Yu. N. Drozhzhinov

Homogeneous generalized functions with respect to one-parametric group

We give the full description of homogeneous generalized functions along the trajectories of arbitrary one-parametric multiplicative group of linear transformations whose generator matrix has eigenvalues with positive real parts.We also study the problem of extension of such functionals from the space of test functions vanishing at the origin up to the whole space S(ℝn), and discuss the conditions of uniqueness of such extension.

Asymptotically Homogeneous Generalized Functions and Some of Their Applications

A brief description is given of generalized functions that are asymptotically homogeneous at the origin with respect to a multiplicative one-parameter transformation group such that the real parts of all eigenvalues of the infinitesimal matrix are positive. The generalized functions that are homogeneous with respect to such a group are described in full. Examples of the application of such functions in mathematical physics are given; in particular, they can be used to construct asymptotically homogeneous solutions of differential equations whose symbols are homogeneous polynomials with respect to such a group, as well as to study the singularities of holomorphic functions in tubular domains over cones.

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

We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case.