Boundedness of the Density of Wiener Functionals with Low Regularity

Source abstract

In this paper we prove that if a Wiener functional $F$ is $1+δ$-differentiable in the Malliavin sense for some $δ>0$ and satisfies the nondegeneracy condition, then the density of the law of $F$ exists, is bounded and is Hölder continuous of every exponent $β<δ$. This gives a partial positive answer to a problem proposed by Watanabe [13]; to date, this seems to be the best result available for the boundedness of the density of Wiener functionals. The approach is based on the fractional order Sobolev spaces associated with a probability measure and on the Riesz transform.