On the growth rate of powers of a strongly Kreiss bounded operator on an Lp-space

Let T be a strongly Kreiss bounded linear operator on Lp. We obtain a bound on the rate of growth of the norms of the powers of T. The bound is optimal with respect to the polynomial scale. The proof makes use of Fourier multipliers, in particular of the Littlewood–Paley inequalities on arbitrary intervals as initiated by Rubio de Francia and developed by Kislyakov and Parilov.