The permanence of $\mathcal{R}$-boundedness and property($\alpha$) under interpolation and applications to parabolic systems | Graduate School of Mathematical Sciences, The University of Tokyo

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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
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Abstract:
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

Mathematics Subject Classification (2010): Primary 46B70, 47F05; Secondary 35K41, 35K46

Received: 2012-04-06

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

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