  In this paper we aim at employing a compactness criterion of Da Prato,
Malliavin, Nualart for square integrable Brownian functionals to construct
unique strong solutions of SDE's under an integrability condition on the drift
coefficient. The obtained solutions turn out to be Malliavin differentiable and
are used to derive a Bismut-Elworthy-Li formula for solutions of the Kolmogorov
equation.
