Given a random variable F regular enough in the sense of the Malliavin
calculus, we are
able to measure the distance between its law and any probability measure
with a density
function which is continuous, bounded, strictly positive on an interval in
the real line and
admits finite variance. The bounds are given in terms of the Malliavin
derivative of F. Our
approach is based on the theory of It?o diffusions and the stochastic
calculus of variations.
Several examples are considered in order to illustrate our general
results.