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 φk be a sequence of 'test' functions such that φk→φ0 as k→∞ in some function space. Assume that the following limit exists: 1/ρ(k) (f(kt), φk(t))→c where ρ(k) is a regularly varying function. Find conditions under which the limit 1/ρ(k) (f(kt), φ(t))→cφ, k→∞, exists for all test functions φ. 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