R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type

The property of maximal $Lp$-regularity for parabolic evolution equations is investigated via the concept of $mathcal R$-sectorial operators and operator-valued Fourier multipliers. As application, we consider the $Lq$-realization of an elliptic boundary value problem of order $2m$ with operator-valued coefficients subject to general boundary conditions. We show that there is maximal $Lp$-$Lq$-regularity for the solution of the associated Cauchy problem provided the top order coefficients are bounded and uniformly continuous.