Tauberian theorem for generalized multiplicative convolutions

The following problem is discussed. Let $f$ be a generalized function of slow growth with support on the positive semi-axis, and let $\varphi_k$ be a sequence of “test” functions such that $\varphi_k\to\varphi_0$ as $k\to+\infty$ in some function space. Assume that the following limit exists: $\frac1{\rho(k)}(f(kt),\varphi_k(t))\to c$ where $\rho(k)$ is a regularly varying function. Find conditions under which the limit $\frac1{\rho(k)}(f(kt),\varphi(t))\to c_\varphi$, $k\to+\infty$, exists for all test functions $\varphi$. We state and prove theorems that solve this problem and apply them to the problem of existence of quasi-asymptotics for the solution of an ordinary differential equation with variable coefficients. We prove Abelian and Tauberian theorems for a wide class of integral transformations of distributions, for example, the generalized Stieltjes integral transformation.