  We aim at estimating the invariant density associated to a stochastic
differential equation with jumps in low dimension, which is for $d=1$ and
$d=2$. We consider a class of jump diffusion processes whose invariant density
belongs to some H\"older space. Firstly, in dimension one, we show that the
kernel density estimator achieves the convergence rate $\frac{1}{T}$, which is
the optimal rate in the absence of jumps. This improves the convergence rate
obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor
index for $d=1$ and is equal to $\frac{\log T}{T}$ for $d=2$. Secondly, we show
that is not possible to find an estimator with faster rates of estimation.
Indeed, we get some lower bounds with the same rates $\{\frac{1}{T},\frac{\log
T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain
the asymptotic normality of the estimator in the one-dimensional case.
