This should roughly contain the contents of chapter 8.
Part of Lemma 8.0.1.
Part of Lemma 8.0.1: Equation (8.0.1).
Note that the norm ||ϕ||_C^τ
is normalized by definition, i.e., it is R ^ τ
times the best
Hölder constant of ϕ
, so the Lean statement corresponds to the blueprint statement.
We assume that ϕ
is globally Hölder for simplicity, but this is equivalent to being
Hölder on ball z R
as ϕ
is supported there.
Part of Lemma 8.0.1: sup norm control in Equation (8.0.2). Note that it only uses the sup
norm of ϕ
, no need for a Hölder control.
Auxiliary lemma: part of the Lipschitz control in Equation (8.0.2), when the distance between
the points is at most R
.
Part of Lemma 8.0.1: Lipschitz norm control in Equation (8.0.2). Note that it only uses the sup
norm of ϕ
, no need for a Hölder control.
Proposition 2.0.5.