A pseudodifferential calculus, introduced by Melrose, built from vector fields tangent to the boundary (the b-tangent bundle) that organizes symbols and operators adapted to manifolds with boundary and to degeneracies at the boundary.

Principle

Replace the standard tangent bundle by the b-tangent bundle and use symbols homogeneous with respect to rescaled covariables so that operators generated by b-vector fields have a closed algebra with calculable mapping and index properties.

Demonstration

On a compact manifold with boundary, the b-Laplacian (constructed from b-vector fields) models problems with cylindrical ends; the b-calculus produces parametrices and precise mapping properties on weighted Sobolev spaces adapted to the boundary behaviour.

Misapplication

Treating an operator with boundary-degenerate coefficients as if it belonged to the standard pseudodifferential calculus (ignoring the b-structure) leads to incorrect parametrices, lost uniform estimates near the boundary, and misidentified Fredholm properties.

Consequence

Provides a functional framework to construct parametrices, prove Fredholmness and compute indices for elliptic b-operators, and to analyze asymptotics of solutions near the boundary using weighted spaces and the b-symbol.

Reversal

If one abandons the b-perspective and uses the classical calculus instead, boundary degeneracies appear singular and essential spectral and index information may be lost; conversely, introducing b-structures where unnecessary complicates symbol classes without gain.

Boundary

Targets differential and pseudodifferential operators generated by vector fields tangent to the boundary on manifolds with boundary or cylindrical ends; does not automatically treat corners, conic singularities, or scattering-type asymptotics without further modification.

Semantic Tension

The b-calculus sits near other calculi (scattering calculus, cusp calculus, Boutet de Monvel algebra); tension arises when deciding whether the boundary singularity is best handled by tangential rescaling (b) or by transmission-type boundary operator algebras.

Synthesis

B-Calculus reframes analysis on manifolds with boundary by privileging b-vector fields and rescaled symbols, yielding an algebraic and microlocal toolkit that captures boundary degeneracies, produces parametrices on weighted spaces, and clarifies index and asymptotic behaviour.