Manuscript XII-D@1 · XII-D.tex · section · 308a2080

The actual local source kernel

D3: the completed-kernel cone calculation › The actual local source kernel · lines 854–886

XII-D@1#eq:j-WF, XII-D@1#eq:t-kernel

Let $j_{D} (y; y_{1}, y_{2})$ be the symmetric bilinear kernel defined by

⟨ j_{D} (y; \cdot, \cdot), γ_{1} \otimes γ_{2} ⟩ = J_{D} (γ_{1}, γ_{2}) (y) .

It is a finite sum of differentiated diagonal deltas with smooth coefficients built from $σ_{ε}, σ_{ε}^{'}, σ_{ε}^{''}$ . Consequently

supp j_{D} WF (j_{D}) \subset {y = y_{1} = y_{2}, t_{y} \geq 0}, \subset {(y, q; y_{1}, p_{1}; y_{2}, p_{2}) : q + p_{1} + p_{2} = 0} ∖ 0.

Flatness at zero means that these are ambient smooth coefficients. No step function is inserted.

The polarized completed kernel is

t_{D} (y_{1}, y_{2}, x) = π_{y *} [B_{f, g} (y, x)_{μν} j_{D}^{μν} (y; y_{1}, y_{2})] .

Displayed formulas in this passage

XII-D@1#eq:j-WF · line 866
$supp j_{D} WF (j_{D}) \subset {y = y_{1} = y_{2}, t_{y} \geq 0}, \subset {(y, q; y_{1}, p_{1}; y_{2}, p_{2}) : q + p_{1} + p_{2} = 0} ∖ 0.$
XII-D@1#eq:t-kernel · line 878
$t_{D} (y_{1}, y_{2}, x) = π_{y *} [B_{f, g} (y, x)_{μν} j_{D}^{μν} (y; y_{1}, y_{2})] .$
unlabeled · line 858
$⟨ j_{D} (y; \cdot, \cdot), γ_{1} \otimes γ_{2} ⟩ = J_{D} (γ_{1}, γ_{2}) (y) .$