Growth rate of positive Kreiss bounded $C_0$-semigroups on $L^p$

Abstract

In this paper, we compare several Cesàro and Kreiss type boundedness conditions for a C 0 -semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on an ordered Banach lattice. Furthermore, we prove that a Kreiss bounded positive C 0 -semigroup on L p , 1 ≤ p < +∞ has a growth rate O t/ log(t) max(1/p,1/p ) if p > 1 and O(t 1− ) for some ∈ (0, 1) if p = 1, improving previous estimates.