Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$.
Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that
$\limsup_{\omega,t \in\Omega\times[S,T]} \sum_i^m \sum_j^n | g_{ij}(\omega,t)|<\infty$ and
$\int_S^T E|G(\omega,t)^6|dt<\infty.$
I have to prove that:
$E\left|\int_S^T G(\omega,t)dW_t\right|^6 \le 15^3(T-S)^2\int_S^TE|G(\omega,t)^6|dt<\infty.$
I've also a hint:
$\int_S^T \int_\Omega H(\omega,t)^4 K(\omega,t)^2dtdP(\omega) \le \left\{\int_S^T \int_\Omega H(\omega,t)^6 dtdP(\omega) \right\}^{4/6} \left\{ \int_S^T \int_\Omega K(\omega,t)^6dtdP(\omega))\right\}^{2/6}.$
My idea was to use Itō's isometry in order to pass from $dW_t$ in $dt$ but I don't know if it's possible with $6$ at the exponent. Maybe a change of variable? Anyway I can't figure out where the coefficient $15^3(T-S)^2$ came from...
Thank you for your help
EDIT: I found a very interesting article from Novikov called "On moment inequalities and identities for stochastic integrals", which analyses a very similar case. I had not time to properly study this work but the key was applying the Itō's formula to a specific function.