A new approach to maximal \(L_p\)-regularity (Q2708564)

In this article a new characterization of maximal \(L_p\)-regularity is given mainly in terms of \(R\)-boundedness. The first result is that an infinitesimal generator \(A\) of a bounded analytic semigroup \(T_t\) on a UMD-space has maximal \(L_p\)-regularity if and only if one of the sets \(\{\lambda(\lambda- A)^{-1}:\lambda\in i\mathbb{R},\lambda\neq 0\}\), \(\{T_t,tAT_t: t> 0\}\) or \(\{T_z:|\text{arg }z|\leq \delta\}\) is \(R\)-bounded. This result is a consequence of Mihlin's multiplier theorem for operator-valued multiplier functions. One of the important results in applications is the one, inspired by the Dore-Venni theorem on the sum \(A+B\) of two operators \(A\), which is \(R\)-sectorial of type \(\theta_A\), and \(B\) with an \(H_\infty(\Sigma(\theta_B))\)-functional calculus, where \(R\)-sectorial means sectorial with the replacement of the boundedness of \(\{\lambda R(\lambda, A)\}\) by its \(R\)-boundness. The results are applied to Volterra equations, equations of second-order in time, nonautonomous equations, etc. A number of other important applications are given, including the ones to perturbation theorems, analytic semigroups with a Gaussian estimate and contraction semigroups on \(L_q(\Omega, \mu)\), \(1< q< \infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].