We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on \(\textrm{L}^p_{\sigma } (\mathbb {R}^d)\) , \(d \ge 2\) , for . These include optimal resolvent bounds, the property of maximal \(\textrm{L}^q\) -regularity, the boundedness of its \(\textrm{H}^{\infty }\) -calculus, and characterizations of fractional power domains. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.
