Analyticity and Discrete Maximal Regularity on Lp-Spaces

Abstract

bounded and analytic on L r for all r strictly between p and q. This is a discrete analogue of the well-known corresponding result for analytic semigroups (e tA ). As recently shown by the author, the analyticity of T is a necessary condition for the maximal regularity of the discrete time evolution equation u n+1 &Tu n = f n for all n # Z + , u 0 =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 Pru ss. For the Poisson bound result of Coulhon and Duong and Hieber and Pru ss we give a slight improvement and a short proof. Academic Press

Subordinated discrete semigroups of operators

“…Actually, Lyubich's solution in [19] to Zemánek's question, which involves certain fractional Volterra-type operators, also seems to be a special case of Corollary 1.4 in view of the fact (see [21]) that the operator T 1 := I − V is a Kreiss operator in the spaces L p (0, 1), 1 ≤ p ≤ ∞. The following fundamental theorem on Ritt operators will be an important tool throughout the paper (for details of the theorem and further developments on Ritt operators, see [2,3,23,24,16,27,19] and their references). For θ ∈ (0, π) define the sectors Λ θ := {z ∈ C : z = 0, | Arg z| < θ} and Λ θ := {0}∪{z ∈ C : | Arg z| ≤ θ}.…”

Square Functions for Ritt Operators on Noncommutative $L^p$-Spaces

“…Details and complements can be found in [ALM], [Bl1], [Bl2], [LM2], [Lyu], [NaZ], [Nev] and [Vit]. Let X be a Banach space.…”

Maximal regularity in l spaces for discrete time fractional shifted equations

“…First results concerning maximal regularity for discrete time abstract Cauchy problems in Banach spaces are due to Blunck ([7], [8]). Kalton and Portal [21] considered maximal regularity in p spaces for the critical cases p = 1, ∞.…”