Manuscript IX-Free@1 · IX-Free.tex · lemma · 33b20e96

Lemma: Full support of the mixed kernel

F2: evaluated scalar descent · lines 670–723 · proof included or linked

IX-Free@1#lem:fullsupport, IX-Free@1#eq:boxK

The distribution in (eq:K) satisfies

supp K = M_{+}^{2} .

Proof. Write $a = t - s$ and $z = x - y$ . The Fourier normalization is

Δ_{0}^{+} (a, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2∣ p ∣} e^{i p \cdot z - i ∣ p ∣ a} .

Applying $□_{x}$ to (eq:K) is legitimate distributionally. The two terms without the explicit factor $t$ are homogeneous wave solutions. Moreover,

(- \partial_{t}^{2} - ∣ p ∣^{2}) (t e^{- i ∣ p ∣ (t - s)}) = 2 i ∣ p ∣ e^{- i ∣ p ∣ (t - s)} .

Consequently

□_{x} K (x, y) = - \frac{i}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z - i ∣ p ∣ a} = \partial_{a} Δ_{0}^{+} (a, z) .

For completeness, damping the radial integral by $e^{- δ ∣ p ∣}$ , $δ > 0$ , and performing its angular integration gives

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

On the open set $a \neq = 0$ , $∣ z ∣^{2} \neq = a^{2}$ , its limiting derivative is

\partial_{a} Δ_{0}^{+} = \frac{a}{2 π ^{2} ( ∣ z ∣ ^{2} - a ^{2} ) ^{2}},

which is nonzero. That open set is dense. Its inverse image in $M_{+}^{2}$ is also dense, since the difference map is open.

Thus the right-hand side of (eq:boxK) has full support on $M_{+}^{2}$ . A differential operator cannot enlarge distributional support, so $K$ has full support.

Displayed formulas in this passage

IX-Free@1#eq:boxK · line 695
$□_{x} K (x, y) = - \frac{i}{2} \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot z - i ∣ p ∣ a} = \partial_{a} Δ_{0}^{+} (a, z) .$
unlabeled · line 673
$supp K = M_{+}^{2} .$
unlabeled · line 681
$Δ_{0}^{+} (a, z) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2∣ p ∣} e^{i p \cdot z - i ∣ p ∣ a} .$
unlabeled · line 689
$(- \partial_{t}^{2} - ∣ p ∣^{2}) (t e^{- i ∣ p ∣ (t - s)}) = 2 i ∣ p ∣ e^{- i ∣ p ∣ (t - s)} .$
unlabeled · line 706
$Δ_{0, δ}^{+} (a, z) = \frac{1}{4 π ^{2}} \frac{1}{∣ z ∣ ^{2} - ( a - i δ ) ^{2}} .$
unlabeled · line 713
$\partial_{a} Δ_{0}^{+} = \frac{a}{2 π ^{2} ( ∣ z ∣ ^{2} - a ^{2} ) ^{2}},$

Lemma: Full support of the mixed kernel

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