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

Abstract

Let 1<p<∞ and let T be a positive strongly Kreiss bounded operator on an L^p-space. We prove that there exist constants C,κ>0 such that |T^N| ≤ C ( log(N+1) )^κ, N ≥ 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 ℓ^p-estimate for the orbit on intervals of length of order √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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BibTeX RIS

BibTeX

@misc{arnold2026positive,
  title = {{Positive strongly Kreiss bounded operators on $L^p$-spaces have logarithmic power growth}},
  author = {Loris Arnold},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27771},
  eprint = {2609.27771},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27771},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Positive strongly Kreiss bounded operators on L^p-spaces have logarithmic power growth
AU  - Loris Arnold
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27771
UR  - https://arxiv.org/abs/2609.27771
AB  - Let 1<p<∞ and let T be a positive strongly Kreiss bounded operator on an L^p-space. We prove that there exist constants C,κ>0 such that |T^N| ≤ C ( log(N+1) )^κ, N ≥ 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 ℓ^p-estimate for the orbit on intervals of length of order √N; a positive vector-valued lifting of T then produces a self-improving estimate which can be iterated down to the logarithmic scale.
ER  -

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