Pseudo-differential operator

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In mathematical analysis, a pseudo-differential operator extends the idea of a differential operator. They are widely used in partial differential equations and quantum field theory, including models on non-Archimedean spaces. The subject began in the 1960s with researchers like Kohn, Nirenberg, Hörmander, Unterberger and Bokobza and played a role in proofs related to the Atiyah–Singer index theorem.

How they work
- For a ordinary differential operator with constant coefficients, you can write it using the Fourier transform: applying the operator is like multiplying the Fourier transform by a polynomial P(ξ) and then taking the inverse transform.
- A pseudo-differential operator generalizes this by using a symbol P(x, ξ) that depends on both position x and frequency ξ. It acts on a function u by mixing its Fourier transform û(ξ) with this symbol.
- One common way to write it is: P(x, D)u(x) = (2π)^{-n} ∫ e^{i x·ξ} P(x, ξ) û(ξ) dξ.

Symbol classes and order
- If P(x, ξ) is smooth in both variables and satisfies bounds on its derivatives, it belongs to a Hörmander symbol class, written S^m_{1,0}. The number m is the order of the operator.
- Roughly, P grows like ⟨ξ⟩^m as |ξ| becomes large, and derivatives of P decay accordingly.

Key properties
- Differential operators with smooth coefficients are included as pseudo-differential operators (they have a polynomial symbol in ξ).
- The product of two pseudo-differential operators is again a pseudo-differential operator, and its symbol can be computed from the symbols of the factors.
- The adjoint (and transpose) of a pseudo-differential operator is also a pseudo-differential operator.
- If a differential operator is elliptic (its symbol is invertible for large ξ), its inverse is a pseudo-differential operator of the opposite order. This helps solve many linear elliptic equations.

Local versus pseudo-local
- Differential operators are local: they depend only on values near a point.
- Pseudo-differential operators are pseudo-local: they don’t create new singularities where the input was smooth, though they can affect behavior in a broader way.

Kernels and microlocal view
- Pseudo-differential operators can be represented by kernels K(x,y). The nature of the singularity of K on the diagonal (where x = y) reflects the operator’s order.
- If the symbol satisfies certain conditions with m ≤ 0, the kernel behaves like a singular integral kernel.
- Much of the analysis of these operators reduces to studying their symbols, which is the focus of microlocal analysis.

In short, pseudo-differential operators generalize differential operators by using symbols that depend on both position and frequency, allowing for powerful techniques to solve equations and understand how waves and signals behave at a very fine, "local in frequency" level.