The Stokes operator in weighted L q -spaces II: weighted resolvent estimates and maximal L p -regularity

Abstract

In this paper we establish a general weighted L q -theory of the Stokes operator \({\mathcal{A}}_{q,\omega}\) in the whole space, the half space and a bounded domain for general Muckenhoupt weights \(\omega \in A_q\). We show weighted L q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator \({\mathcal{A}}_{q,\omega}\) generates a bounded analytic semigroup but even yield the maximal L p -regularity of \({\mathcal{A}}_{q,\omega}\) in the respective weighted L q -spaces for arbitrary Muckenhoupt weights \(\omega \in A_q\). This conclusion is archived by combining a recent characterisation of maximal L p -regularity by \({\mathcal{R}}\)-bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L p -regularity. Preprint (1999)] with the fact that for L q -spaces \({\mathcal{R}}\) -boundedness is implied by weighted estimates.