Pseudodifferential calculi on the half-line respecting prescribed asymptotic types (Q1882520)

The authors set up scales of Sobolev spaces on the half-line \(\mathbb R_+\) with prescribed conormal asymptotic expansions \[ u(t) \sim \sum_{j=1}^M\sum_{k=0}^{m_j}c_{j,k}\log^k(t)t^{p_j} \] as \(t \to 0\) for the functions \(u\), and construct appropriate calculi of pseudodifferential operators which leave these spaces invariant. The ellipticity condition for these operators is related to the prescribed asymptotics, and elliptic operators are shown to admit parametrices within the calculus, and give rise to Fredholm operators in the Sobolev spaces. The case of Taylor expansions is a special case and leads to the standard smooth Sobolev spaces on the half-axis and to pseudodifferential operators with the transmission property. The method makes use of Mellin pseudodifferential operators near the origin and control on the complete sequence of Mellin symbols. The classes of operators in this work are subcalculi of Schulze's cone algebra on the half-axis.