A Tauberian theorem for quasiasymptotic decompositions of measures with supports in the positive octant

The paper is devoted to a multidimensional Tauberian theorem of Hardy–Littlewood type for quasiasymptotic expansions of measures concentrated in the positive octant. Here the quasiasymptotic expansion is assumed to be local, i.e., its terms are generalized functions concentrated at the origin. The asymptotic behavior of the remainder is estimated with respect to the scale of regularly varying (self-similar) functions along trajectories defined by one-parameter groups of automorphisms of the cone in which the measure is concentrated. The case of one variable is investigated in more detail; in particular, a Hardy–Littlewood type theorem is proved for generalized functions that are nonnegative measures for large values of the argument.