  A notion of local $U$-statistic process is introduced and central limit
theorems in various norms are obtained for it. This involves the development of
several inequalities for $U$-processes that may be useful in other contexts.
This local $U$-statistic process is based on an estimator of the density of a
function of several sample variables proposed by Frees [J. Amer. Statist.
Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth central
limit theorems in the sup and in the $L_p$ norms are obtained for these
estimators.
