Manuscript XV-C@1 · XV-C.tex · proposition · 5afd124d

Proposition: Local distributional order

The supported-sector criterion in evaluable form › A finite-order bound for the resulting realization · lines 1088–1121 · proof included or linked

XV-C@1#prop:order

Suppose, on a relatively compact tangent neighborhood, coefficient extensions can be chosen with a common estimate

∣ ⟨ b_{γ}, ψ ⟩ ∣ \leq C_{γ} p_{k} (ψ), ∣ γ ∣ \leq r .

Then the corresponding realization satisfies a local estimate of total derivative order at most $k + r$ .

Proof. For a test $ϕ$ in a fixed product neighborhood, put

ψ_{γ} (x) = \partial_{q}^{γ} ϕ (x, 0) .

Tangential derivatives of $ψ_{γ}$ through order $k$ are bounded by derivatives of $ϕ$ through total order $k + ∣ γ ∣ \leq k + r$ . Thus

∣ ⟨ t, ϕ ⟩ ∣ \leq ∣ γ ∣ \leq r \sum C_{γ} p_{k} (ψ_{γ}) \leq ∣ γ ∣ \leq r \sum C_{γ} p_{k + r} (ϕ),

with the last seminorm taken in the product coordinates.

Displayed formulas in this passage

unlabeled · line 1092
$∣ ⟨ b_{γ}, ψ ⟩ ∣ \leq C_{γ} p_{k} (ψ), ∣ γ ∣ \leq r .$
unlabeled · line 1102
$ψ_{γ} (x) = \partial_{q}^{γ} ϕ (x, 0) .$
unlabeled · line 1109
$∣ ⟨ t, ϕ ⟩ ∣ \leq ∣ γ ∣ \leq r \sum C_{γ} p_{k} (ψ_{γ}) \leq ∣ γ ∣ \leq r \sum C_{γ} p_{k + r} (ϕ),$