Manuscript XX-F@1 · XX-F.tex · section · de0bd24f

A weaker explicit infrared domain for the original Newton map

A weaker explicit infrared domain for the original Newton map · lines 869–895

XX-F@1#eq:weighted

Write $⟨ x ⟩ = 1 + ∣ x ∣$ . For $q > 1$ , define decreasing-source seminorms

∥ b ∥_{q, m; J} = j + ∣ α ∣ \leq m max t \in J, x sup ⟨ x ⟩^{q} ∣ \partial_{t}^{j} \partial_{x}^{α} b (t, x)∣, J ⋐ I .

Sources are smooth and possess the finite seminorms needed for each conclusion. These conditions are weaker than requiring every Schwartz seminorm to be finite.

Let

N b = \int P_{i} (x - y) b^{i} (t, y) d^{3} y .

For the rods, $b^{ia} = (\calCz)^{ia}$ .

Displayed formulas in this passage

XX-F@1#eq:weighted · line 874
$∥ b ∥_{q, m; J} = j + ∣ α ∣ \leq m max t \in J, x sup ⟨ x ⟩^{q} ∣ \partial_{t}^{j} \partial_{x}^{α} b (t, x)∣, J ⋐ I .$
unlabeled · line 890
$N b = \int P_{i} (x - y) b^{i} (t, y) d^{3} y .$

Nearby in this section

theorem: Theorem: Weighted native Newton realization
section: A weaker explicit infrared domain for the original Newton map
proposition: Proposition: The absolute-convergence threshold cannot be lowered universally
section: A weaker explicit infrared domain for the original Newton map