Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Scale identities and the direct convergence target

Scale identities and the direct convergence target · lines 1243–1266

Em01-C@3#eq:profiles

Define the compact profiles

ϑ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h) = - div (h ρ (h)), ζ (h) = 2^{8} ρ (2 h) - ρ (h) .

Then

δ \partial_{δ} ρ_{δ} = ϑ_{δ}, ρ_{δ /2} - ρ_{δ} = ζ_{δ} .

Both have integral zero. Their moments satisfy

m_{γ} (ϑ) = ∣ γ ∣ m_{γ} (ρ), m_{γ} (ζ) = (2^{- ∣ γ ∣} - 1) m_{γ} (ρ) .

The first sign follows directly by integration by parts.

Displayed formulas in this passage

Em01-C@3#eq:profiles · line 1254
$δ \partial_{δ} ρ_{δ} = ϑ_{δ}, ρ_{δ /2} - ρ_{δ} = ζ_{δ} .$
unlabeled · line 1247
$ϑ (h) = - 8 ρ (h) - h \cdot \nabla ρ (h) = - div (h ρ (h)), ζ (h) = 2^{8} ρ (2 h) - ρ (h) .$
unlabeled · line 1261
$m_{γ} (ϑ) = ∣ γ ∣ m_{γ} (ρ), m_{γ} (ζ) = (2^{- ∣ γ ∣} - 1) m_{γ} (ρ) .$