The total variation distance between two double Wiener-Itô integrals

Using an approach recently developed by  Nourdin and Poly (2013), we improve the rate in an inequality for the total variation distance between two double Wiener-Itô integrals originally due to  Davydov and Martynova (1987). An application to the rate of convergence of a functional of a correlated two-dimensional fractional Brownian motion towards the Rosenblatt random variable is then given, following a previous study by  Maejima and Tudor (2012).