  For a Kahler manifold X, we study a space of test functions W* which is a
complex version of H1. We prove for W* the classical results of the theory of
Dirichlet spaces: the functions in W* are defined up to a pluripolar set and
the functional capacity associated to W* tests the pluripolar sets. This
functional capacity is a Choquet capacity.
  The space W* is not reflexive and the smooth functions are not dense in it
for the strong topology. So the classical tools of potential theory do not
apply here. We use instead pluripotential theory and Dirichlet spaces
associated to a current.
