| Abstract: | We consider pseudodifferential operators on the half-axis of the form  where \documentclass{article}\pagestyle{empty}\begin{document}$ u(z) = \int\limits_0^\infty {{\rm t}^{{\rm z - 1}} u(t)} $\end{document} is the MELLIN transform of u and a(t, z) satisfies suitable smoothness properties in t and holomorphy and growth properties in z in some strip around the line Re z = 1/2. (1) is called pseudodifferential operator of MELLIN type or shortly MELLIN operator with the symbol a(t, z). For example, FUCHS ian differential operators, singular integral operators and integral operators with fixed singularities can be written in this form. In the paper we give a new composition theorem for MELLIN operators which has a natural extension to operators with symbols meromorphic in a left half-plane. The theorem can be used in the construction of left parametrices modulo compact operators in weighted SOBOLEV spaces. This approach yields rather precise results on the complete asymptotics of solutions at the point t = 0 for an equation a(t, δ) u = f when the right-hand side f has a prescribed asymptotical behaviour at t = 0. The results are extended to pseudodifferential equations of MELLIN type on a finite interval as well as to systems of such equations. |
