Gauge fixing and regularity of axially symmetric and axistationary second order perturbations around spherical backgrounds

Perturbation theory in geometric theories of gravitation is a gauge theory of symmetric tensors defined on a Lorentzian manifold (the background spacetime). The gauge freedom makes uniqueness problems in perturbation theory particularly hard as one needs to understand in depth the process of gauge fixing before attempting any uniqueness proof. This is the first paper of a series of two aimed at deriving an existence and uniqueness result for rigidly rotating stars to second order in perturbation theory in General Relativity. A necessary step is to show the existence of a suitable choice of gauge and to understand the differentiability and regularity properties of the resulting gauge tensors in some "canonical form", particularly at the centre of the star. With a wider range of applications in mind, in this paper we analyse the fixing and regularity problem in a more general setting. In particular we tackle the problem of the Hodge-type decomposition into scalar, vector and tensor components on spheres of symmetric and axially symmetric tensors with finite differentiability down to the origin, exploiting a strategy in which the loss of differentiability is as low as possible. Our primary interest, and main result, is to show that stationary and axially symmetric second order perturbations around static and spherically symmetric background configurations can indeed be rendered in the usual "canonical form" used in the literature while loosing only one degree of differentiability and keeping all relevant quantities bounded near the origin.