zbMATH — the first resource for mathematics

Maximal regularity of discrete and continuous time evolution equations. (English) Zbl 0981.39009
The author considers the maximal regularity problem for the discrete time evolution equation \( u_{n+1}- T u_n=f_n\) for all \(n \in N_0\), \(u_0=0\), where \(T\) is a bounded operator on a UMD space \(X\). The maximal regularity of \(T\) is characterized by two types of conditions: the first one by \(R\)-boundedness properties of the discrete time semigroup \(\{T^n\}_{n \in N_0}\) and for the resolvent \(R(\lambda, T)\); the second one by the maximal regularity of the continuous time evolution equation \(u'(t)- A u(t)=f(t)\) for all \(t >0\), \(u(0)=0\), where \(A := T-I\).

39A12 Discrete version of topics in analysis
42A45 Multipliers in one variable harmonic analysis
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI