  In this paper, we address the problem of estimating a multidimensional
density $f$ by using indirect observations from the statistical model
$Y=X+\varepsilon$. Here, $\varepsilon$ is a measurement error independent of
the random vector $X$ of interest, and having a known density with respect to
the Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under
$L_p$-losses when the error $\varepsilon$ has a characteristic function with a
polynomial decay. To achieve this goal, we first construct a kernel estimator
of $f$ which is fully data driven. Then, we derive for it an oracle inequality
under very mild assumptions on the characteristic function of the error
$\varepsilon$. As a consequence, we get minimax adaptive upper bounds over a
large scale of anisotropic Nikolskii classes and we prove that our estimator is
asymptotically rate optimal when $p\in[2,+\infty]$. Furthermore, our estimation
procedure adapts automatically to the possible independence structure of $f$
and this allows us to improve significantly the accuracy of estimation.
