Distributions asymptotically homogeneous along the trajectories determined by one-parameter groups (Q2901877)

Let \(U = \left\{U_k, k>0\right\}\) be a continuous multiplicative one-parameter group of linear transformations of \({\mathbb R}^n\) and suppose that \({\mathcal F}\) is a space of test functions which is invariant under \(U_k.\) Then \(f\in {\mathcal F}^\prime\) is said to be asymptotically homogeneous on \({\mathcal F}\) along \(U\) if for some continuous and positive function \(\rho\) and for some \(g\in {\mathcal F}^\prime\) NEWLINE\[NEWLINE \lim_{k\to +\infty}\frac{1}{\rho(k)}\left(f(U_kt), \psi(t)\right) = \left(g(t), \psi(t)\right) NEWLINE\]NEWLINE for every \(\psi \in {\mathcal F}\).NEWLINENEWLINEThe present paper gives a complete description of the asymptotically homogeneous tempered distributions in the case that \(U_k = e^{E \ln k}\) where \(E\) is a linear transformation of \({\mathbb R}^n\) with the property that the real parts of all eigenvalues are positive. The main results are contained in Section 4 and the basic tool is a generalized spherical representation of distributions, which permits to reduce the problem to the study of radial asymptotic properties of distributions defined on special spaces of test functions. This representation is based on the use of generalized spherical coordinates, defined by the formula NEWLINE\[NEWLINE t = \mu(r,e) = U_re,\;e\in \Gamma,\;r > 0, NEWLINE\]NEWLINE where \(\Gamma\) is a closed surface surrounding the origin in such a way that each trajectory defined by \(U\) intersects \(\Gamma\) transversally and only once.NEWLINENEWLINELet \(\zeta\) denote the map sending the function \(\varphi\in {\mathcal S}({\mathbb R}^n)\) to \(\psi(r,e):=\varphi(U_re)\). In Section 1, the authors introduce some special spaces of test functions and distributions in order to describe the image of the map \(\zeta\) and justify the corresponding change of variables formula for distributions. In particular, the condition that \(f\in {\mathcal S}^\prime({\mathbb R}^n)\) is asymptotically homogeneous along \(U\) is equivalent to the fact that its generalized spherical representation is asymptotically homogeneous over the space \(\zeta({\mathcal S}({\mathbb R}^n))\). The technical part of the paper is contained in Sections 2 and 3, which aims to study the distributions asymptotically homogeneous on the spaces of test functions introduced in Section 1.