Solution of a Loewner Chain Equation in Several Complex Variables

Proposition 2.1 will show that a polynomially bounded solution of (1.1) can be recovered from its first n0 coefficients and the solution of (1.2). Conversely,

Theorem 2.8 will show that by finding polynomially bounded solutions to the first n0 coefficient equations (1.4) we can find a solution of (1.1). These results generalize Poreda [17, Theorem 4.1 and Theorem 4.4]. Finally, after a discussion about the existence of polynomially bounded solutions to the coefficient equations we will obtain the main result, Theorem 2.11, that guarantees the existence of a Loewner chain solution for (1.1). We also consider what happens to the various classes of univalent ma

Proposition 2.1. If f (z, t) is a polynomially bounded solution of (1.1) such that f (z, t) = etA

Lemma 2.2. If Qk ∈Pk (Cn) then the following identities hold for t ∈R etAetBkQk e−tAz
k = Qk zk
(2.3) etAQk e−tAz
k = e−tBkQ zk
(2.4)

Lemma 2.3. If Fk, Gk : [0, ∞) →Pk (Cn), k = 2, . . . , m are solutions of (1.4) and the limits f (z, s) := lim t→∞etA

Lemma 2.4. If P is a polynomial such that P (t) ≥0 for t ≥s then ˆ ∞ s P (t) e(t−s)A ∥v (z, s, t)∥n0+1 (1 −∥v (z, s, t)∥)2 dt ≤ Qǫ,A,P (s) (1 −∥z∥)2 k+(A) m(A) +ǫ , ǫ > 0 where Qǫ,A,P is a polynomial of the same degree as P.

Lemma 2.6. If Fk : [0, ∞) →Pk (Cn), k = 2, . . . , m are polynomially bounded and the limit f (z, s) = lim t→∞etA

Corollary 2.7. All polynomially bounded solutions of (1.1) are Loewner chains.

Theorem 2.8. If Fk, k = 2, . . . , n0 are polynomially bounded solutions of (1.4) then g (z, s) := lim t→∞etA

Theorem 2.11. The equation (1.1) always has a polynomially bounded Loewner chain solution that is uniquely determined by the values of F ≤ k zk, 0
, k = 2, . . . , n0, which can be prescribed arbitrarily. Furthermore, if A + ¯A is nonresonant then the solution can be chosen to be bounded.

Lemma 2.14. Every sequence of Loewner chains {fk (z, t)} such that Dfk (0, t) = etA and e−tAfk (z, t) ≤CrP (t) , ∥z∥≤r < 1, t ≥0, where P (t) is a polynomial, has a subsequence that converges locally uniformly on Bn to a polynomially bounded Loewner chain f (z, t) for t ≥0.

Theorem 2.15. If F ⊂Qn0 k=2 P ≤ k Pk (Cn)
is bounded (compact) then SF A (Bn) is normal (compact). Furthermore, given ǫ > 0 there exists a constant Cǫ,A,F such that ∥f (z)∥≤ Cǫ,A,F (1 −∥z∥)2 k+(A) m(A) +ǫ , f ∈SF A (Bn) .

Theorem 3.1. ˆSA (Bn) is compact if and only if A is nonresonant.

Corollary 2.7). By Remark 2.10 this generalizes [6, Corollary 4.8]. On the other hand, if A is resonant, there either is no holomorphic solution (for example if H2 /∈B2 P2 (Cn)
) or the holomorphic solutions (in fact, biholomorphic) are not unique.

Proposition 3.5. Let A = diag (1, λ), Reλ ≥1. Define Φα,β : S∗B1
→ˆSA B2
by Φα,β (f) (z) =  f (z1) , f (z1) z1 α (f ′ (z1))β z2  . If α ∈[0, Reλ] and β ∈[0, 1/2] such that α + β ≤Reλ then Φα,β

Proposition 3.8. Let f : Bn →Cn be a holomorphic mapping and f (z) = z + ∞ X k=2 Fk zk
. Then f is A-asymptotically spirallike if and only if there exists h ∈HA (Bn) such that (3.3) f (z) = lim t→∞etA

Theorem 3.5] one sees that h ∈HA (Bn) and that v is the solution of (1.2). We have f (z) = lim t→∞etAν (f (z) , 0, t) = lim t→∞etAf (v (z, 0, t)) locally uniformly on Bn. Like in the proof of Proposition 2.1 we also see that lim t→∞etAf (v (z, 0, t)) = lim t→∞etA

Lemma 3.10. Let A = diag (λ1, . . . , λn) and h ∈HA (Bn). If v = (v1, . . . , vn) is the solution of (1.6) then ∥vi (z, t)∥≤C      e−Reλit , Reλi < 2m (A) (1 + t)e−Reλit , Reλi = 2m (A) e−2m(A)t , Reλi > 2m (A) where C is a constant that depends on A, λi and ∥z∥.

Lemma 3.11. Let λ ∈C such that Reλ ≥0, a ∈C and h : [0, ∞) →C such that |h (t)| ≤C, t ≥0. If (3.4) lim t→∞ ˆ t 0 esλ (h (s) + a) ds = 0 then |a| ≤C.

Proposition 3.12. Suppose that A is normal, nonresonant and n0 = 2. Then Sa A (Bn) is a normal family. Furthermore, if f ∈Sa A (Bn) has h ∈HA (Bn) as an infinitesimal generator (see Proposition 3.8) then f can be embedded as the first element of a bounded Loewner chain with infinitesimal generator h.