Gabriela Kohr

3. Parametric representation on the unit ball
4. Parametric representation on the unit ball
Lemma
Poreda-Szadkowska, 1990: Let f ∈ S∗ (Bn ) and h(z) = [Df (z)] −1f (z).
Then f (z) = lim e
t→∞ t v(z, t) locally uniformly on Bn , where v(z, t) is the
unique solution of the initial value problem
∂v
∂t
= −h(v), ∀ t ≥ 0, v(z, 0) = z,
i.e. f has parametric representation. In addition,
z
≤ f (z) ≤
(1 + z) 2
z
(1 − z) 2 , z ∈ Bn .
• J.A. Pfaltzgraff (1991); R. Barnard, C. FitzGerald and S. Gong
(1991); M. Elin, S. Reich, D. Shoikhet, 2000.
• If f ∈ S then f has parametric representation, i.e. there exists a
Loewner chain f (z, t) such that f = f (·, 0).
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 36 / 62