Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: Scalar existence and support

Flat Cauchy construction and coefficient equations › Scalar equations, with every switch term · lines 843–866 · proof included or linked

X-A-ST@1#prop:scalarconstruction

Back to manuscript

Equation (eq:scalarsolution) is the unique formal solution of $E_{g} [q] ρ_{g} = 0$ with the free input's Cauchy data and zero positive-order data. Its positive-order sources are smooth and supported in $supp g$ , and

supp ρ_{j, g} \subset J_{η}^{+} (supp g) \cap M_{T_{*}}, j = 1, 2.

Proof. At orders one and two the equations are respectively

P_{m} ρ_{1, g} = - V_{1, g} [h] ϕ_{0}, P_{m} ρ_{2, g} = - V_{1, g} [h] ρ_{1, g} - V_{2, g} [h, k] ϕ_{0} .

Apply Lemma lem:cauchy. Every coefficient of the source operators contains $g$ or a derivative of $g$ .

Displayed formulas in this passage

unlabeled · line 850
$supp ρ_{j, g} \subset J_{η}^{+} (supp g) \cap M_{T_{*}}, j = 1, 2.$
unlabeled · line 858
$P_{m} ρ_{1, g} = - V_{1, g} [h] ϕ_{0}, P_{m} ρ_{2, g} = - V_{1, g} [h] ρ_{1, g} - V_{2, g} [h, k] ϕ_{0} .$

Proposition: Scalar existence and support

Displayed formulas in this passage

Nearby in this section