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.