On a Problem of Multidimensional Tauberian Theory

In many Tauberian theorems, the asymptotic properties of functions were investigated with respect to a predefined function (usually in the scale of regularly varying functions). In this paper, we address an alternative problem: Given a generalized function, does it have asymptotics with respect to some regularly varying function? We find necessary and sufficient conditions for the existence of quasiasymptotics of those generalized functions whose Laplace transforms have a bounded argument in a tube domain over the positive orthant. Moreover, we point out a regularly varying function with respect to which quasiasymptotics exists. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays emanating from the origin behaves as a regularly varying function. We use the obtained results to find the quasiasymptotics of the generalized Cauchy problem for convolution equations whose kernels are passive operators.