15, 23 (2021). https://doi.org/10.1007/s11785-020-01060-9

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Keywords

  • Hardy–Littlewood
  • Kallman–Rota
  • and Landau–Kolmogorov inequalities

Mathematics Subject Classification

  • Primary 47A30
  • 34L40
  • Secondary 47B25
  • 47B44