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Fundamental solution by division of a fourier symbol

Example

Assume Countable Choice and write $D = d / d x$ . The integrable function

$E (x) = \frac{1}{2} e^{- ∣ x ∣}$

defines a tempered fundamental solution for $1 - D^{2}$ on $R$ :

$(1 - D^{2}) E = δ_{0}, F E (ξ) = \frac{1}{1 + 4 π^{2} ξ^{2}} .$

Facts & Assumptions

Given: Countable Choice and the $2 π$ Fourier convention.

[F1]

The distributional transform of an $L^{1}$ function is represented by its integral transform (Fourier transform agrees with l one and plancherel transforms).

[F2]

The symbol identity is $F (P (D) u) = P (2 π i ξ) F u$ (Constant coefficient differential operators become polynomial multipliers).

[F3]

Fourier transformation is injective on $S^{'}$ , and $F δ_{0} = 1$ (Fourier transform is a topological automorphism of tempered distributions, Fourier transform of delta constants plane waves and polynomials).

Verification

technique · explicit integral transform and symbol multiplication

1.1

Since $E \in L^{1} (R)$ , [F1] applies. Split at zero and use the elementary decaying exponential antiderivative.

F1algebra

$\begin{aligned} \hat{E} (ξ) & = \frac{1}{2} (\int_{0}^{\infty} e^{- (1 + 2 π i ξ) x} d x + \int_{0}^{\infty} e^{- (1 - 2 π i ξ) x} d x) \\ = \frac{1}{2} (\frac{1}{1 + 2 π i ξ} + \frac{1}{1 - 2 π i ξ}) = \frac{1}{1 + 4 π^{2} ξ^{2}} . \end{aligned}$

[F1, algebra]

2.1

For $P (z) = 1 - z^{2}$ , apply the Fourier-symbol identity to step 1.1.

F2step 1.1

$F ((1 - D^{2}) E) = (1 - (2 π i ξ)^{2}) \hat{E} (ξ) = 1.$

But $1 = F δ_{0}$ by [F3], so injectivity yields $(1 - D^{2}) E = δ_{0}$ . [F2, F3, step 1.1]

3.1

The denominator is strictly positive on the real frequency axis, so this particular division produces a smooth bounded multiplier. The calculation does not assert that an arbitrary polynomial symbol can be divided in $S^{'}$ , nor any general PDE existence or regularity theorem. Countable Choice is used only through [F1]–[F3].

givenstep 2.1∎

Depends on

Used by

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Sources

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)