The unique solvability of a linear half-space problem is obtained in the Hölder classes of functions in an arbitrary finite time interval. The problem arises as a result of the linearization of a free boundary problem for the Navier–Stokes system governing the unsteady motion of a finite compressible liquid mass. The boundary conditions in the linear problem is noncoercive because of the surface tension on the free boundary. This fact is the main difficulty in the study of the problem, the equation being a parabolic system in the sense of Petrovskii with respect to the components of the velocity vector field. The principal idea of the investigation of the noncoercive linear problem is to reduce it to a parabolic problem corresponding to the zero surface tension and to analyze integral convolution operators arising in this reduction.