## The permanence of $\mathcal{R}$-boundedness and property($\alpha$) under interpolation and applications to parabolic systems

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 3, Page 359--407.

Kaip, Mario; Saal, J\"{u}rgen
The permanence of $\mathcal{R}$-boundedness and property($\alpha$) under interpolation and applications to parabolic systems
This note consists of two parts. In the first part we consider the behavior of $\mathcal{R}$-boundedness, $\mathcal{R}$-sectoriality, and property($\alpha$) under the interpolation of Banach spaces. In a general setting we prove that for interpolation functors of type $h$ the $\mathcal{R}$-boundedness, the $\mathcal{R}$-sectoriality, and the property($\alpha$) preserve under interpolation. In particular, this is true for the standard real and complex interpolation methods. (Partly, these results were indicated in \cite{kalton06}, however, with just a very brief outline of their proofs.) The second part represents an application of the first part. We prove $\mathcal R$-sectoriality, or equivalently, maximal $L^p$-regularity for a general class of parabolic systems on interpolation spaces including scales of Besov- and Bessel-potential spaces over $\R^n$.
Keywords: Interpolation, $\mathcal{R}$-boundedness, maximal regularity, parabolic systemS