A linear operator defined by a phase-space symbol a(x, ξ) via an oscillatory integral or quantization rule that generalizes differential operators by allowing nonpolynomial dependence on the frequency variable, encoding microlocal and nonlocal behavior used in PDE and spectral analysis.

Principle

Encode an operator by a symbol on cotangent space and propagate singularities and regularity via symbol estimates (symbol classes S^m and calculus rules) rather than purely local differential coefficients.

Demonstration

On R^n the standard Kohn–Nirenberg quantization sends a symbol a(x, ξ) in S^m to (Op(a)u)(x)=∫ e^{i x·ξ} a(x, ξ) û(ξ) dξ; when a(x, ξ)=|ξ|^2 this recovers the Laplacian, while nonpolynomial a produce fractional Laplacians or smoothing operators depending on decay in ξ.

Misapplication

Treating any operator defined by a kernel or integral as a pseudodifferential operator without verifying symbol regularity and proper phase-space estimates, leading to false conclusions about mapping properties or microlocal behavior.

Consequence

When the symbol satisfies appropriate estimates the operator inherits mapping properties (L^2-boundedness, Sobolev regularity shifts, elliptic parametrices) and one can compose, take adjoints, and compute principal symbols to analyze singularities.

Reversal

A local differential operator is the restricted case with polynomial symbol in ξ; inverting the perspective yields purely local algebraic coefficient calculus rather than microlocal frequency calculus.

Boundary

Applies to operators whose kernels admit phase-space symbol descriptions; excludes general Fourier integral operators without symbol quantization, operators on nonmanifold settings without adapted quantizations, or symbols outside established symbol classes unless explicitly extended.

Semantic Tension

Competes with Fourier integral operators: both use phase-space but FI operators emphasize global phase and propagation, while pseudodifferential operators emphasize local symbol expansions and calculi; conflating them obscures propagation versus microlocal regularity distinctions.

Synthesis

A pseudodifferential operator is the microlocal quantization of a phase-space symbol giving a calculus that extends differential operators to nonpolynomial frequency dependence, enabling precise control of regularity, composition, and singularity propagation in PDE and spectral problems.