Abstract
The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ℝ+, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ℝ+ ∋ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.
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Dorschfeldt, C., Schulze, BW. Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach. Ann Glob Anal Geom 12, 135–171 (1994). https://doi.org/10.1007/BF02108294
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DOI: https://doi.org/10.1007/BF02108294