  We present moment inequalities for completely degenerate Banach space valued
(generalized) U-statistics of arbitrary order. The estimates involve suprema of
empirical processes which, in the real-valued case, can be replaced by simpler
norms of the kernel matrix (i.e., norms of some multilinear operators
associated with the kernel matrix). As a corollary, we derive tail inequalities
for U-statistics with bounded kernels and for some multiple stochastic
integrals.
