Asymptotically homogeneous solutions to differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter group (Q483599)

Let \(U = (U_k: k > 0)\) be a multiplicative one-parameter group of linear transformations on \({\mathbb R}^n\) such that all eigenvalues of the generator of the group are positive and let \(P(t)\) be a polynomial such that \(P(U_k t) = t^q P(t)\) for some \(q\in {\mathbb R}.\) Given \(g\in{\mathcal S}^\prime({\mathbb R})\) asymptotically homogeneous at infinity along the group \(U,\) the paper deals with conditions to guarantee that the differential equation \(P(\partial) u = g\) admits a solution which is also asymptotically homogeneous at infinity along the group.