  We estimate linear functionals in the classical deconvolution problem by
kernel estimators. We obtain a uniform central limit theorem with
$\sqrt{n}$-rate on the assumption that the smoothness of the functionals is
larger than the ill-posedness of the problem, which is given by the polynomial
decay rate of the characteristic function of the error. The limit distribution
is a generalized Brownian bridge with a covariance structure that depends on
the characteristic function of the error and on the functionals. The proposed
estimators are optimal in the sense of semiparametric efficiency. The class of
linear functionals is wide enough to incorporate the estimation of distribution
functions. The proofs are based on smoothed empirical processes and mapping
properties of the deconvolution operator.
