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1 2 t0 := d z. π ZD is the area with respect to Lebesgue measure. The set N t := (t0, t1, t¯1, t2, t¯2, t3, t¯3,... ) R+ C , ± ∈ × with t¯k denoting the complex conjugate of tk, are local co-ordinates on the manifold (moduli space) of smooth closed contours , cf. [18]. The other set of (natural) co-ordinatesC is given by the coefficients of the normalised Riemann mapping. So, we have two different sets of co-ordinates for , as shown below: C The first co-ordinate R CN The first co-ordinate respects the +∗ respects the ♠6 × i❘❘ conformal radius ♠♠ ❘❘❘ interior area ♠♠♠ ❘❘❘ ♠♠♠ ❘❘❘ ♠♠♠ ❘❘ Aut( ) Der+( ) t ∂t0 , ∂t , ∂t¯ k N O | O { ±}|{ k k } ∈ Hence, the tangent space to permits also two descriptions, namely, as [3, 6, 9, 18] C C Der0( ) := z [[z]]∂z and ∂t0 , ∂t , ∂t¯ k N O { k k } ∈ n+1 d N Define ℓn := z dz , n , which span the positive part of the Witt algebra, i.e. for n, m Z − ∈ ∈ [ℓ ,ℓ ]=(m n)ℓ . m n − m+n The ∂tk can be determined either by specific boundary variations of the domain, which do only change one harmonic moment at the time, cf. [9, Formula (2.11)], or in terms of the Faber polynomials [18]. Combining results in [9] with our considerations, the relation for the different tangent vectors is given by
Proposition 1.1. The vector fields ∂tk k>1 on and the operators ℓk k>1 are related by { } C { } 1 l ∂tk tl = ξ− δn(ξ) dξ = δkl, kπ I∂Dc | | where 1 dz δn(ξ) := ∂n − (ℓkG0)(z,ξ) , 2πi I z ∞ c c and ∂n is the normal derivative on the boundary ∂D with respect to ξ ∂D , and G0(x, ξ) is the Dirichlet Green function associated to the Dirichlet problem in∈Dc. Krichever, Marshakov, Mineev-Weinstein, Wiegmann and Zabrodin [7, 13, 22], in dif- ferent constellations, defined the logarithm of a τ-function which for a contour C = ∂D is given by [7, 18]
1 1 1 2 2 R ln(τ): C 2 ln d zd w . C ∋ 7→ −π ZD ZD z − w ∈
3
The τ-function connects complex analysis with the dispersionless hierarchies and inte- grable systems [7] A key result, which expresses the Riemann mapping in terms of the τ-function, is the following Theorem 1.2 ([7, 18]). Let g : Cˆ D Cˆ D be the conformal map, normalised by \ → \ g( )= and g′( ) > 0. Then the following formula holds: ∞ ∞ ∞ 2 k 2 1 ∂ ln(τ) ∞ z− ∂ ln(τ) ln(g(z)) = ln(z) 2 . − 2 ∂t − k ∂t0∂tn 0 Xk=1 This formula would be a key to describe the solution to the conformal welding problem (c.f. [21]) associated to Malliavin’s canonic diffusion [12] within the framework of Loewner- Kufarev equation, which would be a future work. Another interpretation of the τ-function, given by Takhtajan [18, Corollary 3.10], is that it is a K¨ahler potential of a Hermitian metric on a, a> 0. Kirillov and Juriev [8], defined a two-parameter familyC (h, c) of K¨ahler potentials K e h,c on the determinant line bundle Det∗ over the (Sato)-Segal-Wilson Grassmannian, where h is the highest-weight and c the central charge of the CFT. For h = 0 and c = 1, i.e. the free Boson, one has [8, 5] for the metric
K0,1(f) g0,1(f) = e− dλdλ¯ where λ is the co-ordinate in the fibre over the schlicht function f. In terms of the Grunsky matrix Zf , associated to an f , we obtain the relation between the K¨ahler potential, for h =0,c = 1, and the τ-function,∈ S as ln(τ)(f) ln det(1 Z Z¯ ). ∼ − f f The following diagram summarises our discussion so-far:
R K¨ahler potential O c●● ln det(1 Z Z¯ ) ●● − f f ln(τ) ●● ●● Aut( ) o / Det∗ / Det∗ O C |M π Z τSW Aut+( ) o / M := Zf : f GrSW / C[[t1, t2, t3,... ]] = ∗ O S Krichever { ∈S} ⊂ H0 where Z is the Grunsky matrix, cf. [8] and 0∗ the charge 0 sector of the boson Fock space [6, p.279]. For the second square from theH left, we should note that the Krichever mapping does not distinguish and algebro-geometrically. Namely, the Krichever em- bedding of a Riemann mappingS usesC only the negative part of the Grunsky coefficients b m, n, m, n =1, 2,... but not b0,0. One finds from the defining equation of the Grunsky − − coefficients that b0,0 is the only entry of the Grunsky matrix which depends on the con- formal radius. Consequently, Krichver’s embedding forgets about the conformal radius. But in order to keep track of the modulus of the derivative of the normalised Riemann mapping, we put that information into the determinant line bundle. This is the mapping Det∗ M . C →The structure| of the rest of the paper is as follows: In Section 2 we establish a relation between the theory of Laplacian growth models, and their integrable structure, with a class of random matrices and second order free probability. We succinctly summarise it in a dictionary. In Section 3 we consider controlled Loewner-Kufarev equations and recall the necessary facts. Then, in Section 4, we give several estimates for the Grunsky coefficients associated to solutions to a controlled Loewner-Kufarev equation. Proofs of 4 several estimates which need results from [1] are relegated to Appendix A. Finally, we prove Theorem 3.3 in Section 5.
2. Integrability and Higher Order Free Probability Another motivation in the works of Marshakov et al. was the close connection random matrix theory red has with (Laplacian) growth models and integrable hierarchies. Takebe, Teo and Marshakov discussed the geometric meaning of the eigenvalue distribution in the large N limit of normal random matrices in conjunction with the one variable reduction via the Loewner equation [17]. In [1] we established and discussed a relation between CFT and free probability theory. Here we briefly present a novel connection between integrable hierarchies, large N limits of Gaussian random matrices and second (higher) order free probability [2]. First, consider, cf. [18, p. 42], j(z)j(w) hh ii the normalised current two-point functions for free Bososn on Cˆ D, and the analogous correlation function with the Dirichlet boundary conditions, i.e. \ j(z)j(w) . hh iiDBC Further, let, cf. [2, p. 11], M( 1 , 1 ) G(z,w) := z w , zw 1 1 be the second order Cauchy transform, and M( z , w ) the second order moments. We obtain Theorem 2.1. Assume that the second order free cumulants R(z,w) vanish, i.e. we have an integrability /zero-curvature condition, such as for e.g. the Gaussian and Wishart random matrices. Then the tensor corresponding to the second order Cauchy transform G(z,w)dz dw is given by the Ward identity ⊗ (2.1) G(z,w)dz dw = j(z)j(w) j(z)j(w) ⊗ hh ii− hh iiDBC G (z)G (w) 1 (2.2) = ′ ′ dz dw (G(z) G(w))2 − (z w)2 ⊗ − − ∞ 2 m 1 n 1 ∂ ln(τ) (2.3) = z− − w− − dz dw ∂t ∂t ⊗ m,nX=1 m n where G(z) is the first order Cauchy transform. This suggests us to make a dictionary translating languages from a integrable system and free probability. Table 1 is an attempt to list objects in these fields sharing the same algebraic relations. From the above it follows now, that general higher order free (local) cumulants are given by Ward identities of n-point functions of the twisted Boson field over arbitrary Riemann surfaces.
3. Controlled Loewner-Kufarev Equations The connection with the Loewner equation, the class of schlicht functions and integrable systems was established by Takebe, Teo and Zabrodin [17]. They showed that both the chordal and the radial Loewner equation give consistency conditions of such integrable hierarchies. A particularly important class of such consistency conditions can be obtained from specific control functions. 5
Table 1.
Laplacian Growth Model Free Probability Theory (dToda hierachy)
Exterior of the domain D, c Dc Spectral measure Harmonic moments of D,
1 n ∂ log τ vn = z zdz = , 2πi Z∂D ∂tn 2 ∂ log τ ? v0 = (log z )zdz = π ZD | | ∂t0 [18, Corollary 3.3] Derivative of harmonic moments, The second order limit moments, ∂vm for m, n > 1 αm,n for m, n > 1 ∂tn The first order Cauchy transform, Riemann mapping, M( 1 ) G : Dc Dc G(x)= x → x Normalised reduced 2-point current correlation function of free bosons on P1 parametrised by C = ∂D, G(z,w)= (z)(w) (z)(w) hh ii−hh iiDBC ∂2G (z,w) ∂2G (z,w) = 0 DBC ∂z∂w − ∂z∂w ∞ 2 The second order Cauchy transform, m 1 n 1 ∂ ln(τ) = z− − w− − , 1 1 ∂tm∂tn M( , ) m,nX=1 G(x, y)= x y xy where G0(z,w) is the Green function associated with the ∂-Laplacian 1 = ∂ ∂ △0 −2 ∗ ∗ and GDBC is the Green function c associated to 0 on D [18, Theorem△ 3.9] Normalised 1-point current correlation function of free bosons on CP1 parametrised by C = ∂D,
∞ n 1 ∂ log τ ? (z) = z− − hh ii − ∂t Xn=1 n [18, Theorem 3.1]
In the previous paper [1], the authors introduced the notion of a solution to the con- trolled Loewner-Kufarev equation (see [1, Definition 2.1])
(3.1) df (z)= zf ′(z) dx (t) + dξ(x, z) , f (z) z D t t { 0 t} 0 ≡ ∈ 6 where D = z < 1 is the unit disc in the complex plane, x : [0, T ] R, {| | } 0 → x , x , : [0, T ] C 1 2 ··· → are given continuous functions of bounded variation, called the driving functions. We define ∞ n ξ(x, z)t := xn(t)z . Xn=1 In the current paper, we determine a class of driving functions for which we establish the continuity of the solution, as a curve embedded in the (Sato)-Segal-Wilson Grassmannian, with respect to time. For this reason, we introduce the following class of controlled Loewner-Kufarev equations. Let us first recall from the monograph by Lyons and Qian [11, Section 2.2] a few basic notions. For a fixed T > 0, let ∆ : (s, t):0 6 s 6 t 6 T be the two-simplex. T { } Definition 3.1 ((see [11, Section 2.2])). A continuous function ω : (s, t):0 6 s 6 t< + R , { ∞} → + is called a control function if it satisfies super-additivity: ω(s,u)+ ω(u, t) 6 ω(s, t), for all 0 6 s 6 u 6 t, and vanishes on the diagonal, i.e. ω(t, t) = 0, for all t [0, T ]. ∈ Now, let V be a Banach space and X : [0, T ] V , T > 0, be a path such that → X X 6 ω(s, t), 0 6 s 6 t 6 T, | t − s| for a control function ω : ∆T R+. Then X is called a Lipschitz path controlled by ω. We have the following →
Definition 3.2. Let ω be a control function. The driven Loewner-Kufarev equation‡ (3.1) is controlled by ω if for any n N, p =1, , n and i1, , ip N with i1 + + ip = n, we have ∈ ··· ··· ∈ ··· n nx0(t) i1x0(u1) ipx0(up) ω(0, t) e e− dxi1 (u1) e− dxip (up) 6 , Z06u1< nx0(t) i1x0(u1) ipx0(up) e e− dxi1 (u1) e− dxip (up) Z06u1< ‡ Here we avoid the conflicting use of the word “controlled”, which has two meanings, cf. [11] p. 16. 7 (see [1, Definition 3.1] or [16, Section 2]). Any bounded univalent function f : D C with f(0) = 0 and ∂f(D) being a Jordan curve, is embedded into Gr via → H f W := span 1 Q f (1/z) 1 > Gr 7→ f { }∪{ n ◦ ◦ |S }n 1 ∈ (see [1, Sections 3.2 and 3.3]), where Qn is the n-th Faber polynomial associated to f. Note that f extends to a continuous function on D by Caratheodory’s Extension The- orem for holomorphic functions. Let H1/2 = H1/2(S1) be the Sobolev space on S1 endowed with the inner product 1 2 n n 1/2 given by h, g H / = h0g0 + n Z n hngn for h = n Z hnz ,g = n Z gnz H . h i ∈ | | ∈ ∈ ∈ Assume that f extends to a holomorphicP function onP an open neighbourhoodP of D. Then 1/2 span 1 Q f (1/z) 1 > H and we consider the orthogonal projection { }∪{ n ◦ ◦ |S }n 1 ⊂ H1/2 1/2 1/2 1/2 P : H W , where W := span 1 Q f (1/z) 1 > f → f f { }∪{ n ◦ ◦ |S }n 1 rather than the orthogonal projection H W . → f Let ω be a control function and let f 6 6 be a univalent solution to the Loewner- { t}0 t T Kufarev equation controlled by ω. Suppose that each ft extends to a holomorphic function on an open neighbourhood of D. With these assumptions, our main result is the following 1 Theorem 3.3. Suppose that ω(0, T ) < 8 . Then there exists a constant c = c(T ) > 0 such that P P 6 cω(s, t) k ft − fs kop for every 0 6 s < t 6 T , where is the operator norm. k•kop Thus we obtained a continuity result with respect to the time-variable of the solution embedded into the Grassmannian in which the modulus of continuity is measured by the control function ω. Let us point out that the assumption of the existence of an analytic 1 continuations of ft’s across S is extrinsic and should be discussed in detail in the future. 4. Auxiliary Estimates Along Controlled Loewner-Kufarev Equations 4.1. Controlling Loewner-Kufarev equation by its driving function. We shall begin with a prominent example of a control function as follows. Example 4.1. If y : [0, + ) C, is continuous and of bounded variation, then we have ∞ → n y 1-var(s,t) := sup yui yui−1 < + k k N | − | ∞ n ; Xi=1 s=u0 Example 4.2. Let n N and y1, ,yn : [0, + ) C be continuous and controlled by a control function ω. Then∈ we have··· ∞ → m ω(s, t)n sup dy1(u1) dyn(un) 6 N m ; Zri−16u1< In fact, we shall prove this by induction on n. The case for n = 1 is clear by definition. Consider the case for n 1. Putting ω (t) := ω(s, t), we find that the total variation − s measure dyn on [s, + ) is smaller than the Lebesgue-Stieltjes measure dωs associated with ω on| [s,| + ), in∞ the sense of dy 6 dω for any Borel set B [s, + ). s ∞ B | n| B s ⊂ ∞ Therefore we have R R ri dy1(u1) dyn(un) 6 dy1(u1) dyn 1(un 1) dyn(un) − − Zri−16u1< m m n n n ω(ri 1,ri) 6 ω(ri 1,ri) 6 ω(s, t) , − − Xi=1 Xi=1 and hence we get the above inequality. Proposition 4.2. Let ω and ω be two control functions. Let x : [0, + ) R be a 0 0 ∞ → continuous function controlled by ω0 and with x0(0) = 0. Then ω0(s,t) (i) ω′(s, t) := e (ω0(s, t)+ ω(s, t)), for 0 6 s 6 t defines a control function. Let n N and y , ,y : [0, + ) C, be continuous functions controlled by ω. Then ∈ 1 ··· n ∞ → (ii) we have n nx0(t) ω′(0, t) e dy1(u1) dyn(un) 6 . Z ··· n! 06u1< nx0(t) nx0(s) e dy1(u1) dyn(un) e dy1(u1) dyn(un) Z06u1< ω0(s,u) ω0(u,t) ω′(s,u)+ ω′(u, t) = e ω′′(s,u) + e ω′′(u, t) ω0(s,t) ω0(s,u) ω0(s,t) ω0(u,t) ω0(s,t) = e e − ω′′(s,u) + e − ω′′(u, t) . By the super-additivity and non-negativity of ω0, we have ω0(s,u) ω0(s, t) 6 0 and ω (u, t) ω (s, t) 6 0. Therefore, by using the non-negativity and super-additivity− for 0 − 0 ω′′, we get ω0(s,t) ω0(s,t) ω′(s,u)+ ω′(u, t) 6 e ω′′(s,u)+ ω′′(u, t) 6 e ω′′(s, t)= ω′(s, t). nx0(t) x0(t) x0(0) n nω0(0,t) (ii) Since x0 = 0, we have e = (e − ) 6 e . On the other hand, by Example 4.2 we have that ω(0, t)n dy1(u1) dyn(un) 6 . Z ··· n! 06u1< (iii) Let 0 6 s 6 t 6 T be arbitrary. Then we have nx0(t) nx0(s) e dy1(u1) dyn(un) e dy1(u1) dyn(un) Z06u1< Corollary 4.3. Let ω0 and ω be two control functions. Consider the Loewner-Kufarev equation (3.1) and suppose that the following two conditions hold: (i) x0 is controlled by ω0; N n n n C (ii) for every n , there exist continuous functions y1 ,y2 , ,yn : [0, T ] controlled by ∈ω such that ··· → n n n xn(t)= dy1 (s1)dy2 (s2) dyn(sn), 0 6 t 6 T. Z06s1 Corollary 4.4. Let ω be a control function and ft 06t6T be a solution to the Loewner- Kufarev equation controlled by ω. If ω(0, T ) < 1 then{ } f (D) is bounded for any t [0, T ]. 4 t ∈ 4.2. Some analytic aspects of Grunsky coefficients. Let S,S′ Z be countably ⊂ infinite subsets, and A =(ai,j)i S,j S′ be an S S′-matrix. For each sequence x =(xj)j S′ ∈ ∈ × ∈ of complex numbers, we define a sequence TAx = ((TAx)i)i S by (TAx)i := j S′ aijxj if ∈ ∈ it converges for all i S. We will still denote TAx by Ax when it is defined.P ∈ 2 Let ℓ2(S) be the Hilbert space consisting of all sequences a =(ai)i S such that i S ai < ∈ ∈ | | + , with the Hermitian inner product a, b 2 = i S aibi, for a =(ai)i S, b =P (bi)i S ∞ h i ∈ ∈ ∈ ∈ ℓ2(S). The associated norm will be denoted by P 2. For each s R, the space k•k ∈ s 2 s 2 ℓ2(S) := a =(an)n S : (1 + n ) an < + ∈ | | ∞ n Xn S o ∈ is a Hilbert space under the Hermitian inner product given by a, b := max 1, n 2sa b h i2,s { | |} n n Xn S ∈ 10 s for a =(ai)i S, b =(bi)i S ℓ2(S). The associated norm will be denoted by 2,s. Let us recall∈ a classical∈ and∈ well-known result from the theory of univalentk•k functions. For the definition and properties of Grunsky coefficients, see [19, Chapter 2, Section 2], [20, Section 2.2] or [1, Definition A.1 and Proposition A.2]. Theorem 4.5 ((Grunsky’s inequality [15, Theorem 3.2])). Let f : D C be a univalent → functions with f(0) = 0, and let (bm,n)m,n6 1 be the Grunsky coefficients associated to f. − Then for any m N and λ m,λ m+1, ,λ 1 C, it holds that ∈ − − ··· − ∈ 1 1 2 2 − − λk ( k) bk,lλl 6 | | . − ( k) kX6 1 l=Xm kX= m − − − − This can be rephrased with our notation as follows: Let B := ( m( n) b m,n)m N,n N, − − ∈ ∈− where bm,n for m, n 6 1 are Grunsky coefficients associated top a univalent function f on D such that f(0) = 0. − Corollary 4.6. (i) B : ℓ2( N) ℓ2(N) and is a bounded linear operator with the operator norm satisfying− B →6 1. k k (ii) B∗ : ℓ (N) ℓ ( N) and is a bounded linear operator with the operator norm 2 → 2 − satisfying B∗ 6 1. k k N N (iii) The bounded linear operator 1+ BtBt∗ : ℓ2( ) ℓ2( ) is injective and has a dense image. → Proof. (i) For each a =( , a 3, a 2, a 1) ℓ2( N), we have by Theorem 4.5, ··· − − − ∈ − ∞ ∞ ∞ 2 √ √ Ba 2 = nk b n, ka k nl b n, la l k k − − − − − − Xn=1 Xk=1 Xl=1 ∞ ∞ 2 ∞ √n a 2 √ n 2 = n b n, k( k a k) 6 | − | = a 2. − − − n k k Xn=1 Xk=1 Xn=1 (ii) Since the Grunsky matrix (bm,n)m,n6 1 is symmetric: bm,n = bn,m for all m, n 6 1, the assertion is proved similarly to (i). − − (iii) The injectivity is clear since the adjoint operator of B is B∗. Then the second assertion is also clear since 1 + BB∗ is self-adjoint. Remark 4.1. The semi-infinite matrix defined by B1 := (√mnb m, n)m,n N is called the − − ∈ Grunsky operator, and then the Grunsky inequality (Theorem 4.5) shows that B1 is a bounded operator on ℓ2(N) with operator norm 6 1. This operator, together with three additional Grunsky operators, are known to play a fundamental role in the study of the geometry of the universal Teichm¨uller space. For details, cf. the papers by Takhtajan- Teo [19] or Krushkal [10]. In the sequel, we fix a control function ω, and a solution ft 06t6T to a Loewner-Kufarev equation controlled by ω. We denote by b (t) for m, n{6 } 1 the Grunsky coefficients m,n − associated with ft, and Bt := m( n) b m,n(t) N N, − − m ,n p ∈ ∈− Bt∗ := ( n)m bn, m(t) N N. − n ,m p − ∈− ∈ 1 N It is clear that the linear operator (1 + BtBt∗)− : Im(1 + BtBt∗) ℓ2( ) is bounded. 1 N → Therefore by Corollary 4.6–(iii), (1 + BtBt∗)− extends to ℓ2( ) and the extension will be denoted by At : ℓ2(N) ℓ2(N). In particular, it is easy to see that At 6 1, holds for the operator norm. → k k 11 We shall exhibit the indices which parametrise our operators in order to help under- standing the following: 3 2 1 1 2 3 ··· − − − ··· ...... 1 B1, 3 B1, 2 B1, 1 . . . . . ··· − − − . . . . 2 B2, 3 B2, 2 B2, 1 ··· − − − 3 B∗ 3,1 B∗ 3,2 B∗ 3,3 B = , B∗ = − − − − ··· , 3 B3, 3 B3, 2 B3, 1 ··· − − − 2 B∗ 2,1 B∗ 2,2 B∗ 2,3 . − − − − ··· . . . . . . . . . . . . . . 1 B∗ 1,1 B∗ 1,2 B∗ 1,3 − − − − ··· 1 2 3 1 2 3 ··· ··· 1 1 ∗ ∗ ∗ ··· ∗ ∗ ∗ ··· 2 1 2 BB∗ = ∗ ∗ ∗ ··· , A =(I + BB∗)− = ∗ ∗ ∗ ··· 3 3 . ∗. ∗. ∗. ···. . ∗. ∗. ∗. ···. . . . . .. . . . . .. and 3 2 1 ··· − − − ...... 3 (B∗AB) 3, 3 (B∗AB) 3, 2 (B∗AB) 3, 1 B∗AB = . − ··· − − − − − − 2 (B∗AB) 2, 3 (B∗AB) 2, 2 (B∗AB) 2, 1 − ··· − − − − − − 1 (B∗AB) 1, 3 (B∗AB) 1, 2 (B∗AB) 1, 1 − ··· − − − − − − The following is a consequence from [1, Theorem 2.12] and will be proved in Section A.2. Corollary 4.7. Let ω be a control function, and f 6 6 be a solution to the Loewner- { t}0 t T Kufarev equation controlled by ω. Let b m, n(t), n, m N be the Grunsky coefficients − − ∈ associated to ft, for 0 6 t 6 T . Then for any 0 6 s 6 t 6 T and n, m N with n + m > 3, we have ∈ ω(0,t)2 (i) b 1, 1(t) 6 and b 1, 1(t) b 1, 1(s) 6 ω(s, t)ω(0, T ). | − − | 2 | − − − − − | (8ω(0, t))m+n (ii) b m, n(t) 6 . | − − | 16(m + n)(m + n 1)(m + n 2) − − m+n 1 ω(s, t)(8ω(0, T )) − (iii) b m, n(t) b m, n(s) 6 . | − − − − − | 16(m + n 1)(m + n 2) − − Along the Loewner-Kufarev equation controlled by ω, we obtain the following 1 6 6 Corollary 4.8. If ω(0, T ) < 8 , then for 0 s < t T , (i) B∗ B∗ = B B 6 cω(s, t), k t − s k k t − sk (ii) A A 6 2cω(s, t), k t − sk 8ω(0,T ) where c := 1 (8ω(0,T ))2 > 0. − 12 Proof. (i) By Corollary 4.7–(ii), we have ∞ ∞ 2 2 Bt Bs 6 (Bt Bs)n, m k − k | − − | Xn=1 mX=1 ∞ ∞ 2 = √nm(b n. m(t) b n, m(s)) | − − − − − | Xn=1 mX=1 ∞ ∞ √nmω(s, t)(8ω(0, T ))n+m 1 2 6 − 16(n + m 1)(n + m 2) Xn=1 mX=1 − − ω(s, t) 2 ∞ 2 8ω(0, T ) 2 6 (8ω(0, T ))2n = ω(s, t) . 8ω(0, T ) 1 (8ω(0, T ))2 Xn=1 − (ii) Since 1 1 A A =(1+ B B∗)− (1 + B B∗)− t − s t t − s s 1 1 = (1+ B B∗)− (B B∗ B B∗)(1 + B B∗)− t t s s − t t s s 1 1 = (1+ B B∗)− (B B )B∗(1 + B B∗)− t t s − t t s s 1 1 (1 + B B∗)− B (B∗ B∗)(1 + B B∗)− , − t t s t − s s s we have A A 6 B B + B∗ B∗ =2 B B . k t − sk k s − tk k s − t k k t − sk Finally, define Λ = (Λm,n)m Z,n Z by Λm,n := √m δm, n + δm,0δ0,n for m N and n N, that is, ∈ ∈ − ∈ ∈ − . .. √3 √2 √1 (4.1) Λ= 1 . √1 √2 √ 3 . .. It is clear that Λ : ℓ1/2(Z) ℓ (Z) and is a continuous linear isomorphism. 2 → 2 5. Proof of Theorem 3.3 1 Let ω be a control function such that ω(0, T ) < , and let f 6 6 be a univalent 8 { t}0 t T solution to the Loewner-Kufarev equation controlled by ω. Suppose further that ft extends to holomorphic functions on open neighbourhoods of D for all t [0, T ]. ∈ 1/2 We then note that for each t [0, T ], it holds that Q (t, f (1/z)) 1 H , where ∈ n t |S ∈ Qn(t, w) is the n-th Faber polynomial associated to ft. Therefore we have 1/2 span 1 Q (t, f (1/z)) 1 > H H. { }∪{ n t |S }n 1 ⊂ ⊂ In particular, we have 1/2 1/2 H W := span 1 Qn(t, ft(1/z)) 1 n>1 ft { }∪{ |S } H span 1 Q (t, f (1/z)) 1 > = W . ⊂ { }∪{ n t |S }n 1 ft 1/2 n n We fix an inner product on H by requiring for h = n Z hnz ,g = n Z gnz − 1/2 ∈ z n ∈ zn ∈ 1/2 ∞ N N H , that h, g H := h0g0 + n=1 n(h ng n + hngn). ThenP √n n 1P √n n h i − − { } ∈ ∪{ }∪{ } ∈ P 13 forms a complete orthonormal system of H1/2. By this, the infinite matrix Λ defined in (4.1) determines a bounded linear isomorphism H1/2 H through the identification 1/2 1/2 Z → H ∼= ℓ2 ( ). Recall that for each univalent function f : D C with f(0) = 0 and an analytic → continuation across S1, the orthogonal projection H1/2 W 1/2 is denoted by P . In → f f order to prove Theorem 3.3, we need to calculate the projection operator Pf . For this, we shall consider first the following change of basis. 1 1 Let w (z) := Q f(z− ), for z S and n N. Then we have n n ◦ ∈ ∈ w w w ( , 3 , 2 , 1 ) ··· √3 √2 √1 3 2 1 −1 −2 −3 , z , z , z , 1, z , z , z , = √3 √2 √1 √1 √2 √3 ··· ··· ...... 1 0 0 ··· (5.1) 0 1 0 ··· 0 0 1 ··· 0 0 0 × ··· √3 1 b 3, 1 √2 1 b 2, 1 √1 1 b 1, 1 ··· · − − · − − · − − √3 2 b 3, 2 √2 2 b 2, 2 √1 2 b 1, 2 ··· · − − · − − · − − √3 3 b 3, 3 √2 3 b 2, 3 √1 3 b 1, 3 ··· · − − · − − · − − . . . . . . . . . By putting 3 2 1 2 3 z z z z− z− z− z+ := , , , , z := , , , , ··· √3 √2 √1 − √1 √2 √3 ··· and e e √3 1 b 3, 1 √2 1 b 2, 1 √1 1 b 1, 1 ··· · − − · − − · − − √3 2 b 3, 2 √2 2 b 2, 2 √1 2 b 1, 2 B := ··· · − − · − − · − − , √3 3 b 3, 3 √2 3 b 2, 3 √1 3 b 1, 3 ··· · − − · − − · − − . . . . . . . . . (so we have put Bt = ((Bt)n,m)n>1,m6 1 where (Bt)n,m := n( m) bm, n(t) for n > 1 and − − − m 6 1), the equation (5.1) is written in a simpler formp as: − I w3 w2 w1 , , , =( z+ 1 z ) 0 , √ √ √ − ··· 3 2 1 B e e where ...... 1 0 0 ··· 0 1 0 ··· I 0 0 1 ··· 0 = 0 0 0 . ··· B √3 1 b 3, 1 √2 1 b 2, 1 √1 1 b 1, 1 ··· · − − · − − · − − √3 2 b 3, 2 √2 2 b 2, 2 √1 2 b 1, 2 ··· · − − · − − · − − √3 3 b 3, 3 √2 3 b 2, 3 √1 3 b 1, 3 ··· · − − · − − · − − . . . . . . . . . 14 Consider the change of basis I 0 B∗ w3 w2 w1 v1 v2 v3 − ( w 1 v ) :=: ( , , , , 1, , , , ) := ( z+ 1 z ) 0 1 0 √ √ √ √ √ √ − ··· 3 2 1 1 2 3 ··· B 0 I e e e e where we note that the matrix on the right-hand side is non-degenerate, with inverse 1 1 1 I 0 B∗ − I B∗(I + BB∗)− B 0 B∗(I + BB∗)− − − 0 1 0 = 0 1 0 . 1 1 B 0 I (I + BB∗)− B 0 (I + BB∗)− − We note that the identity I 0 B∗ I 0 B∗ ∗ I + B∗B 0 O − − 0 1 0 0 1 0 = 0 1 0 B 0 I B 0 I O 0 I + B∗B 1/2 and the fact that (z+, 1, z ) is a complete orthonormal system in H implies that − H1/2 H1/2 H1/2 =e spane 1, w , w , w , span v , v , v , { 1 2 3 ···} ⊕ { 1 2 3 ···} is an orthogonal decomposition of H1/2. 1 Let A := (I + BB∗)− . Then 1 I 0 B∗ − − (z+, 1, z )=(w, 1, v) 0 1 0 − B 0 I e e e e I B∗AB 0 B∗A − =(w, 1, v) 0 1 0 AB 0 A e e − =(w(I B∗AB) vAB, 1, wB∗A + vA) − − so that e e e e 2 2 z 1 z− (Pf (z+), 1, Pf (z )) := ( , Pf , Pf (z), 1, Pf (z− ), Pf , ) − ··· √2 √2 ··· e e =(w(I B∗AB), 1, wB∗A) − = ((z+ + z B)(I B∗AB), 1, (z+ + z B)B∗A) e − − e − I B∗AB 0 B∗A e e − e e =(z+, 1, z ) 0 1 0 − B(I B∗AB) 0 BB∗A e e − From these, we find that for n > 1, n k k z ∞ z z− Pf = (I B∗AB) k, n + (B(I B∗AB))k, n , √n √ − − − √ − − Xk=1 n k k o n k k z− ∞ z z− Pf = (B∗A) k,n + (BB∗A)k,n , √n √ − √ Xk=1 n k k o from which, the following is immediate: 15 k 1/2 Proposition 5.1. Let h = k Z hkz H . Then ∈ ∈ ∞ ∞ P zn Pf (h)= (I B∗AB) n, k√k hk +(B∗A) n,k√k h k − − − − − √n Xn=1 n Xk=1 o n ∞ ∞ z− + h0 + (B(I B∗AB))n, k√k hk +(BB∗A)n,k√k h k . − − − √n Xn=1 n Xk=1 o Denote by Bt the associated matrix of Grunsky coefficients of ft. k 1/2 Proposition 5.2. Let h = k Z hkz H . Then ∈ ∈ P (h) P (h) P ft − fs ∞ ∞ ∞ n √ √ z = (Bs∗AsBs Bt∗AtBt) n, k k hk + (Bt∗At Bs∗As) n,k k h k − − − − − − √n Xn=1 n Xk=1 Xk=1 o ∞ ∞ + B (I B∗A B ) B (I B∗A B ) √k h t − t t t − s − s s s n, k k Xn=1 n Xk=1 − ∞ n √ z− + (BtBt∗At BsBs∗As)n,k k h k , − − √n Xk=1 o so that 2 P (h) P (h) 1/2 k ft − fs kH ∞ ∞ ∞ 2 √ √ = (Bs∗AsBs Bt∗AtBt) n, k k hk + (Bt∗At Bs∗As) n,k k h k − − − − − − Xn=1 Xk=1 Xk=1 ∞ ∞ √ + Bt(I Bt∗AtBt) Bs(I Bs∗AsBs) n, k k hk − − − − Xn=1 Xk=1 ∞ 2 √ + (BtBt∗At BsBs∗As)n,k k h k . − − Xk=1 We are now in a position to prove Theorem 3.3. Proof of Theorem 3.3. By Proposition 5.2, we have 2 P (h) P (h) 1/2 6 2(I + II)+3(III + IV + V ), k ft − fs kH where ∞ ∞ 2 √ I := (Bs∗AsBs Bt∗AtBt) n, k k hk , − − − Xn=1 Xk=1 ∞ ∞ 2 √ II := (Bt∗At Bs∗As) n,k k h k , − − − Xn=1 Xk=1 ∞ ∞ 2 III := (Bt Bs)n, k√k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ IV := (BtBt∗AtBt BsBs∗AsBs)n, k k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ V := (BtBt∗At BsBs∗As)n,k k h k . − − Xn=1 Xk=1 Estimate of I. 16 ∞ ∞ 2 √ I = (Bs∗AsBs Bt∗AtBt) n, k k hk − − − Xn=1 Xk=1 We shall note that As At = At[(I + BtBt∗) (I + BsBs∗)]As = At(BtBt∗ BsBs∗)As and hence we obtain the− following identity: − − B∗A B B∗A B s s s − t t t =(B∗ B∗)A B + B∗(A A )B + B∗A (B B ). s − t s s t s − t s t t s − t According to this decomposition, I can be estimated as I 6 3(I1 + I2 + I3), where 2 ∞ ∞ √ I1 := ((Bs∗ Bt∗)AsBs) n, k k hk , − − − Xn=1 Xk=1 2 ∞ ∞ √ I2 := (Bt∗(As At)Bs) n, k k hk , − − − Xn=1 Xk=1 ∞ ∞ 2 √ I3 := (Bt∗At(Bs Bt)) n, k k hk . − − − Xn=1 Xk=1 Each of which is estimated as follows: By Corollaries 4.6 and 4.8, we have 2 2 2 2 2 I 6 B B A B Λh 6 c ω(s, t) h 1/2 . 1 k s − tk k s sk k kH 11 k kH for some constant c11 > 0. Similarly, we have 2 2 2 2 2 2 I 6 B∗ A A B Λh 6 c ω(s, t) h 1/2 , 2 k t k k s − tk k tk k kH 12 k kH 2 2 2 2 2 I 6 B A B B Λh 6 c ω(s, t) h 1/2 3 k t tk k s − tk k kH 13 k kH for some constants c12,c13 > 0. Combining these together, we obtain 2 2 I 6 c ω(s, t) h 1 2 1 1 k kH / for some constant c1 > 0. Estimate of II. ∞ ∞ 2 √ II = (Bt∗At Bs∗As) n,k k h k − − − Xn=1 Xk=1 According to the identity B∗A B∗A =(B∗ B∗)A + B∗(A A ), t t − s s t − s t s t − s we estimate II as II 6 2(II1 + II2), where 2 ∞ ∞ √ II1 = ((Bt∗ Bs∗)At) n,k k h k , − − − Xn=1 Xk=1 ∞ ∞ 2 √ II2 = (Bs∗(At As)) n,k k h k . − − − Xn=1 Xk=1 17 By Corollaries 4.6 and 4.8, we have 2 2 2 2 2 II 6 B∗ B∗ A Λh 6 c ω(s, t) h 1/2 , 1 k t − s k k tk k kH 21 k kH 2 2 2 2 2 II 6 B∗ A A Λh 6 c ω(s, t) h 1/2 , 2 k s k k t − sk k kH 22 k kH for some constant c21,c22 > 0. Therefore we have obtained 2 2 II 6 c ω(s, t) h 1 2 2 k kH / for some c2 > 0. Estimate of III. ∞ ∞ 2 III = (Bt Bs)n, k√k hk − − Xn=1 Xk=1 is estimated by using Corollary 4.8 as 2 2 2 2 III 6 B B Λh 6 c ω(s, t) h 1/2 k t − sk k kH 3 k kH for some constant c3 > 0. Estimate of IV . 2 ∞ ∞ √ IV = (BtBt∗AtBt BsBs∗AsBs)n, k k hk − − Xn=1 Xk=1 Along the decomposition B B∗A B B B∗A B t t t t − s s s s =(B B )B∗A B + B (B∗A B B∗A B ) t − s t t t s t t t − s s s =(B B )B∗A B + B (B∗ B∗)A B + B B∗A B (B∗ B∗)A B t − s t t t s t − s t t s s s s s − t t t + B B∗A (B B )B∗A B + B B∗A (B B ), s s s s − t t t t s s s t − s the quantity IV is estimated as IV 6 5(IV1 + IV2 + IV3 + IV4 + IV5), where ∞ ∞ 2 √ IV1 = ((Bt Bs)Bt∗AtBt)n, k k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ IV2 = (Bs(Bt∗ Bs∗)AtBt)n, k k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ IV3 = (BsBs∗AsBs(Bs∗ Bt∗)AtBt)n, k k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ IV4 = (BsBs∗As(Bs Bt)Bt∗AtBt)n, k k hk , − − Xn=1 Xk=1 ∞ ∞ 2 √ IV5 = (BsBs∗As(Bt Bs))n, k k hk . − − Xn=1 Xk=1 By using Corollaries 4.6 and 4.8, it is easy to see that 2 2 IV 6 c ω(s, t) h 1 2 for i =1, 2, 3, 4, 5, i 5i k kH / for some constants c51,c52,c53,c54,c55 > 0. Therefore we get 2 2 IV 6 c ω(s, t) h 1/2 5 k kH 18 for some constant c5 > 0. Estimate of V . ∞ ∞ 2 √ V = (BtBt∗At BsBs∗As)n,k k h k − − Xn=1 Xk=1 Along the decomposition B B∗A B B∗A t t t − s s s =(B B )B∗A + B (B∗ B∗)A + B B∗(A A ), t − s t t s t − s t s s t − s the quantity V is estimated as V 6 3(V1 + V2 + V3), where 2 ∞ ∞ √ V1 = ((Bt Bs)Bt∗At)n,k k h k − − Xn=1 Xk=1 ∞ ∞ 2 √ V2 = (Bs(Bt∗ Bs∗)At)n,k k h k , − − Xn=1 Xk=1 ∞ ∞ 2 √ V3 = (BsBs∗(At As))n,k k h k . − − Xn=1 Xk=1 By Corollaries 4.6 and 4.8, we have 2 2 2 2 2 V 6 B B B∗A Λh 6 c ω(s, t) h 1 2 , 1 k t − sk k t tk k kH 51 k kH / 2 2 2 2 2 2 V 6 B B∗ B∗ A Λh 6 c ω(s, t) h 1/2 , 2 k sk k t − s k k tk k kH 52 k kH 2 2 2 2 2 V 6 B B∗ A A Λh 6 c ω(s, t) h 1/2 , 3 k s s k k t − sk k kH 53 k kH for some constants c51,c52,c53 > 0. Combining these estimates, we get 2 2 V 6 c ω(s, t) h 1/2 . 5 k kH for some constant c5 > 0. Now by combining the estimates for I, II, III, IV and V , we obtain the assertion. Appendix A. n A.1. Proof of Corollary 4.4. We write ft(z) as ft(z)= C(t) n∞=1 cn(t)z , and then it is n enough to show that supz D n∞=1 cn(t)z < + , for each t P[0, T ]. By [1, Theorem 2.8], ∈ | | ∞ ∈ P n ω(0, t)n cn(t) 6 w(n)i1, ,ip , | | ··· n! Xp=1 i1, X,ip N: ··· ∈ i1+ +ip=n ··· e where w(n)i1, ,ip = (n i1)+1 (n (i1 + i2))+1 (n (i1 + + ip 1))+1 ··· { − }{ − }···{ − ··· − } 6 (n 1)+1 (n 2)+1 (n (p 1))+1 e { − }{ − }···{ − − } n 1 n 1 = n(n 1) (n p)= n − p! 6 n2 − p!. − ··· − p 19 Therefore, n n n 1 ω(0, t) c (t) 6 n2 − p! 1 | n | n! Xp=1 i1, X,ip N: ··· ∈ i1+ +ip=n ··· n n n 1 ω(0, t) n 1 = n2 − p! − n! p 1 Xp=1 − n n 1 n n 1 6 n2 − ω(0, t) − p 1 Xp=1 − n 1 n 1 n = n4 − ω(0, t) =4− n(4ω(0, t)) . n 1 ∞ 6 ∞ Hence, supz D n=1 cn(t)z n=1 cn(t) < + if ω(0, T ) < 4 . ∈ | | | | ∞ P P A.2. Proof of Corollary 4.7. (i) is immediate from Definition 3.2 since b 1, 1(t) is explicitly given by − − t 2x0(t) 2x0(u) b 1, 1(t)= e e− dx2(u) − − − Z0 (see [1, Proposition 2.10–(ii)]). (ii) Since the Loewner-Kufarev equation is controlled by ω, we have that t n+m (n+m)x0(t) (p + q)! ω(0, t) e ((xip xi2 xi1 ) ¡ (xjq xj2 xj1 ))(u)dxk(u) 6 Z0 ··· ··· p!q! (n + m)! (for the notation ¡, see [1, Definition 2.11]) if k +(i1 + + ip)+(j1 + + jq)= n + m. Then by [1, Theorem 2.12], we get ··· ··· b m, n(t) | − − | n+m 2 m i n j − − − (p + q)! ω(0, t)n+m 6 wi1, ,ip;j1, ,jq ··· ··· p!q! (n + m)! Xk=2 16Xi wi1, ,ip;j1, ,jq =(m i1)(m (i1 + i2)) (m (i1 + + ip)) ··· ··· − − ··· − ··· (n j )(n (j + j )) (n (j + + j )) × − 1 − 1 2 ··· − 1 ··· q 6 i(m 1)(m 2) (m (p 1)) − − ··· − − j(n 1)(n 2) (n (q 1)) × − − ··· − − m 1 n 1 = ij − (p 1)! − (q 1)!. p 1 − q 1 − − − 20 In the second and third term, we have (n (j1+ +jq)) = k m,(m (i1+ +ip)) = k n, so that − ··· − − ··· − n 1 w∅;j1, ,jq 6 (k m) − (q 1)!, ··· − q 1 − − m 1 wi1, ,iq;∅ 6 (k n) − (p 1)!. ··· − p 1 − − So we have b m, n(t) | − − | n+m 2 m i n j − − − m 1 n 1 6 ij − − (p 1)!(q 1)! p 1 q 1 − − Xk=2 16Xi Xk=2 16Xi (p+q)! p+q m+n 2 In the first term of the last equation, we have 6 2 6 2 − , (p 1)!(q 1)! 6 p!q! − − (p + q 1)! 6 (m + n 3)!, and − − m i 1 n j 1 1= − − , 1= − − . p 1 q 1 i1, X,ip N: j1, X,jq N: ··· ∈ − ··· ∈ − i1+ +ip=m i j1+ +jq=n j ··· − ··· − 21 For the second and the third terms, we have (q 1)! 6 (n+m 3)! and (p 1)! 6 (n+m 3)! respectively. Then we obtain − − − − b m, n(t) | − − | m+n 2 n+m 2 (m + n 3)!2 − − 6 ω(0, t)n+m − ij (n + m)! n Xk=2 16Xi m i 1 − − m 1 m i 1 2(m 1) i − − − = − − , p (m i 1) p m 1 pX′=0 ′ − − − ′ − n j 1 − − n 1 n j 1 2(n 1) j − − − = − − q (n j 1) q n 1 qX′=0 ′ − − − ′ − give b m, n(t) | − − | m+n 2 n+m 2 (m + n 3)!2 − − 2(m 1) i 2(n 1) j 6 ω(0, t)n+m − ij − − − − (n + m)! m 1 n 1 n Xk=2 16Xi n j 1 n+1 We shall use here the identity (n +1 j) − = which enables us to conclude j=k − k 1 k+1 P − m 1 2(m 1) − 2(m 1) i − j 1 2m 1 i − − = (2(m 1)+1 j) − = − . m 1 − − m 1 m Xi=1 − jX=m − 22 Hence, we conclude that b m, n(t) | − − | m+n 2 (m + n 3)!2 − 2m 1 2n 1 2n 1 2m 1 6 ω(0, t)m+n − − − + − + − (m + n)! n m n n m o m+n 2 (m + n 3)!2 − 2m 1 2n 1 6 ω(0, t)m+n − − +1 − +1 (m + n)! n m on n o ω(0, t)m+n2m+n 2(22m 122n 1) (8ω(0, t))m+n 6 − − − = . (m + n)(m + n 1)(m + n 2) 16(m + n)(m + n 1)(m + n 2) − − − − (iii) Since the Loewner-Kufarev equation is controlled by ω, we have t u (n+m)x0(t) kx0(v) e d((xip xi2 xi1 ) ¡ (xjq xj2 xj1 ))(u) e− dxk(v) Z0 ··· ··· Z0 s u (n+m)x0(s) kx0(v) e d((xip xi2 xi1 ) ¡ (xjq xj2 xj1 ))(u) e− dxk(v) − Z0 ··· ··· Z0 n+m 1 (p + q)! ω(s, t)ω(0, T ) − 6 (n + m) . p!q! (n + m)! Now the remaining case is the same as (i). Acknowledgment. The first author was supported by JSPS KAKENHI Grant Number 15K17562. The second author was partially supported by the ERC advanced grant “Non- commutative distributions in free probability”. Both authors thank Theo Sturm for the hospitality he granted to T.A. at the University of Bonn. T.A. thanks Roland Speicher for the hospitality offered him in Saarbr¨ucken. R.F. thanks Fukuoka University and the MPI in Bonn for its hospitality, Roland Speicher for his support from which this paper substantially profited. References [1] Amaba, T. and Friedrich, R. 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