  We show that a compact weighted extremal Kahler manifold (as defined by the
third named author) has coercive weighted Mabuchi energy with respect to a
maximal complex torus in the reduced group of complex automorphisms. This
provides a vast extension and a unification of a number of results concerning
Kahler metrics satisfying special curvature conditions, including constant
scalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitons
and their weighted extensions.
  Our result implies the strict positivity of the weighted Donaldson-Futaki
invariant of any non-product equivariant smooth K\"ahler test configuration
with reduced central fibre, a property also known as weighted K-polystability
on such test configurations. For a class of fibre-bundles, we use our result in
conjunction with the recent results of Chen-Cheng, He, and Han-Li in order to
characterize the existence of extremal Kahler metrics and Calabi-Yau cones
associated to the total space, in terms of the coercivity of the weighted
Mabuchi energy of the fibre. In particular, this yields an existence result for
Sasaki-Einstein metrics on Fano toric fibrations, extending the results of
Futaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case of
Fano projective line bundles.
