Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Wavefront bounds and the separation of hyperbolic and elliptic support

Wavefront bounds and the separation of hyperbolic and elliptic support · lines 1187–1211

XX-B@1#eq:spkernel, XX-B@1#eq:spWF

A retarded Green distribution can have a smooth tail inside the light cone. The wavefront bound does not assert support only on the null cone.

By contrast, the spatial rod kernel is

K_{sp} (τ, x; τ^{'}, y) = - \frac{δ ( τ - τ ^{'} )}{4 π \abs x - y} .

At distinct spatial endpoints,

WF (K_{sp}) = {(τ, x; τ, y; λ, 0; - λ, 0) : λ \neq = 0}, x \neq = y .

Indeed the nonzero smooth spatial factor preserves the wavefront set of the time delta.

Displayed formulas in this passage

XX-B@1#eq:spkernel · line 1194
$K_{sp} (τ, x; τ^{'}, y) = - \frac{δ ( τ - τ ^{'} )}{4 π \abs x - y} .$
XX-B@1#eq:spWF · line 1201
$WF (K_{sp}) = {(τ, x; τ, y; λ, 0; - λ, 0) : λ \neq = 0}, x \neq = y .$