The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals

The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the determinant of its iterated Malliavin matrices. As an application we show that the vector is absolutely continuous if and only if its components are proportional.