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The Weiss conjecture and weak norms

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Bernhard Hermann Haak

Optimization and Control · Submitted 22 Jun 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent.

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arXiv ID1206.5109 (version 1)
Pith IDVNMLIMC2, Pith's permanent record of this paper and its review
CitationBernhard Hermann Haak. "The Weiss conjecture and weak norms." arXiv:1206.5109 (2012). https://pith.science/paper/VNMLIMC2
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,444 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.27 · mentions code or data availability 0.01.

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In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent. result 0.95 0.93 0.86 asserted 1.00
s1 Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$. result 0.87 0.92 0.24 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $Re(\lambda)^\einhalb C(\lambda+A)^{-1}, \qquad Re(\lambda)>0$ on the complex right half plane and weak Lebesgue $L^{2,\infty}$--admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the 'Weiss conjecture' within the scale of Lorentz spaces $L^{p,q}$.

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