Weighted norm estimates and maximal regularity

Abstract

We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate for the semigroup $(e^{tA})$ and it is strictly weaker than the assumption that the $e^{tA}$ are integral operators whose kernels satisfy Gaussian estimates. As an application we present new results for the maximal regularity of Schr\"odinger operators with singular potentials, elliptic higher order operators with bounded measurable coefficients, and elliptic second order operators with singular lower order terms. Moreover, we prove a similar result for maximal regularity of the discrete time evolution equation $ u_{n+1} - Tu_n = f_n ,$ $n\in\mathbb N_0 ,$ $u_0=0 $.