Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck¹

¹ Département des Mathématiques Université de Cergy-Pontoise 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise, France

Studia Mathematica, Tome 146 (2001) no. 2, pp. 157-176

Résumé

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 $l_p(X)$.

Détail

DOI : 10.4064/sm146-2-3

Keywords: consider maximal regularity problem discrete time evolution equation tu mathbb where bounded operator umd space characterize discrete maximal regularity types conditions firstly r boundedness properties discrete time semigroup mathbb resolvent lambda secondly maximal regularity continuous time evolution equation au where t i recent results weis continuous maximal regularity characterized r boundedness properties continuous time semigroup t i again resolvent lambda important tool prove operator valued mikhlin theorem torus mathbb providing conditions symbol infty mathbb mathfrak associated fourier multiplier bounded