Manuscript IX-D-ST@1 · IX-D-ST.tex · proposition · b1dc631a

Proposition: (TC) for the specified realization

A specified flat realization and the solid-ball correction · lines 1982–2011 · proof included or linked

IX-D-ST@1#prop:flatTC

The actual transpose of (eq:Eflat), with the same pairing convention, satisfies (TC) with $ε_{tr} = 1$ .

Proof. Transpose the compact supported pairing and collect the coefficient of $j (v, z)$ :

(E_{fl}^{*} a) (v, z) = \int_{t \geq v} \int \frac{δ ( t - v - ∣ w - z ∣ )}{4 π ∣ w - z ∣} a (t, w) d^{3} w d t .

At $v = 0$ this is (eq:TC). Equivalently,

(E_{fl}^{*} a) (v, z) = \frac{1}{4 π} \int_{0}^{\infty} r \int_{S^{2}} a (v + r, z + r n) d Ω (n) d r .

On compact parameter sets the effective $r$ -range is bounded; all derivatives pass under the integral. The result is smooth near $Σ$ , and the first-order constant-coefficient operator $C^{*}$ preserves smoothness.

Displayed formulas in this passage

unlabeled · line 1992
$(E_{fl}^{*} a) (v, z) = \int_{t \geq v} \int \frac{δ ( t - v - ∣ w - z ∣ )}{4 π ∣ w - z ∣} a (t, w) d^{3} w d t .$
unlabeled · line 2001
$(E_{fl}^{*} a) (v, z) = \frac{1}{4 π} \int_{0}^{\infty} r \int_{S^{2}} a (v + r, z + r n) d Ω (n) d r .$

Proposition: (TC) for the specified realization

Displayed formulas in this passage

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