Moments of Gaussian chaoses in Banach spaces

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including $L_q$ spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in $L_q$ spaces based on exponential random variables.