Analyticity and Discrete Maximal Regularity on Lp-Spaces

In the first part of this paper, we give the following interpolation result on the analyticity (i.e. the property ‖(T−I) Tn‖⩽C/n for all n∈N) of an operator T on Lp: If T is powerbounded on Lp and Lq as well as analytic on Lp, then T is powerbounded and analytic on Lr for all r strictly between p and q. This is a discrete analogue of the well-known corresponding result for analytic semigroups (etA). As recently shown by the author, the analyticity of T is a necessary condition for the maximal regularity of the discrete time evolution equation un+1−Tun=fn for all n∈Z+, u0=0. In the second part of this paper we establish the following two sufficient conditions for its maximal regularity: T is a subpositive analytic contraction, or T is an integral operator satisfying certain Poisson bounds. These results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u′(t)−Au(t)=f(t) for all t∈R+, u(0)=0, due to Lamberton, Weis, Coulhon and Duong and Hieber and Prüss. For the Poisson bound result of Coulhon and Duong and Hieber and Prüss we give a slight improvement and a short proof.