Maximal regularity for evolution equations in weighted L _p -spaces

Abstract.

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem

$\dot{u}(t) + Au(t) = f(t),\; t > 0,\quad u(0) = 0,$

enjoys maximal regularity in weighted L _p -spaces with weights $\omega(t) = t^{p(1-\mu)}$, where $1/p < \mu$, if and only if it has the property of maximal L _p -regularity. Moreover, it is also shown that the derivation operator $D = d/dt$ admits an ${\cal H}^\infty$-calculus in weighted L _p -spaces.