We consider resolvent estimates for the Stokes equations in a space of bounded functions for a bounded or an exterior domain with smooth boundaries. Unlike on the usual $L^{p}$ space, on $L^{\infty}$ the Helmholtz projection does not act as a bounded operator. We invoke the weighted $L^{\infty}$-estimate for harmonic-pressure and then apply the Masuda-Stewart technique for the Stokes operator. This talk is based on a joint work with Y. Giga (U. Tokyo) and M. Hieber (TU Darmstadt).