  We consider the Stokes resolvent problem in a two-dimensional bounded
Lipschitz domain $\Omega$ subject to homogeneous Dirichlet boundary conditions.
We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition
$\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$.
We further show that the Stokes operator admits the property of maximal
regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then
used to characterize domains of fractional powers of the Stokes operator.
Finally, we give an application to the regularity theory of weak solutions to
the Navier-Stokes equations in bounded planar Lipschitz domains.
