Exponential decay and regularity for SG-elliptic operators with polynomial coefficients

We study the exponential decay and the regularity for solutions of elliptic partial differential equations $Pu=f$, globally defined in $\R^n.$ In particular, we consider linear operators with polynomial coefficients which are SG-elliptic at infinity. Starting from $f$ in the so-called Gelfand-Shilov spaces, the solutions $ u \in \mathcal{S}^{\prime}$ of the equation are proved to belong to the same classes. Proofs are based on a priori estimates and arguments on the Newton polyhedron associated to the operator $P$.