Asymptotics of Solutions to Pseudodifferential Equations of M ELLIN Type

We consider pseudodifferential operators on the half‐axis of the form\documentclass{article}\pagestyle{empty}\begin{document}$$ (1)\;a(t,\partial)u\; = \;\frac{1}{{2\pi i}}\;\int\limits_{} {t^{ - z} a(t,z)\;u(z)\;{\rm dz}},\;u \in C_0^\infty (R^ +), $$\end{document}where \documentclass{article}\pagestyle{empty}\begin{document}$ u(z)\; = \;\int\limits_0^\infty {{\rm t}^{{\rm z - 1}} u(t)} $\end{document} is the M ELLIN 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 M ELLIN type or shortly M ELLIN operator with the symbol a ( t , z ). For example, F UCHS 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 M ELLIN 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 S OBOLEV 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 M ELLIN type on a finite interval as well as to systems of such equations.