  In [16], under mild conditions, a Wiener-Hopf type factorization is derived
for the exponential functional of proper L\'evy processes. In this paper, we
extend this factorization by relaxing a finite moment assumption as well as by
considering the exponential functional for killed L\'evy processes. As a
by-product, we derive some interesting new distributional properties enjoyed by
a large class of this random variable, such as the absolute continuity of its
distribution and the smoothness, boundedness or complete monotonicity of its
density. This type of results is then used to derive similar properties for the
law of maxima and first passage time of some stable L\'evy processes. Thus, for
example, we show that for a large class of stable processes the first passage
time has a bounded and non-increasing density on the positive half-line. We
also generate many instances of integral or power series representations for
the law of the exponential functional of L\'evy processes with one or two-sided
jumps. The proof of our main results requires different devices from the one
developed in [16]. It relies in particular on a generalization of a transform
recently introduced in [8] together with some extensions to killed L\'evy
process of Wiener-Hopf techniques.
