Manuscript IX-C-ST@1 · IX-C-ST.tex · lemma · 892f50c4

Lemma: Full distributional support of K

Concrete relations and evaluated scalar descent · lines 1482–1514 · proof included or linked

IX-C-ST@1#lem:fullsupport

The restriction of (eq:K) to $V^{2}$ has

supp K = V^{2} .

Proof. For $a = t - s$ , distributional differentiation of (eq:K) gives

□_{x} K = \partial_{a} Δ_{0}^{+} (a, x - y) .

The terms without the explicit factor $t$ solve the homogeneous wave equation. The $t$ -term yields the displayed derivative.

Away from $a = 0$ and $∣ x - y ∣^{2} = a^{2}$ , Abel damping of the radial Fourier integral gives

Δ_{0}^{+} (a, z) = \frac{1}{4 π ^{2}} \frac{1}{∣ z ∣ ^{2} - ( a - i 0 ) ^{2}}, \partial_{a} Δ_{0}^{+} = \frac{a}{2 π ^{2} ( ∣ z ∣ ^{2} - a ^{2} ) ^{2}} \neq = 0.

That set is dense in every open product chart. Thus $□_{x} K$ has full support, and a differential operator cannot enlarge support. Hence $K$ has full support as well.

Displayed formulas in this passage

unlabeled · line 1485
$supp K = V^{2} .$
unlabeled · line 1493
$□_{x} K = \partial_{a} Δ_{0}^{+} (a, x - y) .$
unlabeled · line 1503
$Δ_{0}^{+} (a, z) = \frac{1}{4 π ^{2}} \frac{1}{∣ z ∣ ^{2} - ( a - i 0 ) ^{2}}, \partial_{a} Δ_{0}^{+} = \frac{a}{2 π ^{2} ( ∣ z ∣ ^{2} - a ^{2} ) ^{2}} \neq = 0.$

Lemma: Full distributional support of K

Displayed formulas in this passage

Nearby in this section