Bounds for the metric and shift vector on extreme Kerr under linear axially symmetric gravitational perturbations

We present integral estimates for the shift and induced metric in the maximal isothermal gauge, corresponding to linear axially symmetric gravitational perturbations over the Minkowski and extreme Kerr metric. These bounds are a step towards finding pointwise bounds for the perturbations. In particular the presented estimate for the shift includes the horizon for the extreme Kerr case.