Katrin FAESSLER

Modulus of curve families – a ‘quasi-invariant’ for QC maps
Definition
The 4-modulus of a curve family Γ on H1 is given by
M4(Γ) = inf ρ
ρ∈adm(Γ)
4 dµ,
adm(Γ) = {ρ : H 1 → [0, ∞] Borel with
For γ = (z, t) horizontal:
γ ρ dℓ = b
a ρ(γ(s))|˙z(s)| ds.
Cf. work by Markina and Vodop’yanov, and by Tang.
Theorem
γ
H 1
ρ dℓ ≥ 1 for γ ∈ Γ loc. rectif.}.
Let f : Ω → Ω ′ be a QC map between domains in H1 and let Γ be a family
of curves in Ω. Then
M4(f (Γ)) ≤ K((z, t), f ) 2 ρ 4 (z, t) dµ(z, t) for all ρ ∈ adm(Γ).
Ω
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 16 / 23