A microlocal pseudodifferential calculus developed to analyse differential operators on manifolds with edge-type singularities or stratified boundaries; it organizes symbols and operator classes according to a base–fiber splitting near the edge and captures anisotropic behavior transverse and tangential to the edge.

Principle

Decompose the local geometry near an edge into a product of a base (edge) and a fiber (conic or model) direction; construct symbol hierarchies that are pseudodifferential along the base and Mellin- or cone-like in the transverse fiber so that parametrices reflect the mixed regularity.

Demonstration

On a manifold with boundary that locally looks like B × C(F) where B is the edge and C(F) a cone over a fiber F, the edge calculus provides operator classes and parametrices for elliptic edge operators, yielding mapping properties between weighted Sobolev spaces adapted to base and fiber regularity.

Misapplication

Using cone algebra or standard pseudodifferential calculus alone on a genuine edge geometry; such misuse fails to capture the coupling between base tangential pseudodifferential behavior and transverse conic singularities, giving incorrect Fredholm and regularity conclusions.

Consequence

Edge calculus produces precise parametrices, symbolic invariants, and index statements for edge-elliptic operators and gives microlocal descriptions of propagation and regularity along and across the edge; it is essential for resolving boundary-value and spectral problems in stratified geometries.

Reversal

If the singularity reduces to an isolated cone (no base), the edge calculus simplifies to the cone algebra; at the other extreme, on a smooth manifold without stratification it reduces to the standard pseudodifferential calculus.

Boundary

Aimed at manifolds with edge-type or fibered boundary singularities where a clear base–fiber decomposition exists; it does not apply directly to more complicated stratifications without iterated versions or to non-microlocal contexts.

Semantic Tension

Interacts closely with cone algebra and b-calculus; the tension lies in selecting the correct calculus to model a given singular geometry (edge vs cone vs boundary) since each imposes different symbol structures and functional frameworks.

Synthesis

Edge calculus is a hybrid microlocal framework that splits variables into edge (base) and transverse (fiber) directions and assembles pseudodifferential and Mellin-type symbol classes to construct parametrices and capture the anisotropic regularity intrinsic to edge singularities.