Summary: The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of \(\mathcal{R}\)-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal \(L_p -L_q\) regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by I. V. Denisova [Interfaces Free Bound. 2, No. 3, 283–312 (2000; Zbl 0971.35057)] under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in [loc. cit.].