Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Cone kernels, including their tips

Retained internal formulas and local existence › Cone kernels, including their tips · lines 2613–2640

Em01-E@2#eq:Rmeasure

Define the locally finite retarded cone measure by

R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x .

The integrand is locally integrable at $x = 0$ . Away from the cone tip this is the familiar notation

R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}) .

Equation (eq:Rmeasure), rather than an unjustified submersion pullback at the tip, defines the global distribution.

Also define the locally integrable function

B (t, x) = \frac{1}{8 π} 1_{{t \geq ∣ x ∣}} .

Its boundary value on the cone is irrelevant to its distribution. Both $R$ and $B$ have local order zero.

Displayed formulas in this passage

Em01-E@2#eq:Rmeasure · line 2616
$R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x .$
unlabeled · line 2626
$R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}) .$
unlabeled · line 2635
$B (t, x) = \frac{1}{8 π} 1_{{t \geq ∣ x ∣}} .$