  We study the multivariate deconvolution problem of recovering the
distribution of a signal from independent and identically distributed
observations additively contaminated with random errors (noise) from a known
distribution. For errors with independent coordinates having ordinary smooth
densities, we derive an inversion inequality relating the $L^1$-Wasserstein
distance between two distributions of the signal to the $L^1$-distance between
the corresponding mixture densities of the observations. This smoothing
inequality outperforms existing inversion inequalities. As an application of
the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein
deconvolution with Laplace noise in dimension one using a Dirichlet process
mixture of normal densities as a prior measure on the mixing distribution (or
distribution of the signal). We construct an adaptive approximation of the
sampling density by convolving the Laplace density with a well-chosen mixture
of normal densities and show that the posterior measure concentrates around the
sampling density at a nearly minimax rate, up to a log-factor, in the
$L^1$-distance. The same posterior law is also shown to automatically adapt to
the unknown Sobolev regularity of the mixing density, thus leading to a new
Bayesian adaptive estimation procedure for mixing distributions with regular
densities under the $L^1$-Wasserstein metric. We illustrate utility of the
inversion inequality also in a frequentist setting by showing that an
appropriate isotone approximation of the classical kernel deconvolution
estimator attains the minimax rate of convergence for $1$-Wasserstein
deconvolution in any dimension $d\geq 1$, when only a tail condition is
required on the latent mixing density and we derive sharp lower bounds for
these problems
