A pseudodifferential calculus adapted to manifolds with isolated conical singularities that captures the operator behavior near the cone apex by mixing Mellin-type symbols in the radial variable with standard pseudodifferential symbols on the link; designed to construct parametrices and describe mapping properties on weighted Sobolev spaces.

Principle

Replace standard symbol expansions near a smooth point by a combination of radial Mellin symbols and angular pseudodifferential symbols on the link so that the calculus respects the dilation symmetry of the cone and encodes asymptotic expansions at the apex.

Demonstration

For an exact metric cone (0,1)_r × Y with Laplace-type operator, the cone algebra expresses the operator as r^{-2} times a Mellin family in the complex dual of log r coupled with an operator on the link Y; this formulation yields parametrices that control asymptotics of solutions and resolvent expansions.

Misapplication

Applying the standard pseudodifferential calculus ignoring the cone structure near an apex; this misses essential singular terms, yields incorrect parametrices, and fails to capture the correct domain or Fredholm properties.

Consequence

Using cone algebra one obtains precise parametrices, index formulas, and resolvent asymptotics for elliptic cone operators and a clean description of domains in weighted Sobolev scales; it enables treatment of boundary spectra (indicial roots) and self-adjoint extension data.

Reversal

On a smooth manifold without conical singularities the cone algebra reduces to the standard pseudodifferential calculus; conversely, if singularities are more severe (e.g. edges), cone algebra alone is insufficient and must be replaced by a finer calculus.

Boundary

Intended for isolated conical singularities or exact cone neighborhoods; it excludes stratified edges or cusps where the geometry requires edge calculus or cusp calculi and different symbol decompositions.

Semantic Tension

Closely related to b-calculus and edge calculus; semantic tension arises in choosing the most appropriate calculus for a given singular geometry since each emphasizes different variable splittings (radial vs tangential) and different model operators.

Synthesis

The cone algebra is a tailored pseudodifferential framework that blends Mellin analysis in the radial direction with angular pseudodifferential structure on the link, producing parametrices and mapping results that faithfully reflect conical singular asymptotics.