Dirichlet-like space and capacity in complex analysis in several variables

For a Kähler manifold X, we study a space of test functions W∗ which is a complex version of W1,2. 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.