  We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the
domain of attraction of an operator-stable distribution with index
$\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_d) \in (0,2]^d$: in particular, we
allow the scalings to be different along the different coordinates. We prove a
strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function
$G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite
direction or scaling": (i) if $\sum_{i=1}^d \alpha_i^{-1} < 2$ (reminiscent of
Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$,
1962]); (ii) if a certain $local$ condition holds (reminiscent of Doney's
condition [Probab. Theory Relat. Fields $\mathbf{107}$, 1997] when $d=1$). We
also provide uniform bounds on the Green function $G(\mathbf{0},\mathbf{x})$,
sharpening estimates when $\mathbf{x}$ is away from this favorite direction or
scaling. These results improve significantly the existing literature, which was
mostly concerned with the case $\alpha_i\equiv \alpha$, in the favorite
scaling, and has even left aside the case $\alpha\in[1,2)$ with non-zero mean.
Most of our estimates rely on new general (multivariate) local large deviations
results, that were missing in the literature and that are of interest on their
own.
