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

Abstract

Let $1<p<\infty$ and let $T$ be a positive strongly Kreiss bounded operator on an $L^p$-space. We prove that there exist constants $C,\kappa>0$ such that \[ \|T^N\| \leq C \left( \log(N+1) \right)^\kappa, \qquad N \geq 1. \] 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.