The Stokes operator in weighted Lq-spaces II:: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L-q-theory of the Stokes operator A(q,omega) in the whole space, the half space and a bounded domain for general Muckenhoupt weights omega is an element of 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 A(q,omega) generates a bounded analytic semigroup but even yield the maximal L-p- regularity of A(q,omega) in the respective weighted L-q -spaces for arbitrary Muckenhoupt weights omega is an element of A(q). This conclusion is archived by combining a recent characterisation of maximal L-p-regularity by 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 R-boundedness is implied by weighted estimates.