Maximal Lp – Lq-Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Abstract

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal Lp – Lq-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an \({\mathcal{R}}\)-sectorial operator in \(L^{p}_{\sigma}(\Omega)\), \(1 < p < \infty\), of \({\mathcal{R}}\)-angle 0, for bounded or exterior domains of Ω.