Summary: Consider the flow of a compressible Newtonian fluid around or past a rotating rigid obstacle in \({\mathbb R}^3.\) After a coordinate transform to get a problem in a time-independent domain we assume the new system to be stationary, then linearize and – in this paper dealing with the whole space case only – use Fourier transform to prove the existence of solutions \(u\) in \(L^q\)-spaces. However, the solution is constructed first of all in terms of \(g=\mathrm {div}\, u\), explicit in Fourier space, and is in contrast to the incompressible case not based on the heat kernel, but requires the analysis of new multiplier functions related to Bochner-Riesz multipliers and leading to the restriction \(\frac{6}{5}<q<6\).