Sobolev-Wigner spaces

In this paper, we provide new results about the free Malliavin calculus on
the Wigner space first developed in the breakthrough work of Biane and
Speicher. We define in this way the higher-order Malliavin derivatives, and we
study their associated Sobolev-Wigner spaces. Using these definitions, we are
able to obtain a free counterpart of the of the Stroock formula and various
variances identities. As a consequence, we obtain a sophisticated proof a la
Ustunel, Nourdin and Peccati of the product formula between two multiple Wigner
integrals. We also study the commutation relations (of different
significations) on the Wigner space, and we show for example the absence of
non-trivial bounded central Malliavin differentiable functionals and the
absence of non-trivial Malliavin differentiable projections.