Gabriela Kohr
Gabriela Kohr
Gabriela Kohr
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Geometric and analytic aspects of Loewner chains<br />
in C n and complex Banach spaces<br />
<strong>Gabriela</strong> <strong>Kohr</strong><br />
Faculty of Mathematics and Computer Science<br />
Babes¸-Bolyai University<br />
Cluj-Napoca, Romania<br />
Based on joint works with Ian Graham (Toronto), Hidetaka Hamada (Fukuoka),<br />
Mirela <strong>Kohr</strong> (Cluj-Napoca)<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 1 / 62
Non-normalized univalent subordination chains and the<br />
generalized Loewner differential equation on the unit ball in<br />
C n<br />
Solutions for the generalized Loewner differential equation in<br />
C n associated with non-normalized univalent subordination<br />
chains given by a time-dependent linear operator<br />
Univalent subordination chains and the generalized Loewner<br />
differential equation in reflexive complex Banach spaces<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 2 / 62
1. Main contributions<br />
1. Contributions in SCV<br />
• J.A. Pfaltzgraff (1974-1975):<br />
-The first generalizations to higher dimensions of various results in the<br />
theory of Loewner chains.<br />
-Generalizations to higher dimensions of the one variable univalence<br />
and quasiconformal extension results due to J. Becker<br />
1. J.A. Pfaltzgraff, Subordination chains and univalence of holomorphic<br />
mappings in C n , Math. Ann., 210 (1974), 55-68.<br />
2. J.A. Pfaltzgraff, Subordination chains and quasiconformal extension<br />
of holomorphic maps in C n , Ann. Acad. Scie. Fenn. Ser. A I Math.,<br />
1(1975), 13-25.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 3 / 62
1. Main contributions<br />
• T. Poreda (1990):<br />
-The Loewner differential equation on the unit polydisc in C n<br />
-Applications of parametric representation to growth theorems and<br />
coefficient estimates on the unit polydisc<br />
-Extensions to complex Banach spaces<br />
1. T. Poreda, On generalized differential equations in Banach spaces,<br />
Dissertationes Mathematicae, 310 (1991), 1-50.<br />
• M. Chuaqui (1995):<br />
-Growth results for strongly starlike maps on the unit ball in C n by using<br />
the theory of Loewner chains<br />
-Explicit quasiconformal extensions to C n of quasiconformal strongly<br />
starlike mappings on B n<br />
1. M. Chuaqui, Applications of subordination chains to starlike<br />
mappings in C n , Pacif. J. Math., 168(1995), 33-48.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 4 / 62
1. Main contributions<br />
• I. Graham, H. Hamada, G. K., M. <strong>Kohr</strong>, J.A. Pfaltzgraff, T.J.<br />
Suffridge, and P. Duren:<br />
-The Loewner differential equation on the unit ball in C n ;<br />
-Distortion results for mappings with parametric representation on B n<br />
-Quasiconformal extension results and Loewner chains<br />
-Geometric aspects of Loewner chains: asymptotic starlikeness and<br />
asymptotic spirallikeness; Extension operators and Loewner chains<br />
-Extreme points and support points associated to families of<br />
biholomorphic mappings with parametric representation<br />
1. I. Graham and G. <strong>Kohr</strong>, Geometric Function Theory in One and<br />
Higher Dimensions, Marcel Dekker, New York, 2003.<br />
2. I. Graham, H. Hamada, G. <strong>Kohr</strong> and M. <strong>Kohr</strong>, Parametric<br />
representation and asymptotic starlikeness in C n , Proc. AMS, 136<br />
(2008), 3963–3973.<br />
3. P. Duren, I. Graham, H. Hamada and G. <strong>Kohr</strong>, Solutions for the<br />
generalized Loewner differential equation in several complex variables,<br />
Math. Ann. 347 (2010) 411–435.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 5 / 62
1. Main contributions<br />
• M. Elin, S. Reich and D. Shoikhet (2000):<br />
-Parametric representation for spirallike mappings in C n and complex<br />
Banach spaces, by using the nonlinear semigroups theory<br />
1. M. Elin, S. Reich, D. Shoikhet, Complex dynamical systems and the<br />
geometry of domains in Banach spaces, Dissertationes Math.,<br />
427(2004), 1-62.<br />
2. S. Reich and D. Shoikhet, Nonlinear Semigroups, Fixed Points, and<br />
Geometry of Domains in Banach Spaces, Imperial College Press,<br />
London, 2005.<br />
• M. Elin; I. Graham, H. Hamada, and G.K.; J.R. Muir:<br />
1. M. Elin, Extension operators via semigroups, J. Math. Anal. Appl.,<br />
377(2011), 239-250.<br />
2. I. Graham, H. Hamada and G.K., Extension operators and univalent<br />
subordination chains, J. Math. Anal. Appl., 386(2012), 278–289.<br />
3. J.R. Muir, A class of Loewner chain preserving extension operators,<br />
J. Math. Anal. Appl., 337(2008), 862-979.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 6 / 62
1. Main contributions<br />
• F. Bracci, M.D. Contreras, S.Diaz-Madrigal (2008):<br />
-The study of a general version of the Loewner differential equation on<br />
the unit disc and complex hyperbolic manifolds, by using the iteration<br />
and the semigroups theory<br />
1. F. Bracci, M.D. Contreras and S.Diaz-Madrigal, Evolution families<br />
and the Loewner equation I: the unit disk, J. Reine Angew. Math., to<br />
appear.<br />
2. F. Bracci, M.D. Contreras and S.Diaz-Madrigal, Evolution families<br />
and the Loewner equation II: complex hyperbolic manifolds, Math.<br />
Ann., 344(2009), 947-962.<br />
• M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk (2009): a<br />
general version of the notion of Loewner chains and the<br />
correspondence with the evolution families.<br />
1. M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk, Loewner<br />
chains in the unit disk, Rev. Mat. Iberoamer., 26 (2010), 975–1012<br />
2. M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk, Geometry<br />
behind chordal Loewner chains, Complex Analysis and Operator<br />
Theory, 4 (2010), 541–587.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 7 / 62
1. Main contributions<br />
• L. Arosio, F. Bracci, H. Hamada and G.K. (2010):<br />
-Abstract construction of Loewner chains on the unit ball and<br />
hyperbolic manifolds: one-to one correspondence between<br />
L d -Loewner chains and L d -evolution families.<br />
1. L. Arosio, F. Bracci, H. Hamada and G.<strong>Kohr</strong>, An abstract approach<br />
of Loewner’s chains, J. Anal. Math., to appear.<br />
2. H. Hamada, G. K., and J. Muir, Extensions of L d -Loewner chains to<br />
higher dimensions, submitted.<br />
• L. Arosio; M. Vodă (2010): resonances to construct solutions to the<br />
generalized Loewner differential equation on the unit ball in C n .<br />
1. L. Arosio, Resonances in Loewner equations, Advances Math.,<br />
227(2011), 1413-1435.<br />
2. M. Vodă, Solution of a Loewner chain equation in several complex<br />
variables, J. Math. Anal. Appl., 375 (2011), 58-74.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 8 / 62
2. Univalent subordination chains in several complex variables<br />
3. Univalent subordination chains<br />
Subordination chains in C n<br />
Definition<br />
(a) If f , g ∈ H(B n ), we say that f is subordinate to g (written f ≺ g) if<br />
there is a Schwarz mapping v such that f = g ◦ v.<br />
• If g is biholomorphic, then f ≺ g iff f (0) = g(0) and f (B n ) ⊆ g(B n ).<br />
(b) f : B n × [0, ∞) → C n is an univalent subordination chain if:<br />
(i) f (·, t) is biholomorphic on B n , f (0, t) = 0, t ≥ 0;<br />
(ii) f (·, s) ≺ f (·, t) whenever 0 ≤ s ≤ t < ∞.<br />
• f (z, t) = e t<br />
0 A(τ)dτ z + · · · , where A : [0, ∞) → L(C n , C n ) is locally<br />
Lebesgue integrable, m(A(t)) > 0, t ≥ 0, and ∞<br />
0 m(A(τ))dτ = ∞.<br />
Here m(A(t)) = minz=1 Re 〈A(t)(z), z〉.<br />
• f (z, t) = etAz + · · · –A-normalized univalent subordination chain.<br />
• f (z, t) = etz + · · · –usual Loewner chain.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 9 / 62
2. Univalent subordination chains in several complex variables<br />
Remark<br />
The first part of the talk refers to univalent subordination chains<br />
with the normalization Df (0, t) = e t<br />
0 A(τ)dτ for t ≥ 0.<br />
f (·, s) ≺ f (·, t) is equivalent to the existence of a Schwarz mapping<br />
v = v(z, s, t)-transition mapping associated with f (z, t), such that<br />
f (z, s) = f (v(z, s, t), t), z ∈ B n , 0 ≤ s ≤ t < ∞.<br />
(vs,t)-evolution family associated to f (z, t).<br />
Remark<br />
Let f (z, t) be a univalent subordination chain and let v(z, s, t) be the<br />
transition mapping associated with f (z, t). Then<br />
(i) v(·, s, t) is biholomorphic on B n and v(z, s, s) = z, z ∈ B n , s ≥ 0.<br />
(ii) v(z, s, ·) is decreasing on [s, ∞), for all z ∈ B n and s ≥ 0.<br />
(iii) Semigroup property:<br />
v(z, s, τ) = v(v(z, s, t), t, τ), z ∈ B n , 0 ≤ s ≤ t ≤ τ < ∞.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 10 / 62
2. Univalent subordination chains in several complex variables<br />
S(B n ) = {h ∈ H(B n ) : h biholomorphic , h(0) = 0, Dh(0) = In}; S(B 1 ) = S<br />
• Every function f ∈ S can be embedded in a Loewner chain.<br />
• In C n , n ≥ 2, there exist mappings f ∈ S(B n ) that cannot be<br />
embedded in Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is a<br />
normal family on B n .<br />
• Every function f ∈ S has parametric representation, i.e.<br />
f (z) = limt→∞ e t v(z, t), where v = v(z, t) is the unique Lipschitz<br />
continuous solution on [0, ∞) of the initial value problem<br />
∂v<br />
= −vp(v, t) a.e. t ≥ 0, v(z, 0) = z,<br />
∂t<br />
for some choice of p = p(z, t) such that p(·, t) ∈ P for almost all<br />
t ∈ [0, ∞) and p(z, ·) is measurable on [0, ∞) for z ∈ U.<br />
P = {p ∈ H(U) : p(0) = 1, Re p(z) > 0, z ∈ U}.<br />
• S 0 (B 1 ) = S; S 0 (B n ) S(B n ) for n ≥ 2, where S 0 (B n ) is the family of<br />
mappings which have parametric representation.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 11 / 62
2. Univalent subordination chains in several complex variables<br />
• S 0 (B n ) is compact, but S(B n ) is not compact for n ≥ 2.<br />
• There exist mappings f ∈ S 0 (B n ) such that 1<br />
2 D2 f (0) > 2.<br />
• In dimension n ≥ 2, there exist Loewner chains f (z, t) such that<br />
{e −t f (·, t)}t≥0 is not a normal family.<br />
• Let p : U × [0, ∞) → C be a function such that p(·, t) ∈ P, t ≥ 0, and<br />
p(z, ·) is measurable on [0, ∞), z ∈ U. Then there exists a unique<br />
Loewner chain f (z, t) which satisfies the Loewner differential equation<br />
∂f<br />
∂t (z, t) = zf ′ (z, t)p(z, t), a.e. t ≥ 0, ∀z ∈ U.<br />
• In dimension n ≥ 2, the Loewner differential equation<br />
∂f<br />
(z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ U,<br />
∂t<br />
does not have a unique normalized univalent solution f (z, t).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 12 / 62
2. Univalent subordination chains in several complex variables<br />
• Many classical results in the theory of holomorphic mappings in C n<br />
do not hold in infinite dimensional complex Banach spaces.<br />
• Montel’s theorem does not hold in the infinite dimensional setting<br />
(but surprisingly, Vitali’s theorem does).<br />
• On a domain in Cn , any univalent (holomorphic and injective)<br />
mapping into Cn is also biholomorphic. However, this result is no<br />
longer true in infinite dimensional complex Banach spaces. For<br />
example, if f : ℓ2 → ℓ2 is given by f (x) = (x 2 1 , x 3 1 , x 2 2 , x 3 2 , . . .), then f is<br />
univalent on the unit ball of ℓ2, but is not biholomorphic.<br />
• On a domain in C n any univalent mapping is open. Heath and<br />
Suffridge (1980): an example of a univalent mapping on the unit ball B<br />
of a complex Banach space which is not biholomorphic, f (B) contains<br />
an open set, but f (B) is not open.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 13 / 62
2. Univalent subordination chains in several complex variables<br />
• There exist biholomorphic mappings on the unit ball B of an infinite<br />
dimensional complex Banach space X which are not bounded on the<br />
ball Br for r ∈ (0, 1).<br />
• Graham, Hamada, K (2012): Let f (x) = (0, f2(x1), f3(x2), . . .) for<br />
x = (x1, x2, . . .) ∈ ℓ2, where fn+1(xn) = 1<br />
n(n+1) (2xn) n+1 , n ≥ 1. Also, let<br />
F(x) = x + f (x). Then F is biholomorphic on ℓ2 and is not bounded on<br />
Br for r ∈ (2/3, 1).<br />
• L. Harris, S. Reich, D. Shoikhet (2000): Let X be a complex Banach<br />
space and let h : B → X be a holomorphic mapping. If L(h) is finite,<br />
then hs is bounded on B for each s ∈ (0, 1), where hs(z) = h(sz) and<br />
L(h) = lim<br />
s→1 sup Re V (hs).<br />
• It is natural to consider to what extent such phenomena require<br />
changes in the development of Loewner theory in complex Banach<br />
spaces.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 14 / 62
2. Univalent subordination chains in several complex variables<br />
Example<br />
A mapping f ∈ H(B n ) is normalized if f (0) = 0 and Df (0) = In.<br />
LSn-the family of normalized locally biholomorphic maps on B n<br />
• If f ∈ LSn, then the following conditions are equivalent:<br />
(i) f ∈ S ∗ (B n ) (i.e. f is biholomorphic and e −t f (B n ) ⊆ f (B n ) for t ≥ 0);<br />
(ii) f (z, t) = e t f (z) is a Loewner chain.<br />
Example<br />
• Let A ∈ L(C n , C n ) be such that Re 〈A(z), z〉 > 0, z = 0. If f ∈ LSn,<br />
then the following conditions are equivalent:<br />
(i) f is spirallike with respect to A (i.e. f is biholomorphic on B n and<br />
e −tA f (B n ) ⊆ f (B n ) for t ≥ 0);<br />
(ii) f (z, t) = e tA f (z) is an A-univalent subordination chain;<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 15 / 62
2. Univalent subordination chains in several complex variables<br />
I. Graham, H. Hamada, G.K, M.K, 2008:<br />
Example<br />
Let A : [0, ∞) → L(Cn , Cn ) be a measurable mapping such that<br />
m(A(t)) > 0 for t ≥ 0 and ∞<br />
0 m(A(t))dt = ∞. Moreover, assume that<br />
A(·) is uniformly bounded on [0, ∞). If f ∈ LSn, then the following<br />
conditions are equivalent:<br />
(i) f is (generalized) spirallike with respect to A (i.e. f is biholomorphic<br />
on B n and exp(− t<br />
0 A(τ)dτ)f (Bn ) ⊆ f (B n ) for t ≥ 0);<br />
(ii) f (z, t) = e t<br />
0 A(τ)dτ f (z) is a univalent subordination chain;<br />
(iii) ℜ〈[Df (z)] −1 A(t)f (z), z〉 > 0, a.e. t ≥ 0, ∀z ∈ B n \ {0}.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 16 / 62
2. Univalent subordination chains in several complex variables<br />
• f (z, t) univalent subordination chain with Df (0, t) = U(t), t ≥ 0,<br />
where U(t) is the unique locally absolutely continuous solution of<br />
dU/dt = A(t)U a.e. t ≥ 0, U(0) = In.<br />
I. Graham, H. Hamada, G. K, M. K (2008-); L. Arosio; M. Vodă (2011).<br />
L. Arosio, F. Bracci, H.Hamada, G.K (2010): L d -Loewner chains.<br />
Definition<br />
Let X be a complex Banach space. For z ∈ X \ {0}, we define<br />
T (z) = {ℓz ∈ L(X, C) : ℓz(z) = z, ℓz = 1}.<br />
Then T (z) = ∅. If A ∈ L(X), let<br />
m(A) = inf{Re [ℓz(A(z))] : z = 1, ℓz ∈ T (z)}<br />
k(A) = sup{Re [ℓz(A(z))] : z = 1, ℓz ∈ T (z)}<br />
|V (A)| = sup{|ℓz(A(z))| : z = 1, ℓz ∈ T (z)}.<br />
V (A)-the numerical radius of the operator A.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 17 / 62
2. Univalent subordination chains in several complex variables<br />
Definition<br />
Let k+(A) = max{Re λ : λ ∈ σ(A)} be the upper exponential index<br />
(Lyapunov) index of A, where σ(A) is the spectrum of A ∈ L(X).<br />
Let k−(A) = min{Re λ : λ ∈ σ(A)} be the lower exponential index<br />
(Lyapunov) index of A.<br />
Remark<br />
Let A ∈ L(X) be such that m(A) > 0. Then<br />
(i) k−(A) ≤ m(A) ≤ k+(A) ≤ k(A) ≤ |V (A)| ≤ A ≤ e|V (A)|;<br />
;<br />
(ii) For each ω > k+(A), there is δ = δ(ω) > 0 s.t. etA ≤ δeωt , t ≥ 0;<br />
(iii)<br />
e m(A)t ≤ e tA (u) ≤ e k(A)t , t ∈ [0, ∞), u = 1.<br />
k+(A) = limt→∞ ln etA t<br />
(iv) L. Harris, 1971: If Pm : X → X is a homogeneous polynomial<br />
mapping of degree m, then Pm ≤ km|V (Pm)|, where km = m m/(m−1)<br />
when m ≥ 2 and k1 = e (k1 = 2 for complex Hilbert spaces).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 18 / 62
2. Univalent subordination chains in several complex variables<br />
The Carathéodory family N<br />
Definition<br />
N = {h ∈ H(B) : h(0) = 0, Re [ℓz(h(z))] > 0, z ∈ B \ {0}, ℓz ∈ T (z)},<br />
N A = {h ∈ N : Dh(0) = A} and M = N I = {h ∈ N : Dh(0) = I}.<br />
• If n = 1, then f ∈ M iff<br />
f (z)/z ∈ P = {h ∈ H(U) : h(0) = 1, Re h(z) > 0, z ∈ U}.<br />
Remark<br />
L. Harris, S. Reich, D. Shoikhet, 2000: Bloch radii; distortion form of<br />
Schwarz’s lemma; T. Poreda, 2001; M. Elin, S. Reich, D. Shoikhet,<br />
2004: geometric aspects of biholomorphic mappings in complex<br />
Banach spaces; I. Graham, H. Hamada, G. K (2002): compactness of<br />
M when X = C n ; radius problems for compact subsets of S(B n );<br />
applications to Loewner’s theory in C n (2008-).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 19 / 62
2. Univalent subordination chains in several complex variables<br />
J.A. Pfaltzgraff (1974; X = C n ); K. Gurganus (1975):<br />
Lemma<br />
If h ∈ N A, where A ∈ L(X) with m(A) > 0, then<br />
1 − z<br />
Re [ℓz(A(z))]<br />
1 + z ≤ Re [ℓz(h(z))]<br />
1 + z<br />
≤ Re [ℓz(A(z))]<br />
1 − z ,<br />
for all z ∈ B \ {0} and ℓz ∈ T (z).<br />
• H. Hamada, G. K (2002): Any h ∈ N is bounded on Br , r ∈ (0, 1).<br />
I. Graham, H. Hamada, G. K, 2002 (X = C n and Df (0) = In); (2008):<br />
Theorem<br />
Let X be a complex Banach space and A ∈ L(X) be s.t. m(A) > 0.<br />
If f (z) = Az + ∞<br />
m=2 Pm(z) ∈ N A, then<br />
(i) |V (Pm)| ≤ 2|V (A)| and Pm ≤ 4m|V (A)| for m ≥ 2.<br />
(ii) m(A)r(1 − r)/(1 + r) ≤ f (z) ≤ 4|V (A)|r/(1 − r) 2 for z ≤ r < 1.<br />
(iii) N A is a compact family when X = C n .<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 20 / 62
2. Univalent subordination chains in several complex variables<br />
I. Graham, H. Hamada, G.K, M.K. (2008-2011), L. Arosio (2011); M.<br />
Voda (2011): non-normalized univalent subordination chains with<br />
normalization given by a time-dependent linear operator.<br />
Assumption<br />
Let A : [0, ∞) → L(Cn , Cn ) be a measurable mapping such that<br />
(a) m(A(t)) > 0 for t ≥ 0, and ∞<br />
0 m(A(t))dt = ∞;<br />
(b) A(·) is uniformly bounded on [0, ∞);<br />
(c) t<br />
s A(τ)dτ ◦ s<br />
r A(τ)dτ = s<br />
r A(τ)dτ ◦ t<br />
s A(τ)dτ, t ≥ s ≥ r ≥ 0.<br />
(d) There exists δ > 0 such that k+(A(t)) ≤ 2m(A(t)) − δ for t ≥ 0;<br />
Remark<br />
(i) (c) holds if A(t) ≡ A ∈ L(C n , C n ) or if A(t) is diagonal, t ≥ 0.<br />
(ii) A-univalent subordination chains (I. Graham, H. Hamada, G. K,<br />
and M. K, 2008).<br />
(iii) Solutions to the Loewner differential equation generated by<br />
A-univalent subordination chains (P. Duren, I. Graham, H. Hamada,<br />
and G.K, 2010).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 21 / 62
2. Univalent subordination chains in several complex variables<br />
J. Pfaltzgraff (1974); T. Poreda (1990); M. Elin, S. Reich, D. Shoikhet<br />
(2000); I. Graham, H. Hamada, G.K, M.K. (2008); L. Arosio; M. Voda<br />
(2011).<br />
Theorem<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(c) hold. Also let<br />
h : B n × [0, ∞) → C n satisfy the following conditions:<br />
(i) h(·, t) ∈ N and Dh(0, t) = A(t) for t ≥ 0;<br />
(ii) h(z, ·) is measurable on [0, ∞) for each z ∈ B n .<br />
Then for each z ∈ B n and s ≥ 0, the initial value problem<br />
(2.1)<br />
∂v<br />
∂t<br />
= −h(v, t), a.e. t ≥ s, v(z, s, s) = z,<br />
has a unique solution v = v(z, s, t) such that v(·, s, t) is a univalent<br />
Schwarz mapping, v(z, s, ·) is Lipschitz continuous on [s, ∞) locally<br />
uniformly with respect to z ∈ B n and Dv(0, s, t) = exp(− t<br />
s A(τ)dτ).<br />
h(z, t)-generating vector field (Herglotz vector field).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 22 / 62
2. Univalent subordination chains in several complex variables<br />
Theorem<br />
T. Poreda, 1990; Graham, Hamada and K., 2001 (A = In); M. Elin, S.<br />
Reich, D. Shoikhet, 2000; I. Graham, H. Hamada, G.K, M.K., 2008:<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Also let<br />
vs,t(z) = v(z, s, t) be the solution of (2.1). Then the limit<br />
t<br />
(2.2) lim e 0<br />
t→∞ A(τ)dτ vs,t(z) = f (z, s)<br />
exists locally uniformly on B n for s ≥ 0. Moreover, f (z, t) is an univalent<br />
subordination chain and {e − t<br />
0 A(τ)dτ f (·, t)}t≥0 is a normal family on B n<br />
and <br />
t≥0 ft(B n ) = C n . In addition, f (z, ·) is locally Lipschitz continuous<br />
on [0, ∞) locally uniformly with respect to z ∈ B n , and<br />
(2.3)<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀ z ∈ Bn .<br />
The univalent subordination chain f (z, t) given by (2.2) is called<br />
the canonical solution of the Loewner differential equation (2.3).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 23 / 62
2. Univalent subordination chains in several complex variables<br />
J. Pfaltzgraf (1975, A = In); I. Graham, H. Hamada, G.K., M.K., 2008.<br />
Theorem<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let<br />
f = f (z, t) : B n × [0, ∞) → C n be such that f (·, t) ∈ H(B n ), f (0, t) = 0<br />
and Df (0, t) = e t<br />
0 A(τ)dτ for t ≥ 0, and f (z, ·) is locally Lipschitz<br />
continuous on [0, ∞) locally uniformly with respect to z ∈ B n . Assume<br />
that there exists a generating vector field h(z, t) = A(t)z + · · · such that<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀ z ∈ Bn .<br />
Then f (z, t) is a subordination chain. If {e − t<br />
0 A(τ)dτ f (·, t)}t≥0 is a<br />
normal family on B n , then f (z, t) is a univalent subordination chain<br />
which coincides with the canonical solution of (2.3).<br />
g(z, t)-standard solution of (2.3) if g(·, t) ∈ H(B n ), g(0, t) = 0 for<br />
t ≥ 0, g(z, ·) is locally Lipschitz continuous on [0, ∞) locally<br />
uniformly with respect to z ∈ B n , and g(z, t) is a solution of (2.3).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 24 / 62
2. Univalent subordination chains in several complex variables<br />
• The above result does not hold if k+(A) = 2m(A).<br />
Example<br />
Let f : B 2 → C 2 be given by f (z) = (z1, z2 + az 2 1 ) for z = (z1, z2) ∈ B 2 ,<br />
where a ∈ C \ {0}. Also let A ∈ L(C 2 , C 2 ) be given by A(z) = (z1, 2z2)<br />
and f (z, t) = e At f (z). Then k+(A) = 2m(A) and f is spirallike w.r.t. A.<br />
Also f (z, t) is a univalent subordination chain such that {e −At f (·, t)}t≥0<br />
is a normal family on B 2 and<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z, t), t ≥ 0, z ∈ B2 ,<br />
where h(z, t) = (z1, 2z2). Let v(z, s, t) = (e s−t z1, e 2(s−t) z2). Then<br />
v(z, s, t) is the unique solution of the initial value problem<br />
∂v<br />
∂t<br />
= −h(v, t), t ≥ s, v(z, s, s) = z.<br />
But e At v(z, s, t) = (e s z1, e 2s z2) f (z, s) = (e s z1, e 2s (z2 + az 2 1 )).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 25 / 62
2. Univalent subordination chains in several complex variables<br />
Theorem<br />
Let A : [0, ∞) → L(Cn , Cn ) be such that (a)-(d) hold. Let f (z, t) be an<br />
univalent subordination chain such that Df (0, t) = exp( t<br />
0 A(τ)dτ).<br />
Then f (z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly<br />
with respect to z ∈ Bn . Also there is a generating vector field<br />
h = h(z, t) such that Dh(0, t) = A(t) and<br />
Theorem<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀ z ∈ Bn .<br />
I. Graham, H. Hamada, G. K., M. K., 2008: Assume the conditions<br />
(a)-(d) hold. Let f (z, t) = e t<br />
0 A(τ)dτ z + · · · be an univalent<br />
subordination chain. If {e− t<br />
0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn ,<br />
then f (z, s) = limt→∞ e t<br />
0 A(τ)dτ v(z, s, t) locally uniformly on Bn for<br />
s ≥ 0, where v = v(z, s, t) is the transition mapping of f (z, t), and<br />
f (z, t) coincides with the canonical solution of (2.3).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 26 / 62
2. Univalent subordination chains in several complex variables<br />
• T. Poreda, 1990 (starlikeness); M. Elin, S. Reich, D. Shoikhet, 2000<br />
(spirallikeness w.r.t. a linear operator).<br />
Corollary<br />
Assume k+(A) < 2m(A). If f (z, t) is an A-univalent subordination chain<br />
such that {e −tA f (·, t)}t≥0 is a normal family, then<br />
z<br />
(1 + z) 2 ≤ e−tA f (z, t) ≤<br />
z<br />
(1 − z) 2 , z ∈ Bn , t ≥ 0.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 27 / 62
2. Univalent subordination chains in several complex variables<br />
Corollary<br />
I. Graham, G.K, 2003:<br />
(i) If f (z, t) = e t z + · · · is a Loewner chain such that {e −t f (·, t)}t≥0 is a<br />
normal family on B n , then<br />
etz (1 + z) 2 ≤ f (z, t) ≤ etz (1 − z) 2 , z ∈ Bn , t ≥ 0.<br />
(ii) If, in addition, the generating vector field h(z, t) of (2.3) satisfies the<br />
condition <br />
1<br />
<br />
〈h(z, t), z〉 − 1<br />
z2 < 1, z ∈ Bn \ {0},<br />
then<br />
etz 1 + z ≤ f (z, t) ≤ etz 1 − z , z ∈ Bn , t ≥ 0.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 28 / 62
2. Univalent subordination chains in several complex variables<br />
Remark<br />
(a) (i) yields the growth result for S ∗ (B n ): R. Barnard, C. FitzGerald<br />
and S. Gong (1991); J.A. Pfaltzgraff (1991); M. Chuaqui (1995), T. Liu<br />
(1999).<br />
(b) (ii) yields the growth result for K (B n ) may be obtained from (ii): C.<br />
FitzGerald, C. Thomas, S. Gong (1995); P. Liczberski (1998), T. Liu<br />
(1999); H. Hamada, G.K. (2000).<br />
(c) There exist Loewner chains f (z, t) that do not satisfy the growth<br />
result (i) and {e −t f (·, t)}t≥0 is not a normal family on B n , for n ≥ 2.<br />
Example<br />
Let g(z, t) =<br />
<br />
et z1<br />
(1−z1) 2 e<br />
,<br />
t z2<br />
(1−z2) 2<br />
<br />
for z = (z1, z2) ∈ B2 , t ≥ 0. Then g(z, t)<br />
is a Loewner chain and if Φ(z) = (z1, z2 + z2 1 ) then Φ ∈ Aut(C2 ) and<br />
<br />
f (z, t) = Φ(g(z, t)) given by f (z, t) =<br />
is a<br />
<br />
et z1<br />
(1−z1) 2 e<br />
,<br />
t z2<br />
(1−z2) 2 + e2t z2 1<br />
(1−z1) 4<br />
Loewner chain such that f (r, 0) > r/(1 − r) 2 for r ∈ (0, 1) and<br />
{e−tf (·, t)}t≥0 is not a normal family on B2 .<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 29 / 62
2. Univalent subordination chains in several complex variables<br />
Solutions for the generalized Loewner differential equation (2.3):<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ Bn ,<br />
where h(z, t) is a generating vector field such that Dh(0, t) = A(t)<br />
for t ≥ 0.<br />
n = 1: Ch. Pommerenke, 1965; J. Becker (1973-1980);<br />
n ≥ 2: I. Graham, G.K., J.A. Pfaltzgraff, 2005 (A = In); P. Duren, I.<br />
Graham, H. Hamada and G.K, 2010 (k+(A) < 2m(A)); L. Arosio<br />
(2010); M. Voda (2010); H. Hamada (2010): m(A) > 0; I. Graham,<br />
H. Hamada, G.K. (2010): Herglotz-vector fields with respect to a<br />
time-dependent linear operator.<br />
J. Becker: useful to construct the solutions of the Loewner<br />
differential equation which, for fixed t, are holomorphic on a<br />
punctured disc (rather than a disc) with center at zero.<br />
In several complex variables, point singularities of holomorphic<br />
mappings are removable.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 30 / 62
2. Univalent subordination chains in several complex variables<br />
(i) If f (z, t) is a Loewner chain and Φ : C n → C n is a normalized<br />
biholomorphic mapping (i.e. Φ(0) = 0 and DΦ(0) = In), not the<br />
identity, then Φ(f (z, t)) is also a Loewner chain and satisfies the<br />
same Loewner differential equation as f (z, t).<br />
In C n , n ≥ 2, normalized univalent solutions of the generalized<br />
Loewner differential equation need not be unique.<br />
Theorem<br />
I. Graham, G.K., J.A. Pfaltzgraff, 2005; P. Duren, I. Graham, H.<br />
Hamada and G.K, 2010 (A(t) ≡ A); Graham, Hamada, K (2010):<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let<br />
f (z, t) = e t<br />
0 A(τ)dτ z + · · · be the canonical solution of (2.3) and let<br />
g(z, t) be a standard solution of (2.3). Then <br />
t≥0 f (Bn , t) = C n and<br />
there exists a mapping Φ ∈ H(C n ) such that<br />
g(z, t) = Φ(f (z, t)), z ∈ B n , t ≥ 0.<br />
In addition, g(·, t) is univalent on B n iff Φ is biholomorphic in C n .<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 31 / 62
2. Univalent subordination chains in several complex variables<br />
Theorem<br />
P. Duren, I. Graham, H. Hamada and G.K, 2010 (A(t) ≡ A); Graham,<br />
Hamada, K (2010): Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d)<br />
hold. Let f (z, t) = e t<br />
0 A(τ)dτ z + · · · be the canonical solution of (2.3)<br />
and let g(z, t) be a standard solution of (2.3). Assume that<br />
{e − t<br />
0 A(τ)dτ g(·, t)}t≥0 is a normal family on B n . Then there exists<br />
Ψ ∈ L(C n , C n ) such that g(z, t) = Ψ(f (z, t)) for z ∈ B n and t ≥ 0.<br />
Corollary<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let g(z, t) be a<br />
standard solution of the Loewner differential equation (2.3) such that<br />
{e − t<br />
0 A(τ)dτ g(·, t)}t≥0 is a normal family on B n and Dg(0, 0) = In. Then<br />
g(z, s) ≡ f (z, s), where f (z, s) is the canonical solution of (2.3).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 32 / 62
2. Univalent subordination chains in several complex variables<br />
Corollary<br />
Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. The canonical<br />
solution f (z, t) = e t<br />
0 A(τ)dτ z + · · · of the Loewner differential equation<br />
(2.3) is the unique normalized univalent subordination chain solution<br />
such that {e− t<br />
0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn <br />
. In addition,<br />
t≥0 f (Bn , t) = Cn .<br />
Corollary<br />
I. Graham, G.K., J.A. Pfaltzgraff, 2005: Any Loewner chain<br />
g(z, t) = e t z + · · · has the form g(z, t) = Φ(f (z, t)), z ∈ B n , t ≥ 0,<br />
where f (z, t) is a Loewner chain of the distinguished class and<br />
Φ : C n → C n is a normalized biholomorphic mapping (i.e. an<br />
automorphism of C n or a normalized Fatou-Bieberbach map).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 33 / 62
2. Univalent subordination chains in several complex variables<br />
Example<br />
Let n = 2 and let f : B2 → C2 be given by f (z) = (z1, z2 + az2 1 ) for<br />
z = (z1, z2) ∈ B2 , where a ∈ C \ {0}. Also let A ∈ L(C2 , C2 ) be given<br />
by A(z) = (z1, 2z2) and f (z, t) = etAf (z). Then m(A) = 1 and<br />
k+(A) = 2. Also f is spirallike with respect to A and f (z, t) = etAf (z) is<br />
a univalent subordination chain such that<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z), t ≥ 0, z ∈ B2 ,<br />
where h(z) = A(z). Next, let g(z, t) = e tA (z) for z ∈ B 2 and t ≥ 0.<br />
Then g(z, t) is a univalent subordination chain and Dg(0, t) = e tA for<br />
t ≥ 0, {e −tA g(·, t)}t≥0 is a normal family on B 2 and g(z, t) satisfies the<br />
same Loewner differential equation as f (z, t). But, f (z, t) ≡ g(z, t).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 34 / 62
2. Univalent subordination chains in several complex variables<br />
Corollary<br />
T. Poreda, 1991; M. Elin, S. Reich, D. Shoikhet, 2000: Let h ∈ M.<br />
Then the differential equation<br />
Df (z)h(z) = f (z), z ∈ B n ,<br />
has a unique solution f ∈ H(B n ) such that f (0) = 0 and Df (0) = In.<br />
This solution is starlike.<br />
• T. Poreda, 1991 (A < 2m(A)); M. Elin, S. Reich, D. Shoikhet, 2004<br />
(k+(A) < 2k−(A)); Duren, Graham, Hamada, K., 2008.<br />
Corollary<br />
Let h ∈ N A be such that k+(A) < 2m(A). Then the differential equation<br />
Df (z)h(z) = f (z), z ∈ B n ,<br />
has a unique solution f ∈ H(B n ) such that f (0) = 0 and Df (0) = In.<br />
This solution is spirallike w.r.t. A.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 35 / 62
3. Parametric representation on the unit ball<br />
4. Parametric representation on the unit ball<br />
Lemma<br />
Poreda-Szadkowska, 1990: Let f ∈ S∗ (Bn ) and h(z) = [Df (z)] −1f (z).<br />
Then f (z) = lim e<br />
t→∞ t v(z, t) locally uniformly on Bn , where v(z, t) is the<br />
unique solution of the initial value problem<br />
∂v<br />
∂t<br />
= −h(v), ∀ t ≥ 0, v(z, 0) = z,<br />
i.e. f has parametric representation. In addition,<br />
z<br />
≤ f (z) ≤<br />
(1 + z) 2<br />
z<br />
(1 − z) 2 , z ∈ Bn .<br />
• J.A. Pfaltzgraff (1991); R. Barnard, C. FitzGerald and S. Gong<br />
(1991); M. Elin, S. Reich, D. Shoikhet, 2000.<br />
• If f ∈ S then f has parametric representation, i.e. there exists a<br />
Loewner chain f (z, t) such that f = f (·, 0).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 36 / 62
3. Parametric representation on the unit ball<br />
T. Poreda, 1991 (A < 2m(A)); M. Elin, S. Reich, D. Shoikhet,<br />
2004 (k+(A) < 2k−(A)); Duren, Graham, Hamada, K., 2008.<br />
Theorem<br />
Let A ∈ L(C n , C n ) be an operator such that k+(A) < 2m(A). Also let<br />
f : B n → C n be a spirallike mapping with respect to A and let<br />
h(z) = [Df (z)] −1 Af (z) for z ∈ B n . Then f (z, t) = e tA f (z) is a univalent<br />
subordination chain and<br />
lim<br />
t→∞ eAtv(z, s, t) = f (z, s)<br />
locally uniformly on B n , where v(z, s, t) = f −1 (e (s−t)A f (z)). Also<br />
∂f<br />
∂t (z, t) = Df (z, t)h(z), t ≥ 0, z ∈ Bn .<br />
Spirallike mappings w.r.t. A with k+(A) < 2m(A) have a<br />
”generalized” parametric representation.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 37 / 62
3. Parametric representation on the unit ball<br />
• In dimension n ≥ 2, there exist mappings f ∈ S(B n ) which cannot be<br />
imbedded in Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is a<br />
normal family on B n (for example, spirallike mappings).<br />
Definition<br />
Let f ∈ H(B n ) be such that f (0) = 0, Df (0) = In. Also let A ∈ L(C n , C n )<br />
such that k+(A) < 2m(A); f has A-parametric representation (and<br />
denote by f ∈ S 0 A (Bn )) if there exists a generating vector field h(z, t)<br />
such that Dh(0, t) = A for t ≥ 0, and<br />
f (z) = lim<br />
t→∞ e tA v(z, t)<br />
locally uniformly on B n , where v = v(z, t) is the unique Lipschitz<br />
continuous solution on [0, ∞) of the initial value problem<br />
∂v<br />
∂t = −h(v, t), a.e. t ≥ 0, v(z, 0) = z, ∀z ∈ Bn .<br />
• A = In-parametric representation; S 0 In (Bn ) := S 0 (B n ).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 38 / 62
3. Parametric representation on the unit ball<br />
I. Graham, H. Hamada, G.K., 2002 (A = In); I. Graham, H. Hamada,<br />
G.K., M.K., 2008 (k+(A) < 2m(A)); cf. T. Poreda (A = In).<br />
Theorem<br />
Let f ∈ S(B n ) and k+(A) < 2m(A). Then f ∈ S 0 A (Bn ) if and only if there<br />
exists an A-univalent subordination chain f (z, t) such that<br />
{e −tA f (·, t)}t≥0 is a normal family on B n and f = f (·, 0).<br />
Remark<br />
• Ch. Pommerenke (1965): S 0 (B 1 ) = S.<br />
• If n ≥ 2 then S 0 (B n ) S(B n ).<br />
• The most important compact subfamilies of S(B n ): starlikeness,<br />
convexity, close-to-starlikeness, are also subfamilies of S 0 (B n ).<br />
Theorem<br />
S 0 A (Bn ) is a compact family for k+(A) < 2m(A).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 39 / 62
Theorem<br />
3. Parametric representation on the unit ball<br />
I. Graham, H. Hamada, G.K. 2003: Let f ∈ S 0 (B n ). Then<br />
z<br />
≤ f (z) ≤<br />
(1 + z) 2<br />
These estimates are sharp. Also<br />
Moreover, 1<br />
2 D2 f (0) ≤ 8.<br />
Corollary<br />
S 0 (B n ) is compact.<br />
Remark<br />
z<br />
(1 − z) 2 , z ∈ Bn .<br />
<br />
<br />
1<br />
2 〈D2 <br />
<br />
f (0)(v, v), v〉 ≤ 2, v = 1.<br />
There exist mappings f ∈ S 0 (B n ), n ≥ 2, such that (1/2)D 2 f (0) > 2.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 40 / 62
3. Parametric representation on the unit ball<br />
• Asymptotic spirallikeness, a natural generalization of spirallikeness.<br />
Definition<br />
Let Ω ⊆ C n be a domain which contains the origin and let<br />
A ∈ L(C n , C n ) be such that m(A) > 0. We say that Ω is<br />
A-asymptotically spirallike if there exists a mapping<br />
Q = Q(z, t) : Ω × [0, ∞) → C n which satisfies the following conditions:<br />
(i) Q(·, t) is a holomorphic mapping on Ω, Q(0, t) = 0, DQ(0, t) = A,<br />
t ≥ 0, and the family {Q(·, t)}t≥0 is locally uniformly bounded on Ω.<br />
(ii) Q(z, ·) is measurable on [0, ∞) for all z ∈ Ω;<br />
(iii) The initial value problem<br />
∂w<br />
∂t<br />
= −Q(w, t) a.e. t ≥ s, w(z, s, s) = z,<br />
has a unique solution w = w(z, s, t) for each z ∈ Ω and s ≥ 0, such<br />
that w(·, s, t) is a holomorphic mapping of Ω into Ω for t ≥ s, w(z, s, ·)<br />
is locally absolutely continuous on [s, ∞) locally uniformly with respect<br />
to z ∈ Ω for s ≥ 0, and lim<br />
t→∞ e At w(z, 0, t) = z locally uniformly on Ω.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 41 / 62
3. Parametric representation on the unit ball<br />
• A domain Ω ⊆ C n which contains the origin is called asymptotically<br />
spirallike if there exists an operator A ∈ L(C n , C n ) with m(A) > 0 such<br />
that Ω is A-asymptotically spirallike.<br />
Definition<br />
Let f ∈ S(B n ) and let A ∈ L(C n , C n ) be such that m(A) > 0. We say<br />
that f is A-asymptotically spirallike if f (B n ) is an A-asymptotically<br />
spirallike domain.<br />
Remark<br />
If Ω is a spirallike domain with respect to A with m(A) > 0, then Ω is<br />
A-asymptotically spirallike.<br />
• Ω ⊆ C n is In-asymptotically spirallike if and only if Ω is asymptotically<br />
starlike.<br />
Problem Does there exist a connection between A-asymptotic<br />
spirallikeness and A-parametric representation?<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 42 / 62
Theorem<br />
3. Parametric representation on the unit ball<br />
Let A ∈ L(C n , C n ) be an operator with m(A) > 0 and k+(A) < 2m(A).<br />
Also let f ∈ S(B n ). Then f has A-parametric representation iff there is<br />
a univalent subordination chain f (z, t) such that Df (0, t) = e At , t ≥ 0,<br />
{e −At f (·, t)}t≥0 is a normal family on B n and f = f (·, 0).<br />
Corollary<br />
Let f : Bn → Cn be a spirallike mapping with respect to A ∈ L(Cn , Cn )<br />
such that k+(A) < 2m(A). Then f has A-parametric representation. In<br />
addition, there is a univalent subordination chain f (z, t) such that<br />
f (0, t) = 0, Df (0, t) = etA , {e−tAf (·, t)}t≥0 is a normal family on Bn and<br />
f = f (·, 0). Let v(z, s, t) be the transition mapping of f (z, t). Then<br />
f (z) = lim e<br />
t→∞ tA v(z, t) locally uniformly on Bn , where v(z, t) = v(z, 0, t).<br />
• Particular cases of this result were obtained by T. Poreda and A.<br />
Szadkowska, 1990; M. Elin, S. Reich, D. Shoikhet, 2004.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 43 / 62
Theorem<br />
3. Parametric representation on the unit ball<br />
Let A ∈ L(C n , C n ) be such that m(A) > 0 and k+(A) < 2m(A). Also let<br />
f ∈ S(B n ). Then f is A-asymptotically spirallike if and only if f has<br />
A-parametric representation.<br />
Remark<br />
The class of A-asymptotically spirallike mappings with k+(A) < 2m(A)<br />
is compact, however the full class of asymptotically spirallike mappings<br />
is not compact in dimension n ≥ 2.<br />
Indeed, let n = 2 and f : B2 → C2 be given by f (z) = (z1, z2 + az2 1 ) for<br />
z = (z1, z2) ∈ B2 . Also let A ∈ L(C2 , C2 ) be given by A(z) = (z1, 2z2).<br />
Then f is spirallike with respect to A for all a ∈ C. Thus f is also<br />
asymptotically spirallike. But, if z0 = (1/2, 0) then f (z0) = (1/2, a/4)<br />
and f (z0) → ∞ as a → ∞.<br />
• The class of asymptotically starlike mappings is compact.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 44 / 62
Remark<br />
3. Parametric representation on the unit ball<br />
Let f : B n → C n be an A-asymptotically spirallike mapping, where A is<br />
a diagonal matrix whose eigenvalues λ1, . . . , λn satisfy the conditions<br />
ℜλ j = a > 0 for j = 1, . . . , n. Then<br />
Theorem<br />
z<br />
≤ f (z) ≤<br />
(1 + z) 2<br />
z<br />
(1 − z) 2 , z ∈ Bn .<br />
Let f ∈ S(B n ) and let A ∈ L(C n , C n ) be such that A + A ∗ = 2aIn, where<br />
a > 0 and A ∗ is the adjoint of A. Then f is asymptotically spirallike if<br />
and only if f is asymptotically starlike.<br />
Theorem<br />
Let f : B n → C n be a spirallike mapping with respect to an operator<br />
A ∈ L(C n , C n ) such that A + A ∗ = 2aIn, where a > 0. Then f ∈ S 0 (B n ).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 45 / 62
Examples<br />
Example<br />
3. Parametric representation on the unit ball<br />
Let f1, . . . , fn ∈ S be such that f1 is not spirallike and let<br />
f (z) = (f1(z1), . . . , fn(zn)) for z = (z1, . . . , zn) ∈ B n . Then f is<br />
asymptotically starlike, and hence asymptotically spirallike, but is not<br />
spirallike.<br />
Example<br />
Let f1 ∈ S be a function which has a1-parametric representation and<br />
assume f : B n−1 → C n−1 has A-parametric representation. Then<br />
F : B n → C n given by F(z) = (f1(z1), f (z ′ )), z = (z1, z ′ ) ∈ B n , has<br />
Ã-parametric representation where à =<br />
a1 0<br />
0 A<br />
(3.1) max{ℜa1, k+(A)} < 2 min{ℜa1, m(A)},<br />
<br />
. In addition, if<br />
then F is Ã-asymptotically spirallike. If f1 is not spirallike then F is also<br />
not spirallike.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 46 / 62
4. Subordination chains in complex Banach spaces<br />
4. Subordination chains in complex Banach spaces<br />
I. Graham, H. Hamada, G.K, M.K, 2012;<br />
• Let X be a reflexive complex Banach space.<br />
X = C n : J. Pfaltzgraff, 1974 (A = In); T. Poreda, 1990; M. Elin, S.<br />
Reich, D. Shoikhet, 2004; Graham, H. Hamada, G.K, M.K. (2008):<br />
Theorem<br />
Let h(z, t) : B × [0, ∞) → X be a generating vector field:<br />
(i) h(·, t) ∈ N , Dh(0, t) = A ∈ L(X), t ≥ 0, where m(A) > 0.<br />
(ii) h(z, ·) is strongly measurable on [0, ∞) for z ∈ B.<br />
Then for each s ≥ 0 and z ∈ B, the initial value problem<br />
(4.1)<br />
∂v<br />
∂t<br />
= −h(v, t) a.e. t ≥ s, v(z, s, s) = z,<br />
has a unique solution v = v(z, s, t) such that v(·, s, t) is a univalent<br />
Schwarz mapping, v(z, s, ·) is Lipschitz continuous on [s, ∞) uniformly<br />
with respect to z ∈ Br , r ∈ (0, 1), Dv(0, s, t) = exp(−A(t − s)).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 47 / 62
4. Subordination chains in complex Banach spaces<br />
• J. Pfaltzgraff (1975; X = C n ); T. Poreda (1990): h(z, t) generating<br />
vector field in (4.1) such that:<br />
h(z, t) ≤ M(r), z ≤ r < 1, t ≥ 0.<br />
• T. Poreda: h = h(z, t) is continuous on B × [0, ∞)<br />
Theorem<br />
Under the assumptions of the previous result, the following estimates<br />
hold:<br />
and<br />
v(z, s, t)<br />
(1 − v(z, s, t))<br />
e −k(A)(t−s)<br />
z<br />
2 ≤ e−m(A)(t−s)<br />
z<br />
≤<br />
(1 + z) 2<br />
, z ∈ B, t ≥ s ≥ 0,<br />
(1 − z) 2<br />
v(z, s, t)<br />
, z ∈ B, t ≥ s ≥ 0.<br />
(1 + v(z, s, t)) 2<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 48 / 62
Theorem<br />
4. Subordination chains in complex Banach spaces<br />
Let h = h(z, t) : B × [0, ∞) → X be a generating vector field such that<br />
Dh(0, t) = A ∈ L(X) for t ≥ 0, and k+(A) < 2m(A). Also, let<br />
v = v(z, s, t) be the unique Lipschitz continuous solution on [s, ∞) of<br />
the initial value problem (4.1). Then the limit<br />
(4.2) lim<br />
t→∞ e tA v(z, s, t) = f (z, s)<br />
exists uniformly on each closed ball Br for r ∈ (0, 1) and s ≥ 0.<br />
Moreover, f (z, t) is an A-normalized univalent subordination chain.<br />
Theorem<br />
Let A ∈ L(X) be s.t. k+(A) < 2m(A). Also, let f (z, t) be an<br />
A-normalized univalent subordination chain. Assume that for each<br />
r ∈ (0, 1), there is M = M(r, A) > 0 such that e −tA f (z, t) ≤ M,<br />
z ≤ r, t ≥ 0. If v(z, s, t) is the transition mapping of f (z, t), then<br />
f (z, s) = limt→∞ e tA v(z, s, t) uniformly on Br , r ∈ (0, 1).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 49 / 62
Corollary<br />
4. Subordination chains in complex Banach spaces<br />
Let f (z, t) = e t z + · · · be a normalized univalent subordination chain.<br />
Assume that for each r ∈ (0, 1), there is M = M(r) > 0 such that<br />
e −t f (z, t) ≤ M, z ≤ r, t ≥ 0. Then<br />
Corollary<br />
r/(1 + r) 2 ≤ e −t f (z, t) ≤ r/(1 − r) 2 , z = r < 1, t ≥ 0.<br />
Assume X is separable. Let h = h(z, t) : B × [0, ∞) → X be a<br />
generating vector field such that Dh(0, t) = A ∈ L(X) for t ≥ 0, and the<br />
condition k+(A) < 2m(A) holds. Also, let f (z, t) = e tA z + · · · be the<br />
A-normalized univalent subordination chain given by (4.2). Assume<br />
that ∂f<br />
∂t (z, t) exists for t ∈ [0, ∞) \ E and z ∈ Bδ, for some δ ∈ (0, 1),<br />
where E ⊂ [0, ∞) (independent of z) has measure zero. Then<br />
∂f<br />
(z, t) = Df (z, t)h(z, t), t ∈ [0, ∞) \ E, ∀ z ∈ B.<br />
∂t<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 50 / 62
Theorem<br />
4. Subordination chains in complex Banach spaces<br />
(i) Assume that f (z, t) = e tA z + · · · is a standard solution of the<br />
generalized Loewner differential equation<br />
(4.3)<br />
∂f<br />
(z, t) = Df (z, t)h(z, t), t ∈ [0, ∞) \ E, ∀z ∈ B,<br />
∂t<br />
where E is a subset of [0, ∞) of measure zero. Also, assume that for<br />
each r ∈ (0, 1), there is M = M(r, A) > 0 such that<br />
e −tA f (z, t) ≤ M(r, A), z ≤ r, t ≥ 0.<br />
Then f (z, t) is an A-univalent subordination chain and (4.2) holds.<br />
(ii) Assume that g(z, t) is another standard solution of (4.3). If for each<br />
r ∈ (0, 1), there is K = K (r, A) > 0 such that e −tA g(z, t) ≤ K ,<br />
z ≤ r and t ≥ 0, then g(z, t) is a subordination chain and there<br />
exists Ψ ∈ L(X) such that g(z, t) ≡ Ψ(f (z, t)). If, in addition,<br />
Dg(0, 0) = I, then g(z, t) ≡ f (z, t).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 51 / 62
Definition<br />
4. Subordination chains in complex Banach spaces<br />
Let A ∈ L(X) be such that k+(A) < 2m(A) and let f ∈ H(B) be a<br />
normalized mapping. We say that f has A-parametric representation<br />
(f ∈ S0 A (B)) if there exists a generating vector field<br />
h = h(z, t) : B × [0, ∞) → Cn such that Dh(0, t) = A for t ≥ 0, and<br />
f (z) = lim e<br />
t→∞ tA v(z, t) uniformly on each closed ball Br for r ∈ (0, 1),<br />
where v = v(z, t) is the unique Lipschitz continuous solution on [0, ∞)<br />
of the initial value problem<br />
∂v<br />
∂t<br />
= −h(v, t) a.e. t ≥ 0, v(z, 0) = z,<br />
for each z ∈ B.<br />
If A = I and f has I-parametric representation, then we say that f has<br />
parametric representation in the usual sense.<br />
• If f ∈ S0 A (B), then f is univalent.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 52 / 62
Remark<br />
4. Subordination chains in complex Banach spaces<br />
(i) Let S be the family of normalized univalent functions on the unit<br />
disc U in C. It is well known that f ∈ S if and only if there exists a<br />
Loewner chain f (z, t) such that f = f (·, 0).<br />
(ii) In C n , n ≥ 2, such result does not hold for the full family S(B n ) of<br />
normalized biholomorphic mappings on the unit ball B n in C n .<br />
(iii) However, if A ∈ L(C n , C n ) is such that k+(A) < 2m(A), then<br />
f ∈ S(B n ) has A-parametric representation if and only if there exists an<br />
A-normalized univalent subordination chain f (z, t) such that<br />
{e −tA f (·, t)}t≥0 is a normal family on B n and f = f (·, 0).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 53 / 62
Corollary<br />
4. Subordination chains in complex Banach spaces<br />
Let A ∈ L(X) satisfy the condition k+(A) < 2m(A). Also, let f : B → X<br />
be a normalized holomorphic mapping. Then the following assertions<br />
hold:<br />
(i) If f has A-parametric representation, then there exists an<br />
A-normalized univalent subordination chain f (z, t) such that f = f (·, 0),<br />
and for each r ∈ (0, 1), there is M = M(r) > 0 such that<br />
(4.4) e −tA f (z, t) ≤ M(r), z ≤ r, t ≥ 0.<br />
(ii) Conversely, assume that f (z, t) = e tA z + · · · is a standard solution<br />
of the generalized Loewner differential equation (4.3), which satisfies<br />
the condition (4.4), and such that f = f (·, 0). Then f has A-parametric<br />
representation.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 54 / 62
4. Subordination chains in complex Banach spaces<br />
• Applications to spirallikeness and starlikeness<br />
Definition<br />
Let A ∈ L(X) be such that m(A) > 0. Also let Ω be a domain in X<br />
which contains the origin. We say that Ω is spirallike with respect to A if<br />
e −tA (w) ∈ Ω, for all w ∈ Ω and t ≥ 0.<br />
A mapping f ∈ S(B) is said to be spirallike with respect to A if f (B) is a<br />
spirallike domain with respect to A.<br />
Note that if A = I in the above definition, we obtain the usual notion of<br />
starlikeness.<br />
Remark<br />
A normalized locally biholomorphic mapping f on B is spirallike with<br />
respect to A ∈ L(X), m(A) > 0, if and only if (T. Suffridge; K.<br />
Gurganus, 1975)<br />
ℜ[ℓz([Df (z)] −1 Af (z))] > 0, z ∈ B \ {0}, ℓz ∈ T (z).<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 55 / 62
Corollary<br />
4. Subordination chains in complex Banach spaces<br />
Let A ∈ L(X) be such that m(A) > 0. Also, let f : B → X be a<br />
normalized locally biholomorphic mapping. Then f is spirallike with<br />
respect to A if and only if f (z, t) = e tA f (z) is an A-normalized univalent<br />
subordination chain.<br />
• T. Poreda (1999); M. Elin, S. Reich, D. Shoikhet (2004); Graham,<br />
Hamada, K.K. (2008); P. Duren, I. Graham, H. Hamada, G.K. (2010):<br />
Corollary<br />
Let A ∈ L(X) be such that k+(A) < 2m(A) and let f be a spirallike<br />
mapping with respect to A. Then f has A-parametric representation.<br />
Corollary<br />
Let A ∈ L(X) be s.t. the condition k+(A) < 2m(A) holds, and let h ∈ N<br />
be s.t. Dh(0) = A. Then the differential equation Df (z)h(z) = Af (z),<br />
z ∈ B, has a unique solution f ∈ H(B) with f (0) = 0 and Df (0) = I.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 56 / 62
5. Open problems<br />
Conjecture<br />
5. Open problems<br />
Let A ∈ L(X) be such that k+(A) < 2m(A). Also, let f (z, t) be an<br />
A-normalized univalent subordination chain f (z, t) such that the<br />
condition (4.4) holds. Then f = f (·, 0) has A-parametric representation.<br />
• In the case X = C n , every Loewner chain satisfies the generalized<br />
Loewner differential equation (Graham, Hamada, K, 2002).<br />
Problem<br />
Let A ∈ L(X) be such that k+(A) < 2m(A) and let f (z, t) be an<br />
A-normalized univalent subordination chain. Does there exist a<br />
generating vector field h(z, t) = Az + · · · such that<br />
∂f<br />
(z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ B?<br />
∂t<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 57 / 62
Conjecture<br />
5. Open problems<br />
Let A ∈ L(X) be such that k+(A) < 2m(A). If f (z, t) is the A-normalized<br />
univalent subordination chain given by (4.2), then f (·, t) is<br />
biholomorphic on B for t ≥ 0.<br />
• P. Duren, I. Graham, H. Hamada, G.K (2010): positive answer to the<br />
following problem when X = C n .<br />
Problem<br />
Let A ∈ L(X) be such that k+(A) < 2m(A) and let ft(z) = f (z, t) be the<br />
A-normalized univalent subordination chain given by (4.2). Also, let<br />
Ω = <br />
t≥0 ft(B). Is it true that Ω = X?<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 58 / 62
Definition<br />
5. Open problems<br />
Let Ω ⊆ X be a domain which contains the origin and let A ∈ L(X) be<br />
such that m(A) > 0. We say that Ω is A-asymptotically spirallike if<br />
there exists a mapping Q = Q(z, t) : Ω × [0, ∞) → X such that:<br />
(i) Q(·, t) ∈ H(Ω), Q(0, t) = 0, DQ(0, t) = A, t ≥ 0, and for each closed<br />
ball B(z0, r) contained in Ω, there exists some M = M(r, z0) > 0 such<br />
that Q(z, t) ≤ M(r, z0) for z ∈ B(z0, r).<br />
(ii) Q(z, ·) is strongly measurable on [0, ∞), for all z ∈ Ω;<br />
(iii) The initial value problem<br />
∂w<br />
∂t<br />
= −Q(w, t) a.e. t ≥ s, w(z, s, s) = z,<br />
has a unique solution w = w(z, s, t) for each z ∈ Ω and s ≥ 0, such<br />
that w(·, s, t) is a holomorphic mapping of Ω into Ω for t ≥ s, w(z, s, ·)<br />
is strongly locally absolutely continuous on [s, ∞), for all z ∈ Ω and<br />
s ≥ 0, and lim e<br />
t→∞ tA w(z, 0, t) = z uniformly on each closed ball B(z0, r)<br />
contained in Ω.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 59 / 62
Definition<br />
5. Open problems<br />
A mapping f ∈ S(B) is called A-asymptotically spirallike<br />
(asymptotically starlike) if f (B) is an A-asymptotically spirallike<br />
(I-asymptotically spirallike) domain.<br />
Remark<br />
Note that any spirallike mapping with respect to an operator A ∈ L(X)<br />
is A-asymptotically spirallike. Also, any normalized starlike mapping is<br />
asymptotically starlike.<br />
It would be interesting to give an answer to the following question. In<br />
the case X = C n , positive answer: Graham, Hamada, K.K. (2008).<br />
Problem<br />
Does there exist a connection between A-normalized univalent<br />
subordination chains and A-asymptotically spirallike mappings in the<br />
case of reflexive complex Banach spaces?<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 60 / 62
5. Open problems<br />
References<br />
1. P. Duren, I. Graham, H. Hamada, G. <strong>Kohr</strong>: Solutions for the<br />
generalized Loewner differential equation in several complex variables,<br />
Math. Ann., 347, 411-435 (2010).<br />
2. Graham, I., Hamada, H., <strong>Kohr</strong>, G.: Parametric representation of<br />
univalent mappings in several complex variables, Canadian J. Math.,<br />
54, 324-351 (2002).<br />
3. Graham, I., Hamada, H., <strong>Kohr</strong>, G., <strong>Kohr</strong>, M.: Parametric<br />
representation and asymptotic starlikeness in C n , Proc. Amer. Math.<br />
Soc., 136, 3963-3973 (2008).<br />
4. Graham, I., Hamada, H., <strong>Kohr</strong>, G., <strong>Kohr</strong>, M.: Asymptotically<br />
spirallike mappings in several complex variables, J. Anal. Math., 105,<br />
267-302 (2008).<br />
5. Graham, I., Hamada, H., <strong>Kohr</strong>, G., <strong>Kohr</strong>, M.: Spirallike mappings<br />
and univalent subordination chains in C n , Ann. Scuola Norm. Sup.<br />
Pisa-Cl. Sci., 7 (2008), 717-740.<br />
<strong>Gabriela</strong> <strong>Kohr</strong> (UBB Cluj) Geometric and analytic aspects of Loewner chains 61 / 62
5. Open problems<br />
6. Graham, I., <strong>Kohr</strong>, G.: Geometric Function Theory in One and Higher<br />
Dimensions, Marcel Dekker, New York (2003).<br />
7. Graham, I., <strong>Kohr</strong>, G., <strong>Kohr</strong>, M.: Loewner chains and parametric<br />
representation in several complex variables, J. Math. Anal. Appl., 281,<br />
425-438 (2003).<br />
8. Graham, I., <strong>Kohr</strong>, G., Pfaltzgraff, J.A.: The general solution of the<br />
Loewner differential equation on the unit ball in C n , Contemporary<br />
Mathematics, 382, 191-203 (2005).<br />
9. Hamada, H., <strong>Kohr</strong>, G.: Subordination chains and the growth<br />
theorem of spirallike mappings, Mathematica (Cluj), 42 (65), 153-161<br />
(2000).<br />
10. Hamada, H., <strong>Kohr</strong>, G.: An estimate of the growth of spirallike<br />
mappings relative to a diagonal matrix, Ann. Univ. Mariae Curie<br />
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