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arXiv · 2609.27771

Positive strongly Kreiss bounded operators on $L^p$-spaces have logarithmic power growth

Loris Arnold
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Abstract

Let $1 0$ such that \[ \|T^N\| \leq C \left( \log(N+1) \right)^κ, \qquad N \geq 1. \] This answers a question raised by Arnold and Cuny concerning positive strongly Kreiss bounded operators on $L^p$-spaces. It also shows that the polynomial exponent appearing in the corresponding problem of Deng, Lorist and Veraar has infimum zero. The proof is elementary: positivity yields a local $\ell^p$-estimate for the orbit on intervals of length of order $\sqrt N$; a positive vector-valued lifting of $T$ then produces a self-improving estimate which can be iterated down to the logarithmic scale.

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Loris Arnold. 2026-08-16. Positive strongly Kreiss bounded operators on $L^p$-spaces have logarithmic power growth. https://arxiv.org/abs/2609.27771

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