Stoilow Factorization of the Heisenberg Group

It is known, see, for example, [1], that WiOc-homeomorphism f : Q ^ Q', Q, Q' C C, is K -quasiconformal if and only if, when f (z) = ^f (z) a. e. z G Q,                          (1)

∂z ∂z where д = ^(z) (the Beltrami coefficient) is a bounded measurable function satisfying

„ „ . K - 1 , ^ д ^ м C < 1

K +1

The equation (1) is called the Beltrami equation . Next theorem characterizes all solutions of the Beltrami equation.

• ^# The work was supported by the Russian Science Foundation, project № 24-21-00319. © 2025 Dorokhin, D. K.

Theorem 1 (Stoilow Factorization). Let a homeomorphism f (z) E Wo2 be a solution of the Beltrami equation (1) and |^(z)| C k < 1- Also let g(z) E W12 be any other solution of the Beltrami equation (1). Then there exists a holomorphic function Ф : Q‘ ^ C such that

g(z) = Ф(f (z)), z E Q- (2)

Conversely, if Ф is holomorphic on Q‘, then the composition Ф о f is W102-solution of the equation (1) on Q.

Initially, S. Stoilow in the paper [2] showed that if there is a continuous, open mapping f between two Riemann surfaces S and S ^′ for which the preimage of any point is a totally disconnected set, then there exists the Riemann surface S and a homeomorphism h : S ^ S such that f о h ^-1 : S ^ S ‘ is a holomorphic mapping. Theorem has significant generalizations in 2D analysis: every open and discrete map h is topologically equivalent to some analytic function, so h = ^ о f , where f is some homeomorphism, ^ is some holomorphic function (mapping h : Q ^ C is called discrete, if set h ^{- 1} (w) is discrete for every w E C). Y. G. Reshetnyak proved [3] that every mappings with bounded distortion are open and discrete; on the other hand, the quasiconformal mappings are homeomorphic mapping with bounded distortion.

The Beltrami equation has importance for isometrical coordinates on the 2-dimensional manifolds and also in the calculus of variations in minimizing the energy functional for homeomorphisms acted from a unit disk to defined domain [1].

Quasiconformal maps on non-Riemannian structures were first considered by G. D. Mostov because of the classification of metric spaces of constant negative curvature [4]. To prove the rigidity theorem, G. D. Mostow needed quasiconformal transformations of the ideal boundary of some symmetric space [5]. M. L. Gromov, using the Gromov–Hausdorff convergence, proved [6– 8] that the geometry of such an ideal boundary is modeled by a nilpotent group with the Carnot–Caratheodory metric. This was one of the incentives for studying quasiconformal maps on the Carnot groups and more general Carnot spaces [8, 9]. In the paper [10] P. Pansu introduced the concept of differentiability of mappings “in terms” of the Carnot–Caratheodory metric ( P -differentiability). Using the concept of P -differentiability, A. Koranyi and H. M. Reimann [11] systematized analytical methods for studying mappings on the Heisenberg groups H ⁿ , assuming a priori P -differentiability of mappings almost everywhere. The analytical apparatus that allows one to develop the theory of quasiconformal mappings on the Carnot groups under minimal assumptions was developed by S. K. Vodopyanov and his students, see [12–16]. A. Koranyi and H. M. Reimann [11, 17] showed that on the Heisenberg groups quasiconformal mappings can be defined in terms of systems of differential equations similar to the classical Beltrami equations on the plane. It should be noted separately that in the Euclidean case the Beltrami equations exist only in dimension 2; on the Heisenberg groups H ⁿ an analogue of the Beltrami equation exists in all dimensions [11, 17]. For strictly pseudoconvex hypersurfaces the theory of the Beltrami equations was constructed in [18].

In our paper, we proved an analogue of the Stoilow factorization theorem on the first Heisenberg group H1 (Theorem 4). Our proof is based on the use of a formula for the Beltrami coefficient of a composition of quasiconformal mappings and an analogue of the Liouville theorem proved by D. V. Isangulova in the paper [19]. To do this, we derive the notation forms of the corresponding differential operators from the composition of quasiconformal mappings (Lemma 1), using which we calculate the corresponding Beltrami coefficient for compositions of quasiconformal mappings (Lemma 2). The last section of the paper is devoted to the following problem: let two quasiconformal mappings with a common Beltrami coefficient be given; will the images of the corresponding Brownian motions then be equivalent, that is, will these processes be expressed through each other by means of a time change (preserve their tra jectories)? We have proven the following

Theorem 2. Let N _t and M _t be quasi-Brownian motions on the Heisenberg group H ¹ with corresponding quasiconformal maps f and g whose Beltrami coefficients are equal almost everywhere. Then, if the map g о f - 1 is a composition of dilations, rotations, and translations on the Heisenberg group H ¹ , then there exists a time change a(t) such that almost surely N t = M a ( t ) .

2.    Basic Definitions and Known Theorems

The first Heisenberg group H ¹ [11, 17, 19, 20] defined in the standard Euclidean space R ³ with the coordinate system (x,y,t) induced by the coordinate frame (O,e i ,e 2 ,e 3 ) , using the following table of commutators

\ [e i ,e 2 ] = - 4е з ,

([е 1 ,е з ] = [е 2 ,е з ] = 0.

Using the Baker–Campbell–Hausdorff formula and the table (3), we obtain an analytical expression of the left translation of w i * W 2 an arbitrary element W 2 = (x 2 ,y 2 ,t 2 ) G H ¹ and an other arbitrary element w i = (x i ,y i ,t i ) G H a :

w i * W 2 = (x i + X 2 , y i + У 2 , t i + t 2 - 2x i y 2 + 2y i X 2 ).

Using (4), we obtain expressions for the basis of left-invariant vector fields (the Jacobi basis [20]) of the Lie algebra V of the group H ¹ at each point (x,y,t) :

-

∂∂

2x dt, T = at'

X = T + 2У^, Y = F

∂x ∂t ∂y

Left-invariant vector fields X, Y are horizontal. We denote by V 1 the tangent bundle induced by horizontal vector fields X, Y . We have V = V i ® V 2 , where V 2 as tangent subbundle, induced “vertical” field T [11, 20].

We will also use the representation of the Heisenberg group. H ¹ as (z,t) G C x R [11, 17]. Then the group operation is defined as

(z i ,t i ) * (Z 2 ,t2) = (z i + Z 2 , t i + t 2 - 2Im(z i • Z 2 )) ,

where z i = x i + iy i , Z 2 = X 2 + iy 2 .

Action is a one-parameter stretching group b _s : H ¹ ^ H ¹ , s > 0 , set by the law 8 _s (z, t) = ssz, s²^.

Homogeneous norm [17] (the Koranyi norm) p(z,t) = ( | z | ⁴ +1 ² ) ¹ / ⁴ sets the metric on H ¹ : P (( z i >t i ⁾> ^(z 2 ^,t 2 ⁾ = P (^{( z} 2 ^,t 2 ^{) - i} * (z i ,t i⁾) .

Definition 1 [19]. Let Q C H ¹ be an open set. Sobolev space W ^1,q (Q) , where 1 <  q < от , consists of locally integrable functions f : Q ^ R, having weak derivatives Xf and Yf , with norm

Ilfllw 1,q(Q) = IlfIIl«(Q) + ll^hf \\lq(q) < от, where Vhf = (Xf, Yf) is a horizontal gradient. We will denote f G W^cq(^), if f € W1,q(U) for any open set U such that U C Q.

Definition 2 [19, 20]. A mapping f : Q ^ H ¹ belongs to W l^c q (Q, H ¹ ) , if f G L ^q (Q, H ¹ ) and the satisfies:

• (A)    for any w G H a function [f] _w : x G Q ^H P(f (x),w) belongs to W lOc q (Q) ;

• (B)    a family of functions V h [f] _w has a majority belongs to L qo^ Q).

Recall that the differential 1-form т sets the contact structure on a (2n + 1)-dimensional manifold, if т Л (dT)n = 0. On the Heisenberg group H1 the contact structure is determined by the form т = 2xdy — 2ydx + dt.                              (6)

We have ker т = V ₁ .

Definition 3 [17]. A diffeomorphism f : U ^ U‘, U,U‘ C H1, will be called a contact map if it preserves the contact structure, that is f *т = Хт,

where Х = 0 is some function.

Condition (7) can be written in the following equivalent form

( —2f 2 Xf 1 + 2f\Xf ₂ + Xf 3 = 0,                          (8)

I - 2f 2 Yf 1 +2f 1 Yf 2 + Yf ₃ = 0.

Mapping of the Sobolev space f : U ^ U ‘ , U, U ‘ C H ¹ are weak contact, that is, the conditions (8) are fulfilled for them almost everywhere (see, for example, [19]).

For mappings f = (f ₁ ,f ₂ ,f 3 ) : Q ^ H ¹ of the Sobolev space W^ c q (Q, H ¹ ) we define the formal horizontal differential D _h f as a matrix

D f = (Xf 1 Yf 1 }

D h f    XXYf₂) •

It follows from general facts, see, for example, [21], that the horizontal differential D h f generates a linear mapping Df : V ^ V , called a formal differential, which is a homomorphism preserving the grading of the Lie algebra V . The determinant of the matrix Df ( x ) will be called the formal Jacobian of the mapping f and denoted by J f (x) .

Definition 4 [19, Definition 1]. The homeomorphism f : Q ^ H ¹ defined on an open set Q C H, is a quasiconformal mapping if f G W i0c4 (Q; H ¹ ) , and there is a constant K ^ 1 , such that the inequality

W D h f (x) ^ 4 <  KJ f (x)                                   (9)

for almost any x G Q . Consider the following vector fields

Z = 1(X — iY ) and Z = | (X + iY ).                   (10)

Definition 5 [11, Definition 2]. The homeomorphism f = (f ₁ , f 2 , f'3) : Q ^ H ¹ is defined on an open set Q C H ¹ , is a quasiconformal mapping, if f is absolutely continuous, P -dif-ferentiable almost everywhere, preserves orientation and almost everywhere on Q satisfies the Beltrami equation

Zf i = ^Zf i ,

Zf ii = pZfu,

where f i = f i + if 2 and f ii = f s + i | f i | 2 , p is some measurable function, such that ||^ W _M < 1 .

These two definitions of a quasiconformal mappping are equivalent, see [11, 21].

We have [22]

D_hf = f f , ^h      Zf I Zf I

W D h f W = | Zf i | + | Zf i | ,

Zf I Zf I

J f = det I Zf Zf \ 0    0

0 _ I = (| Zf i | 2 - | Zf i | 2 ) 2 . | Zf i | 2 -| Zf i |7

Remark 1. If the Beltrami coefficient of the quasiconformal mapping f is zero almost everywhere, then almost everywhere it holds Zf i = 0 . Then the inequality (9) has the form

| Zf i | 4 c K i Zf i l ⁴ .

This is true for all K ^ 1 , i. e. f is a 1 -quasiconformal (conformal) mapping.

Theorem 3 [19, p. 326] . Any (not necessarily orientation-preserving) 1-quasiconformal map on the Heisenberg group H ¹ is represented as a composition of the following type of mappings:

• (1)    Left translation na(x) = a * x, a G H;

• (2)    Dilation S_s(x) = (sz,s²1), s > 0;

• (3)    Rotation ф_а(х) = (е^гаz,t), a G R;

• ⁽⁴⁾    Inversion ^j(x) = Izi2z-—_t, p^-^;

• (5)    Reflection i(x) = (z,t).

3.    Stoilow Factorization on the Heisenberg Group H1

Lemma 1. Let Q, Q ' , Q '' be domains of H ¹ , g : Q ^ Q ' and h : Q ' ^ Q '' are quasiconformal mappings. Then

z (h i ⁰ g) = (Zh i ⁰ g)Zg i + ( Zh i ◦ g)Zg i ,

Z (h i 0 g) = (Zh i 0 g)Zg i + ( Zh i 0 g^Zg i .

⊳ We have

z (h i ° g) = 2 ( x - iY)(h i ° д') = 2 ( [ x (h i ° g) + y (h x ° g)] + i [ x (h x ° g) - y (h i ° g)] )

h i x ° g)g i x +⁽h i y ° g)g x x +⁽h i t ° g)g 3 x +2y (⁽h i x

° 9)9 i t +⁽h i y ° g^g x 1 +⁽h i t ° g)g 3 1)]

+ [(h 2 x ° 9)9 1 y + (h 2 y ° g)g 2 y + (h 2 t ° g)g 3 y - 2x ( hh _2x ° g)g i _t + (h x y ° g)g 2 1 + (h x t ° g)g 3 1)] + i [^(h 2 x ° g)g i x + ⁽h x y ° g^ x x + ⁽h x 1 ° g^ x +²y (hh_2x ° g)g i _t +^(h 2 y ° g)g _{2 t} +⁽h 2 t ° g"^)] ^- i [ ⁽h i x ° g)g i y +⁽h i y ° g)g x y +⁽h i t ° g)g 3 y ^- 2x ( ⁽h i x ° g)g i _t +⁽h i y ° g)g x t +⁽h i t ° g)g 3 t )]

= -[combine them g i_x + 2yg i_t and g iy — 2xg i_t into Xg i and Yg i , accordingly, for all i = 1,2, 3} = 2 [ (h i x ° g)Xg i +(h i y ° g)Xg x +(h i t ° g)Xg 3 ] + [ (h x x ° g)Yg i +(h x y ° g)Yg x +(h x t ° g)Yg3] + i (h x _{2 x}° g)Xg i + (h x y ° g)Xg x + (h x t ° g)Xg3] — i(h i x °° g)Yg i + (h i y ° g)Yg x + (h i t ° g)Yg3\ = { by the condition of contact, we will Xg 3 and Yg 3 ^ = 2 ([ (Xh i ° g)Xg i + (Yh i ° g)Xg x ] + [ (Xh x ° g)Yg i + (Yh x ° g)Yg x] + i [ (Xh x ° g)Xg i + (Yh x ° g)Xg x ] — i [ (Xh i ° g)Yg i + (Yh i ° g)Yg x ] = 2 ([ (Xh i ° g)Xg i + (Yh i ° g)Xg x ] — i [ (Xh i ° g)Yg i + (Yh i ° g)Yg x ]) = (Xh i ° g)Zg i + (Yh i ° g)Zg x = (( Z + Z ) h i ° g ) Zg i — i (( Z — Z )) h i ° g^Zg x = (Zh i ° g)Zg i + ( Zh i ° g ) Zg i .

Similarly:

Z(h i ° g) = 2 (X + iY)(h i ° g) = ^ ([ X(h i ° g) — Y(h x ° g) ] + i [ X(h x ° g) + Y(h i ° g) ]) = (Zh i ° g)Zg i + (Zhi ° g^Zg i . >

Lemma 2. Let f : Q ^ Qi and g : Q ^ Qx be quasiconformal mappings on the Heisnberg groups H1 with Beltrami coefficients µf and µg , accordingly. Then the composition f ° g i : Qx ^ Qi is an quasiconformal mapping with the Beltrami coefficient

Zg i h f ^— h g h f _O g-1 ° g = = • i-----—

.

Zgi ^{1 —} h f h g

< A mapping f ° g ⁱ is quasiconformal (see, for example, [11]). Let h = f ° g ⁱ , then f i = h i ° g . Using the Lemma 1 to Zf i and Zf i , we get

Zf i = (Zh i ° g)Zg i + (Zh i ° g)Zg i ,

Then

Zf i = (Zh i ° g)Zg i + ZZhi ° g)Zg i .

Zf i Zg i — Zf i Zg i

Zhi ° q = ---=----=----- ,

Zg i Zg i — Zg i Z^

From here we find h h ° g

—        ZfZg i — Zf i Zg i

Zhi ° g =--₌----₌----- .

Zg^g i — ZgZg i

h h ° g = —

ZfiZg i — ZfZg i ZfiZg i    ¹

Zf i Zg i — Zf i Zg i = Zg i h f — h g

—

µ f µ _g

Zg i ^{1 —} h f h g

.

Note that 1 — g f g g = 0 almost everywhere, see Definition 5. The product Zf i Zg is also non-zero almost everywhere. Indeed, if Zf i = 0 , then using the Beltrami equation for f , we get that Zf i = 0 , from which J f = 0 , see Remark 1. However, J f =0 is almost everywhere [21, Theorem 4]. Thus, Zf i =0 is almost everywhere. Similar arguments work for Zg _I . ⊲

As a result, we obtain the expression for the Beltrami coefficient of the inverse mapping (in the Lemma 2, we need to put f (x) = x ):

^ g -i ^о g =

—

Zg I

-7- • ^ g • Zg _I

Theorem 4. Let f : Q ^ Qi and g : Q ^ Q2 be quasiconformal mappings on the Heisenberg group H1 with the Beltrami coefficients µf and µg accordingly. The following conditions are equivalent:

• (1)    gf = gg almost everythere on Q;

• (2)    There is an quasiconformal mapping h : Q2 ^ Qi so that f = h о g.

<1 (1) ^ (2) By the Lemma 2, we consider mappings with almost everywhere equal the Beltrami coefficients. We obtain that the Beltrami coefficient of the function h = f о g ^{— 1} is zero and, therefore, h is a conformal function such that f = h о g , see Remark 1.

• (2)    ^ (1) Let f о g¹ = h. Then by the Lemma 2 we get

  ZgI

  ⁰ = gfeg-¹⁰g = ^

  Zg_I

  
  

  gf^—gg

  ^{1 —}gfg_g •

  
  
  
  
4.    Examples 4.1.    Invariance of the Beltrami coefficient with respect to the conformal mapping. Now we check that the composition from the left to the conformal mapping does not change the Beltrami coefficient of the mapping. Consider the functions g = (tz,t3/3), h = j оg = (3z/(3t|z|2 — it2), —3/(9t|z|4 +13)), where j is the inverse of the Heisenberg group.

Therefore, g f = g g almost everywhere. >

To demonstrate the results obtained, let us consider a number of examples.

Now µ _g :

ZgI       izz

^{g g} = Zg l = - t •

We check that µ _g and µ _h are equal almost everywhere:

—              — 9 tzz + i 3 zz (3 |z| ² — 2 it )

_ Zh i _           (3 t|z| ² —it ² ) ²

^g h = Zhi = 9 t|z| ² — 3 it ² — 9 tzz—i 3 zz (3 |z| ² — 2 it )

I                      (3 t|z| ² —it ² ) ²

— 3tzz + izz(3 | z | ² — 2it)

—izz(3|z| ² + it)            izz

-----------—--------:-------- —--— . (t + i|z| ² )(3|z| ² + it)       t + i|z| ²

— it² — izz(3 | z | ² — 2it)

• 4.2.    The Beltrami coefficient for the inverse function. We show how the Beltrami coefficient for the inverse mapping is expressed in terms of the Beltrami coefficient of the original mapping. Consider the mappings h = (tz, t ³ /3) and h ^{— 1} = (z/ 3/3t, 3/3t ) . Now g h and µ _h -1

• 4.3.    The Beltrami coefficient of the composition of the mappings. Now we show how the Beltrami coefficient for the composition of mappings can be expressed in terms of the Beltrami coefficients of the original mappings. Consider the mappings h = (2z + z, 3t) , g = (tz,t³/3) and g ^{- 1} = (z/(3t) ^{1 / 3} , (3t) ^{1 / 3} ) . Now P h , P g and P_g- i

Zhi       izz             Zh—1      izz gh = Zhi = — t + i|z|2 ,  gh-1 = Zh—1 = 3t — i|z|2 •

On the other hand, by the Lemma 2

µ h - 1

Z (h 1 i )       Z Zhi   \ i-1    t+ + izz izz \     1       izz Z (h - 1 i )       \Zh I Ph)           \t — izz t + i \ z \ 2 J           3t — i \ z \ 2 ’

Zhi   1         Zgi       izz              Zgi 1 izz

^ h = Zhi = 2 ^{, P g} = Zg | = — t + i | z | ^{2 , P g -}1 = Zg ^{- 1} = 3t - i | z | ² ’

Consider the Beltrami coefficient for f = h о g = (2tz + tz, — t³) :

Zfi    t — iz(2z + z)

Pf   Zfi   2t + iz(2z + z)

By the Lemma 2,

_ Z(hi о g) _ / Zg-Z(hi о g) Zg-T

P h P g ¹ \         (^{(3t) 3    (3t) 3}

^{1 -} P h P g-1)       \(3t) - 3 +(3t) - 3

• i zz

• izz

1 izz

2 ~ 3 t-i|z| 2

1 izz

^L ■ 2 3 t + i|z| ²

)

◦ g

A

3t — izz

3t + izz

3 t-i\z\ ² - 2 izz 3 t-ilzl 2 6 t +2 i|z| 2 + i zz 3 t + i|z| ²

)

= t3 — it2|z|2 — 2it2zz g    2t3 + 2it2\z\2 + it2 zz

t — iz(2z + z) 2t + iz(2z + z)

5.    Quasi-Brownian Motion on the Heisenberg Group H1

The property of conformal invariance of the Brownian motion is known: on the plane, the conformal image of the Brownian motion is the Brownian motion with modified time, and in R ⁿ this is possible only if the mapping is a harmonic morphism [23] (see also [24]). A similar result for mappings on the Heisenberg group H ¹ was obtained in [25]. Then the question of describing random processes invariant with respect to quasiconformal mappings of the Heisenberg group is of interest. On the plane, such processes are described in the dissertation [26].

Definition 6 [27]. Let X _t and Y _t be independent standard one-dimensional Brownian motions, and S t = 2 ft(Y s dX_s — X _s dY s ) . The random process M t = (X _t , Y _t , S t ) will be called the horizontal Brownian motion in the Heisenberg group H ¹ .

Definition 7. The process A _t will be called the quasi-Brownian motion on the Heisenberg group H ¹ if there exists a quasiconformal mapping f such that f(A t ) is the horizontal Brownian motion.

• ⊳    Proof of Theorem 2. By the definition of the quasi-Brownian motion

g(N ) = (g о f-1)(Bt), where Bt is the Brownian motion on the Heisenberg group. By Theorem 4.1 of [25] the mapping g of-1 will preserve the trajectories of the Brownian motion if and only if g of-1 is a harmonic morphism on the Heisenberg group, that is, if it is a composition of dilations, rotations, or translations. Note that by Theorem 4 the mapping g о f-1 is conformal and orientationpreserving. Then we get that (g о f-1)(Bt) = Ba(t), where Ba(t) is another independent of Bt Brownian motion with changed time a(t) and Nt = Ma(t). >

Acknowledgement. The author expresses his deep gratitude to S. K. Vodopyanov, D. V. Isan-gulova, N. A. Evseev for attention to the author’s results and help.