We study a random walk Sn on Zd (d≥1), in the domain of attraction of an operator-stable distribution with index α=(α1,…,αd)∈(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(0,x) as ∥x∥→+∞, along the “favorite direction or scaling”: (i) if ∑di=1α−1i<2 (reminiscent of Garsia-Lamperti’s condition when d=1 [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when d=1). We also provide uniform bounds on the Green function G(0,x), sharpening estimates when x is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case αi≡α, in the favorite scaling, and has even left aside the case α∈[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.