Maximal regularity of discrete and continuous time evolution equations
Volume 146 / 2001
Abstract
We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1}-Tu_n=f_n$ for all $n\in {\mathbb N}_0,\ u_0=0$, where $T$ is a bounded operator on a UMD space $X$. We characterize the discrete maximal regularity of $T$ by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup $(T^n)_{n\in {\mathbb N}_0}$ and of the resolvent $R(\lambda , T)$, secondly by the maximal regularity of the continuous time evolution equation $u'(t)-Au(t)=f(t)$ for all $t>0,\ u(0)=0$, where $A:=T-I$. By recent results of Weis, this continuous maximal regularity is characterized by R-boundedness properties of the continuous time semigroup $(e^{t(T-I)})_{t\ge 0}$ and again of the resolvent $R(\lambda , T)$.
As an important tool we prove an operator-valued Mikhlin theorem for the torus ${\mathbb T}$ providing conditions on a symbol $M\in L_\infty ({\mathbb T};{{\mathfrak L}}(X))$ such that the associated Fourier multiplier $T_M$ is bounded on $l_p(X)$.
