Maximal L^p-regularity for parabolic and elliptic equations on the line

Abstract.

Let A be a closed operator on a Banach space X. We study maximal L^p-regularity of the problems

$$ \begin{aligned} u^{\prime} (t) & = Au(t) + f(t)\quad {\hbox{and}}\\ u^{\prime\prime}(t) & = Au(t) + f(t) \end{aligned} $$

on the line. The results are used to solve quasilinear parabolic and elliptic problems on the line.