A symbolic compatibility condition for classical pseudodifferential operators at a hypersurface (or boundary) requiring matching of interior and exterior asymptotic expansions so that operators act smoothly across the interface.

Principle

Symbols of the operator must have prescribed parity or matching expansions in the normal covariable so that boundary traces, Poisson operators and parametrices do not exhibit spurious jumps; the transmission property enforces this compatibility at each homogeneous order.

Demonstration

A classical pseudodifferential operator whose full symbol admits an asymptotic expansion in homogeneous components satisfying the parity conditions in the normal variable will map smooth functions with smooth boundary traces to smooth functions; this is used to place such operators inside the Boutet de Monvel algebra.

Misapplication

Ignoring transmission leads to operators that produce boundary layer terms or mismatched leading coefficients, causing loss of regularity at the boundary and invalidating constructions of parametrices that assume smooth matching.

Consequence

When the transmission property holds, one gets smooth action up to the boundary, valid boundary traces, and compatibility with boundary operator calculi; it is often a prerequisite for formulating elliptic boundary problems in classical frameworks.

Reversal

Failure of transmission yields jump discontinuities or nonclassical boundary asymptotics; such operators may require modified calculi (edge, cusp, or fully nonclassical symbol classes) to capture their behaviour.

Boundary

Relevant for classical pseudodifferential operators on smooth manifolds with hypersurfaces or boundary where symbols admit asymptotic homogeneous expansions; it excludes nonclassical symbols, some Fourier integral operators, and settings with insufficient smoothness to form the expansions.

Semantic Tension

This property is distinct from, but related to, boundary conditions themselves or to notions of ellipticity; tension arises in deciding whether to force symbols to satisfy transmission or to treat the failure by switching to an alternative calculus.

Synthesis

The Transmission Property is the symbol-level matching condition ensuring interior and exterior symbolic expansions agree in the normal direction, which guarantees smooth operator action across interfaces and enables the use of boundary operator algebras for elliptic problems.