Beltrami Equations on Rossi Spheres

Abstract

Beltrami equations ${\overline{L}}_t\left(g\right)=\mu (·,t)L_t\left(g\right)$ on $S^3$ (where $L_t$, $\left|t\right|<1$, are the Rossi operators i.e., $L_t$ spans the globally nonembeddable CR structure $\mathcal{H}\left(t\right)$ on $S^3$ discovered by H. Rossi) are derived such that to describe quasiconformal mappings $f:S^3\to N\subset {\mathbb{C}}^2$ from the Rossi sphere $\left(S^3,\mathcal{H}\left(t\right)\right)$. Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions $g_t$ such that $g_t-v\in W_F^{1,2}\left(S^3,\theta \right)$ with $v\in {\mathrm{CR}}^{\infty }\left(S^3,\mathcal{H}\left(0\right)\right)$.

1. Introduction and Statement of Main Result

Let M be a 3-dimensional nondegenerate CR manifold, equipped with the CR structure $\mathcal{H}$. The global CR embedding problem for M is to find a nondegenerate real hypersurface $N\subset {\mathbb{C}}^2$ and a CR isomorphism of $(M,\mathcal{H})$ onto $(N,T_{1,0}\left(N\right))$, where

$$T_{1,0}\left(N\right)=\left[T\left(N\right)\otimes \mathbb{C}\right]\cap T^{1,0}\left({\mathbb{C}}^2\right)$$

is the CR structure on N induced by the complex structure on ${\mathbb{C}}^2$. H. Rossi has produced (cf. [1]) a 1-parameter family ${\left\{\mathcal{H}\left(t\right)\right\}}_{\left|t\right|<1}$ of strictly pseudoconvex CR structures on the sphere $S^3$ such that none of the CR manifolds $\left(S^3,\mathcal{H}\left(t\right)\right)$, $t\ne 0$ (the Rossi spheres) is globally embeddable (cf. also D.M. Burns [2]). One of the purposes of the present paper is to start studying a natural weakening of the global CR embedding problem, seeking for an at least K-quasiconformal mapping from M onto N. The problem is specialized to

$$(M,\mathcal{H})\in \left\{\left(S^3,\mathcal{H}\left(t\right)\right):\left|t\right|<1\right\}.$$

A quasiconformal mapping $f=\left(f^1,f^2\right):S^3\to N$ (in the sense of A. Koranyi and H.M. Reimann [3]) is in particular a contact transformation of positive dilation $\lambda \left(f\right)>0$, and then a vector bundle morphism ${\mu }_f\left(t\right)=\mu \left(f,\mathcal{H}\left(t\right)\right):\mathcal{H}\left(t\right)\to \mathcal{H}\left(t\right)$ (the complex dilation of f) may be built such that quasiconformality is characterized by the Beltrami equations

$${\overline{L}}_t\left(f^j\right)=\mu (·,t)L_t\left(f^j\right),j\in \{1,2\},\left|t\right|<1,$$

(1)

where the functions $\mu \left(·,t\right):S^3\to \mathbb{C}$ are determined by

$${\mu }_f\left(t\right)L_t=\mu (·,t)L_t,$$

$$L_t=Z+t\overline{Z},Z=\overline{w}\frac{\partial }{\partial z}-\overline{z}\frac{\partial }{\partial w}.$$

Building on an idea by C-Y. Hsiao and P-L. Yung (cf. [4]) we use the canonical CR isomorphism (induced by the Cayley map) $H:U=S^3\backslash \left\{(0,-1)\right\}\approx {\mathbb{H}}_1$ to transform the Beltrami Equation (1) into

$$\overline{u}\overline{V}\left(f\right)=\frac{\lambda (·,t)-t}{1-t\lambda (·,t)}uV\left(f\right),$$

(2)

$$u(\zeta ,t)=\frac{1}{2}\frac{{\left({\left|\zeta \right|}^2-i\tau +1\right)}^2}{{\left|\zeta \right|}^2+i\tau +1},(\zeta ,\tau )\in {\mathbb{H}}_1,$$

$$\lambda (x,t)=\mu \left(H^{-1}\left(x\right),t\right),x\in {\mathbb{H}}_1,\left|t\right|<1,$$

where ${\displaystyle V\equiv \frac{\partial }{\partial \zeta }+i\overline{\zeta }\frac{\partial }{\partial \tau }}$ (so that $\overline{V}$ is the unsolvable Lewy operator). Our main result is as follows.

Theorem 1. 

Let ${\left\{\mu \left(·,t\right)\right\}}_{\left|t\right|<1}$ be a smooth 1-parameter family of measurable functions $\mu \left(·,t\right):S^3\to \mathbb{C}$ of compact support

$$\mathrm{Supp}\left[\mu (·,t)\right]\subset S^3\backslash \left\{(0,-1)\right\},\left|t\right|<1,$$

such that

$$\parallel \mu (·,t){\parallel }_{\infty }={\mathrm{ess}\sup }_{p\in S^3}|\mu (p,t)|<\frac{1-\left|t\right|\sqrt{2}}{\sqrt{2}+\left|t\right|}.$$

Let$v\in {\mathrm{CR}}^{\infty }\left(S^3\right)$be a CR function [i.e., $\overline{Z}\left(v\right)=0$]. Let us set

$$\alpha (x,t)=\frac{\lambda (x,t)-t}{1-t\lambda (x,t)}{\left[\frac{u\left(x\right)}{\left|u\right(x\left)\right|}\right]}^2,x\in {\mathbb{H}}_1,\left|t\right|<1.$$

If one of the following conditions holds,

(i)

$\alpha (·,t)\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$, $\alpha (·,t)V\left(v\circ H^{-1}\right)\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$,

(ii)

$\alpha (·,t)\in D_{-2}$, $\alpha (·,t)V\left(v\circ H^{-1}\right)\in D_{-1}$,

(iii)

$\alpha (·,t)\in D_{-2}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$, $\alpha (·,t)V\left(v\circ H^{-1}\right)\in D_{-1}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$,

then the Beltrami Equation (2) has a unique solution $f_t$ such that $f_t-v\circ H^{-1}\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$. Consequently $g_t=f_t\circ H$ is a solution to

$${\overline{L}}_t\left(g\right)=\mu \left(·,t\right)L_t\left(g\right)$$

(3)

such that $g_t-v\in W_F^{1,2}\left(U,\theta \right)$.

Here the spaces $L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)\subset L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ are

$$L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)=\left\{f\in L^2\left({\mathbb{H}}_1,{\theta }_0\right):\widehat{f}\left(\lambda \right)=0\mathrm{a}.\mathrm{e}.\lambda >0\right\},$$

$$L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)=L^2\left({\mathbb{H}}_1,{\theta }_0\right)\ominus L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right),$$

and $\widehat{f}\left(\lambda \right)$ is the Fourier transform of f at $\lambda \in \mathbb{R}\backslash \left\{0\right\}$. The meaning of the sets ${\left\{D_j\right\}}_{j\in \mathbb{Z}}$ will be explained in Section 3.

The paper is organized as follows.

Section 2.1 is devoted to pseudohermitian geometry on a Rossi sphere $\left(S^3,\mathcal{H}\left(t\right)\right)$. We show that Rossi’s CR structures $\left\{\mathcal{H}\right(t):|t|<1\}$ have the same Levi distribution (i.e., the maximally complex distribution associated to the standard CR structure $\mathcal{H}\left(0\right)=T_{1,0}\left(S^3\right)$) and, therefore, the same contact forms. We compute the pseudohermitian geometric objects of interest (the Tanaka–Webster connection, Fefferman’s metric, etc.) of a Rossi sphere endowed with the canonical contact form $\theta =\frac{i}{2}\left(zd\overline{z}+wd\overline{w}-\overline{z}dz-\overline{w}dw\right)$.

Section 2.2 discusses the Folland–Stein spaces

$$W_H^{1,2}(M,\theta ),W_E^{1,2}(U,{\iota }^{\ast }\theta ),$$

on a strictly pseudoconvex CR manifold $\left(M,T_{1,0}\left(M\right)\right)$, equipped with the positively oriented contact form $\theta$, and $E=\{E_a:1\le a\le 2n\}$ is a $G_{\theta }$-orthonormal (local) frame of the Levi distribution $H\left(M\right)=\mathrm{Re}\left\{T_{1,0}\left(M\right)\oplus T_{0,1}\left(M\right)\right\}$, defined on the open set $\iota :U\subset M$. If U is also the domain of a local coordinate neighborhood $\chi :U\to {\mathbb{R}}^{2n+1}$, then $X\equiv \{{\chi }_{\ast }{E}_a:1\le a\le 2n\}$ is a Hörmander system of vector fields on $\Omega =\chi \left(U\right)$ (e.g., in the sense of [5]) and $W_E^{1,2}(U,\theta )$ are essentially the Sobolev-type spaces $W_X^{1,2}\left(\Omega \right)$ (e.g., in [6,7]). Our Theorem 2 in this section accounts for the fact that solving (3) in $W_F^{1,2}\left(S^3,\theta \right)$ is the same as solving (2) in $W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$.

Section 2.3 discusses the basic differential geometric facts on quasiconformal maps of 3-dimensional nondegenerate CR manifolds and gives a proof of a characterization of K-quasiconformality due to A. Koranyi and H.M. Reimann (cf. [3]) yet proved by them only for the Heisenberg group.

In Section 2.4, we derive the Beltrami equations, describing quasiconformal maps of the Rossi sphere $\left(S^3,\mathcal{H}\left(t\right)\right)$ into a real hypersurface $N\subset {\mathbb{C}}^2$.

Section 3 collects the needed tools of harmonic analysis (e.g., the Bargmann representations of the Heisenberg group ${\mathbb{H}}_1$, the corresponding Fourier transform of $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$, and the orthogonal decomposition $L^2\left({\mathbb{H}}_1,{\theta }_0\right)={⨁}_{k\in \mathbb{Z}}{U}^k$) and complex analysis (e.g., the solution to the inhomogeneous tangential Cauchy–Riemann equations $\overline{V}\left(f\right)=g$ on ${\mathbb{H}}_1$) and provides the proof to Theorem 1.

2. Rossi’s Spheres

2.1. CR Structures, Levi Form, Tanaka–Webster Connection

We review a few notations, conventions and basic results in Cauchy–Riemann and pseudohermitian geometry, by mainly following the monograph [8].

2.1.1. CR Manifolds, Pseudohermitian Structures

Let M be a 3-dimensional, orientable, $C^{\infty }$ manifold. A CR structure on M is a complex line subbundle $\mathcal{H}\subset T\left(M\right)\otimes \mathbb{C}$ such that

$${\mathcal{H}}_x\cap {\overline{\mathcal{H}}}_x=\left(0\right),x\in M.$$

The tangential Cauchy–Riemann operator is the first order differential operator

$${\overline{\partial }}_{\mathcal{H}}:C^1\left(M,\mathbb{C}\right)\to C\left({\overline{\mathcal{H}}}^{\ast }\right),$$

$$\left({\overline{\partial }}_{\mathcal{H}}v\right)\overline{W}=\overline{W}\left(v\right),v\in C^1(M,\mathbb{C}),W\in \mathcal{H}.$$

A CR function on M is a $C^1$ solution v to the tangential CR equations ${\overline{\partial }}_{\mathcal{H}}v=0$. Let ${\mathrm{CR}}^k(M,\mathcal{H})$ be the space of all CR functions of class $C^k$, $k\ge 1$.

Let $H\left(M\right)=\mathrm{Re}\left\{\mathcal{H}\oplus \overline{\mathcal{H}}\right\}$ be the Levi distribution. It carries the complex structure

$$J:H\left(M\right)\to H\left(M\right),J(Z+\overline{Z})=i(Z-\overline{Z}),Z\in \mathcal{H},$$

(with $i=\sqrt{-1}$). The conormal bundle is the real line subbundle $H{\left(M\right)}^{\perp }\subset T^{\ast }\left(M\right)$ given by

$$H{\left(M\right)}_x^{\perp }=\left\{\omega \in T_x^{\ast }\left(M\right):\mathrm{Ker}\left(\omega \right)\supset H_x\right\},x\in M.$$

The conormal bundle is trivial (i.e., $H{\left(M\right)}^{\perp }\approx M\times \mathbb{R}$, a vector bundle isomorphism), and hence it admits globally defined nowhere zero $C^{\infty }$ sections $\theta$, each of which is referred to as a pseudohermitian structure on M. Let $\mathcal{P}=\mathcal{P}(M,\mathcal{H})$ be the set of all pseudohermitian structures on M. For every $\theta \in \mathcal{P}$, the Levi form $G_{\theta }$ is

$$G_{\theta }(X,Y)=\left(d\theta \right)(X,JY),X,Y\in H\left(M\right).$$

The CR structure $\mathcal{H}$ is nondegenerate if the Levi form $G_{\theta }$ is nondegenerate (i.e., $G_{\theta }(X,Y)=0$ for every $Y\in H\left(M\right)$ yields $X=0$) for some $\theta \in \mathcal{P}$. If $\mathcal{H}$ is nondegenerate, then every $\theta \in \mathcal{P}$ is a contact form, i.e., $\Psi =\theta \wedge d\theta$ is a volume form on M, and $\mathcal{P}$ splits into two orientation classes ${\mathcal{P}}_{\pm }={\mathcal{P}}_{\pm }(M,\mathcal{H})$. A contact form $\theta \in {\mathcal{P}}_{+}$ is positively oriented (the Levi form $G_{\theta }$ is positive definite). For every $\theta \in {\mathcal{P}}_{+}$, the Webster metric is the Riemannian metric determined by

$$g_{\theta }(X,Y)=G_{\theta }(X,Y),g_{\theta }(X,T)=0,g_{\theta }(T,T)=1,$$

for any $X,Y\in H\left(M\right)$.

2.1.2. Tanaka–Webster Connection, Canonical Circle Bundle, Fefferman’s Metric

The Tanaka–Webster connection of $(M,\theta )$ is the linear connection ∇ on M uniquely determined by the following axioms: (i) $H\left(M\right)$ is parallel with respect to ∇ i.e., ${\nabla }_{Y}X\in H\left(M\right)$ for any $X\in H\left(M\right)$ and any $Y\in \mathfrak{X}\left(M\right)$, (ii) the complex structure J along $H\left(M\right)$ and the Webster metric $g_{\theta }$ are parallel with respect to ∇ i.e., $\nabla J=0$ and $\nabla g_{\theta }=0$, (iii) the torsion $T_{\nabla }$ is pure, i.e.,

$$T_{\nabla }\left(Z,W\right)=0,T_{\nabla }\left(Z,\overline{W}\right)=2i{G}_{\theta }\left(Z,\overline{W}\right)T,$$

$$\tau \circ J+J\circ \tau =0,\tau \left(Y\right)\equiv T_{\nabla }(T,Y),$$

$$Z,W\in \mathcal{H},Y\in \mathfrak{X}\left(M\right).$$

$\tau$ is the pseudohermitian torsion of the Tanaka–Webster connection ∇. By a result of S.M. Webster (cf., for example, [8]), $\tau$ is self-adjoint (i.e., $g_{\theta }\left(\tau X,Y\right)=g_{\theta }\left(X,\tau Y\right)$) and $\tau \left(\mathcal{H}\right)\subset \overline{\mathcal{H}}$ (in particular, $\tau$ is traceless, i.e., $\mathrm{trace}\left(\tau \right)=0$).

For every $C^1$ vector field X on M, the divergence of X is determined by ${\mathcal{L}}_X\Psi =\mathrm{div}\left(X\right)\Psi$ where ${\mathcal{L}}_X$ denotes the Lie derivative at X. The divergence of a vector field is most easily calculated as the trace of the covariant derivative, with respect to the Tanaka–Webster connection ∇. Indeed (by axiom (ii) above), $\nabla \Psi =0$, and hence,

$$\mathrm{div}\left(X\right)=\mathrm{trace}\left\{Y⟼{\nabla }_{Y}X\right\}.$$

A complex valued p-form $\eta \in {\Omega }^p\left(M\right)=C^{\infty }\left({\Lambda }^p{T}^{\ast }\left(M\right)\otimes \mathbb{C}\right)$ is a $(p,0)$-form if $\overline{\mathcal{H}}\rfloor \eta =0$. Let ${\Lambda }^{p,0}\left(M\right)\to M$ be the relevant vector bundle (so that ${\Omega }^{p,0}\left(M\right)=C^{\infty }\left({\Lambda }^{p,0}\left(M\right)\right)$ is the space of all $(p,0)$-forms on M). Then $K(M,\mathcal{H})={\Lambda }^{n+1,0}\left(M\right)$ is a complex line bundle (the canonical bundle over M). ${\mathbb{R}}_{+}={\mathrm{GL}}^{+}(1,\mathbb{R})$ (the multiplicative positive reals) acts freely on $K_0(M,\mathcal{H})=K(M,\mathcal{H})\backslash \left\{\mathrm{zero}\sec \mathrm{tion}\right\}$, thus organizing the quotient space $C(M,\mathcal{H})=K_0(M,\mathcal{H})/{\mathbb{R}}_{+}$ as the total space of a principal circle bundle $S^1\to C(M,\mathcal{H})\stackrel{\pi }{\to }M$. If $\omega \in K{(M,\mathcal{H})}_x$ with $\omega \ne 0$ then $\left[\omega \right]\in C{\left(M,\mathcal{H}\right)}_x$ denotes the class of $\omega$ $mod{\mathbb{R}}_{+}$. Let us assume that $(M,\mathcal{H})$ is strictly pseudoconvex and let $\theta \in {\mathcal{P}}_{+}(M,\mathcal{H})$. Let $\{T_{\alpha }:1\le \alpha \le n\}\subset C^{\infty }\left(U,\mathcal{H}\right)$ be a local frame of $\mathcal{H}$, defined on the open subset $U\subset M$. Let $T\in \mathfrak{X}\left(M\right)$ be the Reeb vector field of $(M,\theta )$. Let $\{{\theta }^{\alpha }:1\le \alpha \le n\}$ be the complex 1-forms on U determined by

$${\theta }^{\alpha }\left(T_{\beta }\right)={\delta }_{\beta }^{\alpha },{\theta }^{\alpha }\left(T_{\overline{\beta }}\right)=0,{\theta }^{\alpha }\left(T\right)=0.$$

$\{{\theta }^{\alpha }:1\le \alpha \le n\}$ is an admissible coframe. Then

$$\omega =\lambda {\left(\theta \wedge {\theta }^1\wedge \cdots \wedge {\theta }^n\right)}_x$$

for some $\lambda \in \mathbb{C}\backslash \left\{0\right\}$. A local trivialization chart of $C(M,\mathcal{H})$ is

$$\Phi :{\pi }^{-1}\left(U\right)\to U\times S^1,\Phi \left(\left[\omega \right]\right)=\left(x,\frac{\lambda }{\left|\lambda \right|}\right).$$

The Fefferman metric is the Lorentzian metric $F_{\theta }\in \mathrm{Lor}\left[C(M,\mathcal{H})\right]$ given by (cf. [8] pp. 128–129)

$$F_{\theta }={\pi }^{\ast }{\tilde{G}}_{\theta }+2\left({\pi }^{\ast }\theta \right)\odot \sigma ,$$

(4)

where

$$\sigma =\frac{1}{n+2}\left\{d\mathbf{s}+{\pi }^{\ast }\left(i{{\omega }_{\alpha }}^{\alpha }-\frac{i}{2}{g}^{\alpha \overline{\beta }}d{g}_{\alpha \overline{\beta }}-\frac{R}{4(n+1)}\theta \right)\right\}$$

(5)

a connection 1-form on the principal bundle $S^1\to C(M,\mathcal{H})\to M$ (the Graham connection, cf. [9]). As to the notation in (4) and (5), the (degenerate) $(0,2)$-tensor field ${\tilde{G}}_{\theta }$ extends the Levi form $G_{\theta }$ to the whole of $T\left(M\right)$ by requesting that ${\tilde{G}}_{\theta }(T,W)=0$ for any $W\in \mathfrak{X}\left(M\right)$ (and ${\tilde{G}}_{\theta }=G_{\theta }$ on $H\left(M\right)\otimes H\left(M\right)$). Additionally, $\mathbf{s}$ is a local fiber coordinate on $C(M,\mathcal{H})$ [a detailed description of $\mathbf{s}$ for $(M,\mathcal{H})=\left(S^3,\mathcal{H}\left(t\right)\right)$ (a Rossi sphere) is given in Section 2.1.5]. Moreover,

$$g_{\alpha \overline{\beta }}=G_{\theta }\left(T_{\alpha },T_{\overline{\beta }}\right),\left[g^{\alpha \overline{\beta }}\right]=\left[g_{\alpha \overline{\beta }}\right],$$

$$\nabla T_{\alpha }={{\omega }_{\alpha }}^{\beta }{T}_{\beta },R=g^{\alpha \overline{\beta }}{R}_{\alpha \overline{\beta }},$$

and $R_{\alpha \overline{\beta }}$ is the pseudohermitian Ricci tensor (cf. [8], p. 50).

2.1.3. Heisenberg Group, Rossi Spheres

Let ${\mathbb{H}}_1=\mathbb{C}\times \mathbb{R}$ be the Heisenberg group, with the group law

$$(z,t)·(\zeta ,\tau )=\left(z+\zeta ,t+\tau +2\mathrm{Im}\left(z\overline{\zeta }\right)\right),$$

for any $z,\zeta \in \mathbb{C}$ and $t,\tau \in \mathbb{R}$. The complex vector field $V=\partial /\partial \zeta +i\overline{\zeta }\partial /\partial \tau$ spans the left invariant CR structure ${\mathcal{H}}_x=\mathbb{C}{V}_x$, with $x\in {\mathbb{H}}_1$. Here, $\overline{V}$ is the Lewy operator and the tangential CR equations on ${\mathbb{H}}_1$ are $\overline{V}\left(F\right)=0$. For instance, if $F(\zeta ,\tau )=|\zeta {|}^2-i\tau$, then $F\in {\mathrm{CR}}^{\infty }\left({\mathbb{H}}_1,\mathcal{H}\right)$.

Let $S^3=\left\{\left(z,w\right)\in {\mathbb{C}}^2:z\overline{z}+w\overline{w}=1\right\}$ be the standard sphere. The CR structure

$$T_{1,0}\left(S^3\right)=\left[T\left(S^3\right)\otimes \mathbb{C}\right]\cap T^{1,0}\left({\mathbb{C}}^2\right)$$

(the canonical CR structure on $S^3$) is the span of $T_1=\overline{w}\partial /\partial z-\overline{z}\partial /\partial w$. Let $H\left(S^3\right)$ be the Levi distribution of the CR manifold $\left(S^3,T_{1,0}\left(S^3\right)\right)$. Let us set

$$L_t=T_1+t{T}_{\overline{1}},\left|t\right|<1,$$

(6)

$$\mathcal{H}{\left(t\right)}_x=\left\{\lambda L_{t,x}:\lambda \in \mathbb{C}\right\},x\in S^3.$$

Here, $T_{\overline{1}}=\overline{T_1}$. Then, we have the following:

(i)

$\mathcal{H}\left(t\right)$ is a nondegenerate CR structure on $S^3$ [such that $\mathcal{H}\left(0\right)=T_{1,0}\left(S^3\right)$].

(ii)

The Levi distributions of $\left(S^3,\mathcal{H}\left(t\right)\right)$ and $\left(S^3,T_{1,0}\left(S^3\right)\right)$ coincide, i.e.,

$$\mathrm{Re}\left\{\mathcal{H}\left(t\right)\oplus \overline{\mathcal{H}\left(t\right)}\right\}=H\left(S^3\right),\left|t\right|<1.$$

(iii)

The CR manifolds $\left(S^3,\mathcal{H}\left(t\right)\right)$ have the same positively oriented contact forms, i.e.,

$${\mathcal{P}}_{+}\left(S^3,\mathcal{H}\left(t\right)\right)={\mathcal{P}}_{+}\left(S^3,T_{1,0}\left(S^3\right)\right).$$

To prove (i)–(iii), we need some preparation. Let us consider the (real valued) differential 1-form $\theta \in {\Omega }^1\left(S^3\right)$ given by

$$\theta ={\mathbf{j}}^{\ast }\left[\frac{i}{2}\left(zd\overline{z}+wd\overline{w}-\overline{z}dz-\overline{w}dw\right)\right]$$

(7)

(with $\mathbf{j}:S^3\subset {\mathbb{C}}^2$). Then, we have the following:

Step 1. $\theta \in {\mathcal{P}}_{+}\left[S^3,T_{1,0}\left(S^3\right)\right]$, i.e., θ is a positively oriented contact form on $S^3$ with respect to the ordinary CR structure $T_{1,0}\left(S^3\right)$.

Proof. 

For simplicity, we drop $\mathbf{j}$. Then

$$d\theta =i\left(dz\wedge d\overline{z}+dw\wedge d\overline{w}\right)$$

and the Levi form $G_{\theta }$ is

$$G_{\theta }\left(T_1,T_{\overline{1}}\right)=-i\left(d\theta \right)\left(T_1,T_{\overline{1}}\right)=\left(dz\wedge d\overline{z}\right)\left(T_1,T_{\overline{1}}\right)+\left(dw\wedge d\overline{w}\right)\left(T_1,T_{\overline{1}}\right)=$$

$$=\frac{1}{2}\left\{{\left|dz\left(T_1\right)\right|}^2+{\left|dw\left(T_1\right)\right|}^2\right\}=\frac{1}{2}\left\{{\left|z\right|}^2+{\left|w\right|}^2\right\}=\frac{1}{2}>0.$$

□

$\theta$ is referred to as the canonical contact form on $S^3$. The Reeb vector field of $\left(S^3,\theta \right)$ is the nowhere zero globally defined vector field $T\in \mathfrak{X}\left(S^3\right)$ determined by $\theta \left(T\right)=1$ and $T\rfloor d\theta =0$.

Step 2. The Reeb vector field T of $(S^3,\theta )$ is given by

$$T=i\left(z\frac{\partial }{\partial z}+w\frac{\partial }{\partial w}-\overline{z}\frac{\partial }{\partial \overline{z}}-\overline{w}\frac{\partial }{\partial \overline{w}}\right).$$

An adapted coframe is a frame $\left\{{\theta }^1\right\}$ in $T_{1,0}{\left(S^3\right)}^{\ast }$ such that

$${\theta }^1\left(T_1\right)=1,{\theta }^1\left(T_{\overline{1}}\right)=0,{\theta }^1\left(T\right)=0.$$

Step 3. ${\theta }^1=wdz-zdw$ is an adapted coframe on $(S^3,\theta )$.

We may now complete the proof of (i)–(iii). The complex distribution $\mathcal{H}\left(t\right)\oplus \overline{\mathcal{H}\left(t\right)}$ is the span of $\{L_t,{\overline{L}}_t\}$ and then [by (6)] the span of $\{T_1,T_{\overline{1}}\}$. Hence, the CR manifolds ${\left\{\left(S^3,\mathcal{H}\left(t\right)\right)\right\}}_{\left|t\right|<1}$ have the same Levi distribution (i.e., $H\left(S^3\right)=\mathrm{Re}\left\{\mathcal{H}\left(t\right)\oplus \overline{\mathcal{H}\left(t\right)}\right\}$) and therefore, the same pseudohermitian structures (i.e., $\mathcal{P}\left(S^3,T_{1,0}\left(S^3\right)\right)=\mathcal{P}\left(S^3,\mathcal{H}\left(t\right)\right)$).

Step 4. The Levi form of $\left(S^3,\mathcal{H}\left(t\right)\right)$

$$G_{\theta }^t\left(X,Y\right)=\left(d\theta \right)\left(X,J^{t}Y\right),X,Y\in H\left(S^3\right),$$

$$J^t:H\left(S^3\right)\to H\left(S^3\right),J^t\left(Z+\overline{Z}\right)=i\left(Z-\overline{Z}\right),Z\in \mathcal{H}\left(t\right),$$

is given by

$$G_{\theta }^t\left(L_t,{\overline{L}}_t\right)=\frac{1-t^2}{2}>0.$$

(8)

Proof. 

Indeed,

$$G_{\theta }^t\left(L_t,{\overline{L}}_t\right)=$$

[as $J^t{L}_t=i{L}_t$ and $J^t{\overline{L}}_t=-i{\overline{L}}_t$]

$$=-i\left(d\theta \right)\left(L_t,{\overline{L}}_t\right)=$$

$$=\frac{1}{2}\left\{|dz\left(L_t\right){|}^2-|dz\left({\overline{L}}_t\right){|}^2+|dw\left(L_t\right){|}^2-|d\overline{w}\left(L_t\right){|}^2\right\}=$$

$$=\frac{1-t^2}{2}\left({\left|z\right|}^2+{\left|w\right|}^2\right)=\frac{1-t^2}{2}>0$$

proving that $\theta \in {\mathcal{P}}_{+}\left(S^3,\mathcal{H}\left(t\right)\right)$ and then

$${\mathcal{P}}_{+}\left(S^3,\mathcal{H}\left(t\right)\right)={\mathcal{P}}_{+}\left(S^3,\mathcal{H}\left(0\right)\right).$$

□

2.1.4. Tanaka–Webster Connection of a Rossi Sphere

We shall need the following commutation table

$$\left[T,T_1\right]=-2i{T}_1,\left[T,T_{\overline{1}}\right]=2i{T}_{\overline{1}},\left[T_1,T_{\overline{1}}\right]=-iT,$$

(9)

$$\left[T,L_t\right]=-\frac{2i(1+t^2)}{1-t^2}{L}_t+\frac{4it}{1-t^2}{\overline{L}}_t,\left[L_t,{\overline{L}}_t\right]=-i(1-t^2)T.$$

(10)

Let ${\nabla }^t$ be the Tanaka–Webster connection of $\left(S^3,\mathcal{H}\left(t\right),\theta \right)$, where $\theta$ is given by (7), and let ${\omega }_t$ be the connection 1-form associated to the frame $\left\{L_t\right\}\subset C^{\infty }\left(\mathcal{H}\left(t\right)\right)$, i.e.,

$${\nabla }^t{L}_t={\omega }_t\otimes L_t,{\omega }_t={\Gamma }_{11}^1\left(t\right){\theta }_t^1+{\Gamma }_{\overline{1}1}^1\left(t\right){\theta }_t^{\overline{1}}+{\Gamma }_{01}^1\left(t\right)\theta .$$

Here, we have set

$${\theta }_t^1=\frac{1}{1-t^2}\left({\theta }^1-t{\theta }^{\overline{1}}\right),{\theta }_t^{\overline{1}}=\overline{{\theta }_t^1},$$

so that $\left\{{\theta }_t^1\right\}$ is an adapted coframe relative to the CR structure $\mathcal{H}\left(t\right)$, i.e.,

$${\theta }_t^1\left(L_t\right)=1,{\theta }_t^1\left({\overline{L}}_t\right)=0,{\theta }_t^1\left(T\right)=0.$$

By a result in [8] (p. 33),

$${\Gamma }_{11}^1\left(t\right)=g^{1\overline{1}}\left(t\right)\left\{L_t\left(g_{1\overline{1}}\left(t\right)\right)-g_{\theta }^t\left(L_t,\left[L_t,{\overline{L}}_t\right]\right)\right\},$$

(11)

$${\Gamma }_{\overline{1}1}^1\left(t\right)=g^{\overline{1}1}\left(t\right)g_{\theta }^t\left(\left[{\overline{L}}_t,L_t\right],{\overline{L}}_t\right),$$

(12)

$${\Gamma }_{01}^1\left(t\right)=g^{1\overline{1}}\left(t\right)g_{\theta }^t\left(\left[T,L_t\right],{\overline{L}}_t\right),$$

(13)

where

$$g^{1\overline{1}}\left(t\right)=\frac{1}{g_{1\overline{1}}\left(t\right)},g_{1\overline{1}}\left(t\right)=G_{\theta }\left(L_t,{\overline{L}}_t\right)=\frac{1-t^2}{2},$$

and $g_{\theta }^t$ is the Webster metric of $\left(S^3,\mathcal{H}\left(t\right),\theta \right)$, i.e., $g_{\theta }^t(X,Y)=G_{\theta }^t(X,Y)$, $g_{\theta }^t(X,T)=0$ and $g_{\theta }^t(T,T)=1$ for any $X,Y\in H\left(S^3\right)$. Substitution from (9) and (10) into (11)–(13) gives

$${\Gamma }_{11}^1\left(t\right)={\Gamma }_{\overline{1}1}^1\left(t\right)=0,{\Gamma }_{01}^1\left(t\right)=-\frac{2i(1+t^2)}{1-t^2},$$

(14)

$${\omega }_t={\Gamma }_{01}^1\left(t\right)\theta =-\frac{2i(1+t^2)}{1-t^2}\theta .$$

(15)

The pseudohermitian torsion ${\tau }_t$ of ${\nabla }^t$ is given by

$${\tau }_t\left(L_t\right)=A_1^{\overline{1}}\left(t\right){\overline{L}}_t,A_1^{\overline{1}}\left(t\right)=-\frac{4it}{1-t^2}.$$

Let $R^t=R^{{\nabla }^t}$ be the curvature tensor field of ${\nabla }^t$. With the convention in [8], p. 50, the only nonzero component of $R^t$, with respect to the frame $\left\{L_t\right\}$, is

$$R^t\left(L_t,{\overline{L}}_t\right)L_t={{R_1}^1}_{1\overline{1}}\left(t\right),{{R_1}^1}_{1\overline{1}}\left(t\right)=2(1+t^2).$$

(16)

In particular, the pseudohermitian scalar curvature of ${\nabla }^t$ is

$$R\left(t\right)=g^{1\overline{1}}{{R_1}^1}_{1\overline{1}}\left(t\right)=\frac{4(1+t^2)}{1-t^2}.$$

To prove (16), one starts from (cf. [8], p. 51)

$$R^t(X,Y)L_t=2\left(d{\omega }_t\right)(X,Y)L_t.$$

In addition (by taking the exterior differential of (15))

$$d{\omega }_t=-\frac{2i(1+t^2)}{1-t^2}d\theta$$

and hence (as $G_{\theta }^t=-id\theta$ on $\mathcal{H}\left(t\right)\otimes \overline{\mathcal{H}\left(t\right)}$)

$$R^t\left(L_t,{\overline{L}}_t\right)L_t=\frac{4(1+t^2)}{1-t^2}{G}_{\theta }^t\left(L_t,{\overline{L}}_t\right)L_t=2(1+t^2)L_t.$$

2.1.5. Fefferman’s Metric of Rossi’s Sphere

Let $C\left(S^3,T_{1,0}\left(S^3\right)\right)$ be the canonical circle bundle over $(S^3,T_{1,0}\left(S^3\right))$. Then

$$C{\left(S^3,T_{1,0}\left(S^3\right)\right)}_x=\left\{\left[\lambda {\left(\theta \wedge {\theta }^1\right)}_x\right]:\lambda \in \mathbb{C}\backslash \left\{0\right\}\right\},x\in S^3.$$

Let $\eta \in {\Omega }^2\left(S^3\right)=C^{\infty }\left({\Lambda }^2{T}^{\ast }\left(S^3\right)\otimes \mathbb{C}\right)$ be a type $(2,0)$-form relative to the CR structure $\mathcal{H}\left(t\right)$ i.e., $\overline{\mathcal{H}\left(t\right)}\rfloor \eta =0$. Then

$$\eta =h\theta \wedge {\theta }_t^1=\frac{h}{1-t^2}\theta \wedge \left({\theta }^1-t{\theta }^{\overline{1}}\right)$$

for some $h\in C^{\infty }\left(S^3,\mathbb{C}\right)$. Hence, the canonical circle bundle over $\left(S^3,\mathcal{H}\left(t\right)\right)$ is

$$C{\left(S^3,\mathcal{H}\left(t\right)\right)}_x=\left\{\left[\lambda {\left(\theta \wedge {\theta }^1-t\theta \wedge {\theta }^{\overline{1}}\right)}_x\right]:\lambda \in \mathbb{C}\backslash \left\{0\right\}\right\},x\in S^3.$$

Let ${\pi }^t:C\left(S^3,\mathcal{H}\left(t\right)\right)\to S^3$ be the canonical projection. Fefferman’s metric

$$F_{\theta }^t=F\left(\mathcal{H}\left(t\right),\theta \right)\in \mathrm{Lor}\left[C\left(S^3,\mathcal{H}\left(t\right)\right)\right]$$

is

$$F_{\theta }^t={\left({\pi }^t\right)}^{\ast }\widetilde{G_{\theta }^t}+2\left({\left({\pi }^t\right)}^{\ast }\theta \right)\odot {\sigma }_t,$$

$${\sigma }_t=\frac{1}{3}\left\{d{\mathbf{s}}_t+{\left({\pi }^t\right)}^{\ast }\left[i{\omega }_t-\frac{i}{2}{g}^{1\overline{1}}\left(t\right)d{g}_{1\overline{1}}\left(t\right)-\frac{1}{8}R\left(t\right)\theta \right]\right\},$$

$$\widetilde{G_{\theta }^t}(X,Y)=G_{\theta }^t(X,Y),\widetilde{G_{\theta }^t}(T,W)=0,X,Y\in H\left(S^3\right),W\in \mathfrak{X}\left(S^3\right).$$

Additionally, ${\mathbf{s}}_t$ is a local fiber coordinate on $C\left(S^3,\mathcal{H}\left(t\right)\right)$ that we now describe in some detail. Let us consider the $C^{\infty }$ diffeomorphism

$${\Phi }_t:C\left(S^3,\mathcal{H}\left(t\right)\right)\to S^3\times S^1,{\Phi }_t\left(\left[\omega \right]\right)=\left(x,\frac{\alpha }{\left|\alpha \right|}\right),$$

$$\omega =\alpha {\left(\theta \wedge {\theta }^1-t\theta \wedge {\theta }^{\overline{1}}\right)}_x,x\in S^3,\lambda \in \mathbb{C}\backslash \left\{0\right\}.$$

Note that ${\Phi }_t\left(\left[\omega \right]\right)$ is invariant under a transformation ${\alpha }^{\prime }=b\alpha$ with $b\in {\mathbb{R}}_{+}$ hence ${\Phi }_t$ is a well defined $C^{\infty }$ diffeomorphism. For every ${\phi }_0\in \mathbb{R}$, we set

$$U\left({\phi }_0\right)=\left\{e^{i\phi }:|\phi -{\phi }_0|<\pi \right\}\subset S^1,$$

$$\psi :({\phi }_0-\pi ,{\phi }_0+\pi )\to U\left({\phi }_0\right),\psi \left(\phi \right)=e^{i\phi },$$

$$arg:U\left({\phi }_0\right)\to ({\phi }_0-\pi ,{\phi }_0+\pi ),arg={\psi }^{-1},$$

$$\mathcal{U}\left({\phi }_0\right)={\Phi }_t^{-1}\left[S^3\times U\left({\phi }_0\right)\right]\subset C\left(S^3,\mathcal{H}\left(t\right)\right),$$

$${\mathbf{s}}_t={\mathbf{s}}_{t,{\phi }_0}:\mathcal{U}\left({\phi }_0\right)\to \left({\phi }_0-\pi ,{\phi }_0+\pi \right),{\mathbf{s}}_t\left(\left[\omega \right]\right)=arg\left(\frac{\alpha }{\left|\alpha \right|}\right).$$

Lemma 1. 

Fefferman’s metric $F_{\theta }^t$ of Rossi’s sphere $\left(S^3,\mathcal{H}\left(t\right),\theta \right)$ is given by

$$F_{\theta }^t={\left({\pi }^t\right)}^{\ast }\widetilde{G_{\theta }^t}+\frac{2}{3}\left[{\left({\pi }^t\right)}^{\ast }\theta \right]\odot d{\mathbf{s}}_t+\frac{1+t^2}{1-t^2}{\left({\pi }^t\right)}^{\ast }\left(\theta \odot \theta \right),$$

(17)

$$\widetilde{G_{\theta }^t}=\frac{1+t^2}{1-t^2}{\theta }^1\odot {\theta }^{\overline{1}}-\frac{t}{1-t^2}\left\{{\left({\theta }^1\right)}^2+{\left({\theta }^{\overline{1}}\right)}^2\right\}.$$

Proof. 

By (15) and (16)

$${\sigma }_t=\frac{1}{3}d{\mathbf{s}}_t+\frac{1+t^2}{2(1-t^2)}{\pi }^{\ast }\theta$$

[the Graham connection 1-form of $\left(S^3,\mathcal{H}\left(t\right),\theta \right)$] yielding (17). The second statement in Lemma 1 follows from

$$\widetilde{G_{\theta }^t}=2g_{1\overline{1}}\left(t\right){\theta }_t^1\odot {\theta }_t^{\overline{1}}.$$

□

2.1.6. Siegel Domain, Cayley Map

Let

$$\Omega =\{({\zeta }_1,{\zeta }_2)\in {\mathbb{C}}^2:\mathrm{Im}\left({\zeta }_2\right)>|{\zeta }_1{|}^2\}$$

be the Siegel domain. We shall need the CR isomorphisms

$$\mathcal{C}:S^3\backslash \left\{(0,-1)\right\}\to \partial \Omega ,\mathcal{C}(z,w)=\left(\frac{z}{1+w},i\frac{1-w}{1+w}\right),$$

(18)

$$\psi :{\mathbb{H}}_1\to \partial \Omega ,\psi (\zeta ,\tau )=\left(\zeta ,\tau +{i\left|\zeta \right|}^2\right),$$

(19)

$$(z,w)\in S^3,w\ne -1,\zeta \in \mathbb{C},\tau \in \mathbb{R},$$

$$H:U=S^3\backslash \left\{(0,-1)\right\}\to {\mathbb{H}}_1,H={\psi }^{-1}\circ \mathcal{C}.$$

Then (i) for every $(z,w)\in S^3\backslash \left\{(0,-1)\right\}$

$$H(z,w)=\left(\frac{z}{1+w},\frac{2\mathrm{Im}\left(w\right)}{{|1+w|}^2}\right),$$

(20)

(ii) for every $\left|t\right|<1$

$$H_{\ast }{L}_t=u^H{V}^H+t\overline{u^H}\overline{V^H},$$

(21)

where $u\in C^{\infty }\left({\mathbb{H}}_1,\mathbb{C}\right)$ is given by

$$u(\zeta ,\tau )=\frac{1}{2}\frac{{\left({\left|\zeta \right|}^2-i\tau +1\right)}^2}{{\left|\zeta \right|}^2+i\tau +1},(\zeta ,\tau )\in {\mathbb{H}}_1.$$

(22)

Here, $u^H=u\circ H$ and $V^H=V\circ H$. Formula (20) follows from (18) (the restriction of the Cayley map $\mathcal{C}:{\mathbb{C}}^2\backslash \{w+1=0\}\to {\mathbb{C}}^2$ to $S^3\backslash \left\{(0,-1)\right\}$) and (19) (the canonical CR isomorphism of the Heisenberg group ${\mathbb{H}}_1$ onto the boundary of the Siegel domain $\Omega \subset {\mathbb{C}}^2$). Formula (21) follows from

$$\left(d_{(z,w)}H\right)L_{t,(z,w)}=\frac{1+\overline{w}}{{(1+w)}^2}{V}_{H(z,w)}+t\frac{1+w}{{(1+\overline{w})}^2}{\overline{V}}_{H(z,w)}$$

together with the observation that (22) yields ${\displaystyle u\left(H(z,w)\right)=\frac{1+\overline{w}}{{(1+w)}^2}}$.

Recall the CR function $F(\zeta ,\tau )={\left|\zeta \right|}^2-i\tau$. If $Z=T_1$, then

$$H_{\ast }Z=\frac{1+\overline{w}}{{(1+w)}^2}{V}^H$$

(23)

so that

$$f(z,w)=\frac{1-w}{1+w},(z,w)\in U\subset S^3,$$

is a CR function, i.e., $f\in {\mathrm{CR}}^{\infty }(U,\mathcal{H}\left(0\right))$. Indeed, $f=F\circ H$ and then (by (23)) $\overline{Z}\left(f\right)=0$. As to the proof of (23), it follows from

$$H_{\ast }Z=\frac{1+\overline{w}+\rho }{{(1+w)}^2}{V}^H-\frac{i\overline{z}\rho }{{(1+w)}^2\left(1+\overline{w}\right)}{\left(\frac{\partial }{\partial \tau }\right)}^H$$

(24)

where $\rho (z,w)=z\overline{z}+w\overline{w}-1$.

2.2. Folland–Stein Spaces

Let $\left(M,\mathcal{H}\right)$ be a strictly pseudoconvex CR manifold, of CR dimension n, and let $\theta \in {\mathcal{P}}_{+}$ be a positively oriented contact form on M. Let $\Psi =\theta \wedge {\left(d\theta \right)}^n$ and let $L^2(M,\theta )$ consist of all measurable functions $u:M\to \mathbb{R}$ such that

$${\parallel u\parallel }_{L^2}={\parallel u\parallel }_{L^2(M,\theta )}={\left({\int }_M{\left|u\right|}^2\Psi \right)}^{1/2}<\infty .$$

One tacitly identifies functions coinciding almost everywhere. Let $L^2\left(H\left(M\right),\theta \right)$ consist of all sections $X:M\to H\left(M\right)$ such that $G_{\theta }{(X,X)}^{1/2}\in L^2(M,\theta )$, i.e.,

$${\parallel X\parallel }_{L^2}={\parallel X\parallel }_{L^2(H\left(M\right),\theta )}={\left({\int }_M{G}_{\theta }(X,X)\Psi \right)}^{1/2}<\infty .$$

A function $u\in L^2(M,\theta )$ is weakly differentiable along $H\left(M\right)$ if there is $X_u\in L^2\left(H\left(M\right),\theta \right)$ such that

$${\int }_M{G}_{\theta }\left(X_u,Y\right)\Psi =-{\int }_{M}u\mathrm{div}\left(Y\right)\Psi$$

for any $Y\in C_0^{\infty }\left(H\left(M\right)\right)$. Such $X_u$ is uniquely determined, up to a set of measure zero. Let $\mathcal{D}\left({\nabla }^H\right)$ consist of all weakly differentiable $u\in L^2(M,\theta )$ and let us consider the linear operator

$${\nabla }^H:\mathcal{D}\left({\nabla }^H\right)\subset L^2\left(M,\theta \right)\to L^2\left(H\left(M\right),\theta \right)$$

given by ${\nabla }^{H}u\equiv X_u$. Note that $C_0^{\infty }\left(M\right)\subset \mathcal{D}\left({\nabla }^H\right)$ so that ${\nabla }^H$ is densely defined. The Sobolev-type space $W_H^{1,2}(M,\theta )$ is $\mathcal{D}\left({\nabla }^H\right)$ equipped with the norm

$${\parallel u\parallel }_{W_H^{1,2}}={\parallel u\parallel }_{W_H^{1,2}(M,\theta )}={\left({\parallel u\parallel }_{L^2(M,\theta )}^2+{\parallel {\nabla }^{H}u\parallel }_{L^2(H\left(M\right),\theta )}\right)}^{1/2}.$$

Let $E\equiv \{E_a:1\le a\le 2n\}\subset C^{\infty }\left(U,H\left(M\right)\right)$ be a local $G_{\theta }$-orthonormal frame (i.e., $G_{\theta }(E_a,E_b)={\delta }_{ab}$) defined on the open set $U\subset M$. Let $i:U\to M$ be the inclusion. A function $u\in L^2(U,i^{\ast }\theta )$ is weakly E-differentiable if for every $a\in \{1,\cdots ,2n\}$, there is $v_a\in L^2(U,i^{\ast }\theta )$ such that

$${\int }_U{v}_a\phi \Psi =-{\int }_{U}u\left\{E_a\left(\phi \right)+\phi \mathrm{div}\left(E_a\right)\right\}\Psi$$

(25)

for any $\phi \in C_0^{\infty }\left(U\right)$. Such $v_a$ is uniquely determined, up to a set of measure zero, and denoted by $E_a\left(u\right):=v_a$. The Folland–Stein space $W_E^{1,2}(U,i^{\ast }\theta )$ consists of all weakly E-differentiable functions $u\in L^2(U,i^{\ast }\theta )$ and is equipped with the norm

$${\parallel u\parallel }_{W_E^{1,2}}={\parallel u\parallel }_{W_E^{1,2}(U,i^{\ast }\theta )}={\left({\parallel u\parallel }_{L^2(U,i^{\ast }\theta )}^2+\sum _{a=1}^{2n}{\parallel E_a\left(u\right)\parallel }_{L^2(U,i^{\ast }\theta )}^2\right)}^{1/2}.$$

Then, we have the following:

(i)

The restriction map $r_U:W_H^{1,2}(M,\theta )\to W_H^{1,2}(U,i^{\ast }\theta )$ is a bounded linear operator,

(ii)

$W_H^{1,2}(U,i^{\ast }\theta )\approx W_E^{1,2}(U,i^{\ast }\theta )$ (an isomorphism of Banach spaces).

The proof of (i) is straightforward. To prove (ii), note first that

$$W_H^{1,2}(U,i^{\ast }\theta )=W_E^{1,2}(U,i^{\ast }\theta )$$

as vector spaces. Indeed if $u\in W_H^{1,2}(U,i^{\ast }\theta )$ then ${\nabla }^{H}u\in L^2\left(H\left(U\right),i^{\ast }\theta \right)$ is well defined and one may consider the functions

$$v_a:=G_{\theta }\left({\nabla }^{H}u,E_a\right),1\le a\le 2n.$$

Then (by the Cauchy–Schwartz inequality)

$${\int }_U|v_a{|}^2\Psi \le {\int }_U{G}_{\theta }({\nabla }^{H}u,{\nabla }^{H}u)G_{\theta }(E_a,E_a)\Psi ={\parallel {\nabla }^{H}u\parallel }_{L^2}^2<\infty$$

so that $v_a\in L^2\left(U,i^{\ast }\theta \right)$. On the other hand (as u is weakly differentiable along $H\left(U\right)$) for every $\phi \in C_0^{\infty }\left(U\right)$

$${\int }_U{v}_a\phi \Psi ={\int }_U{G}_{\theta }\left({\nabla }^{H}u,\phi E_a\right)\Psi =-{\int }_{U}u\mathrm{div}\left(\phi E_a\right)\Psi$$

so that $u\in W_E^{1,2}(U,i^{\ast }\theta )$. The opposite inclusion $W_H^{1,2}(U,i^{\ast }\theta )\supset W_E^{1,2}(U,i^{\ast }\theta )$ may be proved in the same manner. Next, let us observe that

$${\nabla }^{H}u=\sum _{a=1}^{2n}{E}_a\left(u\right)E_a$$

(26)

for every $u\in W_E^{1,2}(U,\theta )$. Indeed, for every $X\in C_0^{\infty }\left(H\left(U\right)\right)$

$${\int }_U{G}_{\theta }\left({\nabla }^{H}u-\sum _{a=1}^{2n}{E}_a\left(u\right)E_a,X\right)\Psi =-{\int }_{U}u\mathrm{div}\left(X\right)\Psi -\sum _{a=1}^{2n}{\int }_U{E}_a\left(u\right){\phi }_a\Psi =$$

(where we have set ${\phi }_a:=G_{\theta }\left(E_a,X\right)\in C_0^{\infty }\left(U\right)$)

$$=-{\int }_{U}u\mathrm{div}\left(X-\sum _{a=1}^{2n}{\phi }_a{E}_a\right)\Psi =0$$

so that ${\nabla }^H-{\sum }_{a=1}^{2n}{E}_a\left(u\right)E_a$ is orthogonal to $C_0^{\infty }\left(H\left(U\right)\right)$ [a dense subspace of $L^2\left(H\left(U\right),i^{\ast }\theta \right)$]. The identity (26) is proved. Finally, one may check that the identity I of $W_E^{1,2}(U,i^{\ast }\theta )$ preserves the norms. Indeed (by (26)),

$$\parallel {\nabla }^H{u\parallel }_{L^2}^2={\int }_U{G}_{\theta }\left({\nabla }^{H}u,{\nabla }^{H}u\right)\Psi =\sum _{a=1}^{2n}{\int }_U|E_a\left(u\right){|}^2\Psi =\sum _{a=1}^{2n}{\parallel E_a\left(u\right)\parallel }_{L^2}^2.$$

It is customary to endow $\left({\mathbb{H}}_1,V\right)$ with the canonical contact form

$${\theta }_0=d\tau +i\left(\zeta d\overline{\zeta }-\overline{\zeta }d\zeta \right).$$

Then $G_{{\theta }_0}\left(V,\overline{V}\right)=1$. Additionally,

$$H^{\ast }{\theta }_0=\lambda \left(H;\theta ,{\theta }_0\right)\theta ,\lambda \left(H;\theta ,{\theta }_0\right)(z,w)=\frac{2}{|1+w{|}^2}.$$

Let us set as customary $\zeta =\xi +i\eta$ and

$$X=\frac{\partial }{\partial \xi }+2\eta \frac{\partial }{\partial \tau },Y=\frac{\partial }{\partial \eta }-2\xi \frac{\partial }{\partial \tau },$$

so that $V=\frac{1}{2}\left(X-iY\right)$. Then

$$E\equiv \left\{E_1,E_2\right\},E_1=\frac{1}{\sqrt{2}}X,E_2=\frac{1}{\sqrt{2}}Y,$$

is a (globally defined) $G_{{\theta }_0}$-orthonormal frame on ${\mathbb{H}}_1$. The CR isomorphism $H:U\approx {\mathbb{H}}_1$ induces $L^2\left({\mathbb{H}}_1,{\theta }_0\right)\approx L^2\left(U,\theta \right)$ (a vector space isomorphism). Indeed, if $f\in L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and $u=f\circ H$ then [by $H^{\ast }{\Psi }_0={\lambda }^2\Psi$ with $\lambda =\lambda \left(H;\theta ,{\theta }_0\right)$ and ${\Psi }_0={\theta }_0\wedge d{\theta }_0$, $\Psi =\theta \wedge d\theta$]

$${\int }_U{\left|u\right|}^2\Psi ={\int }_U{\left(\frac{\left|u\right|}{\lambda }\right)}^2{H}^{\ast }{\Psi }_0={\int }_{{\mathbb{H}}_1}{\left(\frac{\left|f\right|}{\lambda \circ H^{-1}}\right)}^2{\Psi }_0=$$

$$={\int }_{{\mathbb{H}}_1}\frac{4|f(\zeta ,\tau ){|}^2}{{\left[{\left(1+{\left|\zeta \right|}^2\right)}^2+{\tau }^2\right]}^2}{\Psi }_0(\zeta ,\tau )\le 4\parallel f{\parallel }_{L^2({\mathbb{H}}_1,{\theta }_0)}^2<\infty .$$

As H is a CR isomorphism $U\approx {\mathbb{H}}_1$, Cauchy–Riemann analysis is the same on U and ${\mathbb{H}}_1$. However, H does not preserve the contact forms $\theta$ and ${\theta }_0$ so that $(U,\theta )$ and $({\mathbb{H}}_1,{\theta }_0)$ have rather different pseudohermitian geometries. On the same line of thought, we prove the following.

Theorem 2. 

The map $f⟼f\circ H$ is an isomorphism

$$W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)\approx W_F^{1,2}\left(U,\theta \right).$$

Here, $U=S^3\backslash \left\{(0,-1)\right\}$ while $E\equiv \{E_1,E_2\}\subset C^{\infty }\left(H\left({\mathbb{H}}_1\right)\right)$, respectively $F\equiv \{F_1,F_2\}\subset C^{\infty }\left(H\left(U\right)\right)$, are the canonical $G_{{\theta }_0}$-orthonormal, respectively $G_{\theta }$-orthonormal, frames

$$E_1=\frac{1}{\sqrt{2}}\left(V+\overline{V}\right),E_2=\frac{i}{\sqrt{2}}\left(V-\overline{V}\right),$$

$$F_1=Z+\overline{Z},F_2=i\left(Z-\overline{Z}\right).$$

Lemma 2. 

$\mathrm{div}\left(F_a\right)=0$.

Proof. 

It follows from the fact that the only nonvanishing Christoffel symbol of the Tanaka–Webster connection ∇ of $\left(S^3,\theta \right)$ is ${\Gamma }_{01}^1=-2i$ (itself a consequence of (14) with $t=0$). □

Proof of Theorem 2. 

Given $f\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$, we need to show that for every $a\in \{1,2\}$, there is $v_a\in L^2\left(U,\theta \right)$ such that (by (25) and Lemma 2)

$${\int }_U{v}_a\phi \Psi =-{\int }_U(f\circ H)F_a\left(\phi \right)\Psi$$

(27)

for any $\phi \in C_0^{\infty }\left(U\right)$. The candidate for $v_a$ is, of course, obtained by computing $F_a\left(f\circ H\right)$ when f is smooth.

Lemma 3. 

Let $f\in C^1\left({\mathbb{H}}_1\right)$ and $u=f\circ H$. Then

$$F_1\left(u\right)=i\rho \left(g-\overline{g}\right){\left(\frac{\partial f}{\partial \tau }\right)}^H+\frac{1}{\sqrt{2}}\left(h+\overline{h}\right)E_1{\left(f\right)}^H+\frac{i}{\sqrt{2}}\left(h-\overline{h}\right)E_2{\left(f\right)}^H,$$

(28)

$$F_2\left(u\right)=\rho \left(g+\overline{g}\right){\left(\frac{\partial f}{\partial \tau }\right)}^H-\frac{i}{\sqrt{2}}\left(h-\overline{h}\right)E_1{\left(f\right)}^H+\frac{1}{\sqrt{2}}\left(h+\overline{h}\right)E_2{\left(f\right)}^H,$$

(29)

where

$$g,h:{\mathbb{C}}^2\backslash \{w+1=0\}\to {\mathbb{C}}^2,$$

$$g(z,w)=\frac{z}{\left(1+\overline{w}\right){\left(1+w\right)}^2},h(z,w)=\frac{1+w+\rho (z,w)}{{\left(1+\overline{w}\right)}^2},$$

(30)

and $\rho (z,w)={\left|z\right|}^2+{\left|w\right|}^2-1$.

Proof. 

It follows from (24), and its complex conjugate. □

Let $v_a$ be the (restrictions to U of the) right-hand sides of (28) and (29), respectively. By a change of the variable under the integral sign,

$${\int }_U{v}_a\phi \Psi ={\int }_{{\mathbb{H}}_1}{\left(\frac{v_a\phi }{{\lambda }^2}\right)}^{H^{-1}}{\Psi }_0$$

for every $\phi \in C_0^{\infty }\left(U\right)$. Throughout, $v^{H^{-1}}=v\circ H^{-1}$, and the inverse of $H:U\to {\mathbb{H}}_1$ is

$$H^{-1}(\zeta ,\tau )=\left(\frac{2i\zeta }{\tau +i\left({\left|\zeta \right|}^2+1\right)},-\frac{\tau +i\left({\zeta |}^2-1\right)}{\tau +i\left({\left|\zeta \right|}^2+1\right)}\right).$$

(31)

Next (by the very definition of $v_1$),

$${\int }_U{v}_1\phi \Psi =$$

$$=\frac{1}{\sqrt{2}}{\int }_{{\mathbb{H}}_1}{\left(\frac{\phi }{{\lambda }^2}\right)}^{H^{-1}}\left\{{\left(h+\overline{h}\right)}^{H^{-1}}{E}_1\left(f\right)+i{\left(h-\overline{h}\right)}^{H^{-1}}{E}_2\left(f\right)\right\}{\Psi }_0=$$

(as f is E-differentiable)

$$=-\frac{1}{\sqrt{2}}{\int }_{{\mathbb{H}}_1}f\left\{E_1\left(G_{+}^{H^{-1}}\right)+i{E}_2\left(G_{-}^{H^{-1}}\right)\right\}{\Psi }_0$$

where ${\displaystyle G_{\pm }=\frac{\phi }{{\lambda }^2}\left(h\pm \overline{h}\right)}$. Next $E_1\left(G_{+}^{H^{-1}}\right)+i{E}_2\left(G_{-}^{H^{-1}}\right)$ may be computed from

$$G_{+}+G_{-}=\frac{2\phi h}{{\lambda }^2},G_{+}-G_{-}=\frac{2\phi \overline{h}}{{\lambda }^2},$$

so that

$${\int }_{U}f\phi \Psi =$$

(32)

$$={\int }_{{\mathbb{H}}_1}f\left\{V\left[{\left(\frac{\phi \overline{h}}{{\lambda }^2}\right)}^{H^{-1}}\right]+\overline{V}\left[{\left(\frac{\phi h}{{\lambda }^2}\right)}^{H^{-1}}\right]\right\}{\Psi }_0.$$

Note that (by (30) and (31))

$$h\circ H^{-1}=\frac{1}{2}\frac{{\left(1+\overline{F}\right)}^2}{1+F}.$$

(33)

On the other hand (by (23)),

$${\left(H^{-1}\right)}_{\ast }V={\left(\frac{1}{\overline{h}}Z\right)}^{H^{-1}}.$$

(34)

As F is a CR function, i.e., $\overline{V}\left(F\right)=0$ and $V\left(F\right)=2\overline{\zeta }$ (by (33) and (34) and their complex conjugates)

$$V\left[{\left(\frac{\phi \overline{h}}{{\lambda }^2}\right)}^{H^{-1}}\right]={\overline{h}}^{H^{-1}}V\left[{\left(\frac{\phi }{{\lambda }^2}\right)}^{H^{-1}}\right]+2\overline{\zeta }{\left(\frac{\phi }{{\lambda }^2}\right)}^{H^{-1}}\frac{1+F}{1+\overline{F}},$$

$${\overline{h}}^{H^{-1}}V\left[{\left(\frac{\phi }{{\lambda }^2}\right)}^{H^{-1}}\right]=Z{\left(\frac{\phi }{{\lambda }^2}\right)}^{H^{-1}},$$

and substitution into (32) followed by a change of variable under the integral sign gives

$${\int }_U{v}_1\phi \Psi =-{\int }_{U}u{F}_1\left(\phi \right)\Psi +$$

(35)

$$-{\int }_{U}u\phi \left\{{\lambda }^2{F}_1\left(\frac{1}{{\lambda }^2}\right)+2{\left(\frac{1+\overline{F}}{1+F}\zeta +\frac{1+F}{1+\overline{F}}\overline{\zeta }\right)}^H\right\}\Psi .$$

Finally, by the identities

$${(1+F)}^H=\frac{2}{1+w},{\zeta }^H=\frac{z}{1+w},$$

$$Z\left(\lambda \right)=\frac{\overline{z}\lambda }{1+w},{\left(\frac{1}{{\lambda }^2}\right)}^{H^{-1}}=\frac{4}{{|1+F|}^4},$$

(36)

the last integral in (35) vanishes (yielding (27) with $a=1$). □

The proof of the second relation in (27) is similar. Moreover, $v_a\in L^2(U,\theta )$ because

$$h{E}_a{\left(f\right)}^H\in L^2(U,\theta )$$

(37)

and $L^2(U,\theta )$ is a vector space. As to the proof of (37) (by a change of variable)

$${\int }_U|h{E}_a{\left(f\right)}^H{|}^2\Psi ={\int }_{{\mathbb{H}}_1}{\left(\frac{{\left|h\right|}^2}{{\lambda }^2}\right)}^{H^{-1}}|E_a\left(f\right){|}^2{\Psi }_0=$$

(by (33) and the second identity in (36))

$$={\int }_{{\mathbb{H}}_1}\frac{|E_a\left(f\right){|}^2}{{|1+F|}^2}{\Psi }_0\le {\int }_{{\mathbb{H}}_1}|E_a\left(f\right){|}^2{\Psi }_0<\infty .$$

2.3. Quasiconformal Maps

Let N be a (for now, abstract) strictly pseudoconvex 3-dimensional CR manifold. However, in the applications to come, N will be a strictly pseudoconvex real hypersurface $N\subset {\mathbb{C}}^2$ endowed with the induced CR structure

$$T_{1,0}\left(N\right)=\left[T\left(N\right)\otimes \mathbb{C}\right]\cap T^{1,0}\left({\mathbb{C}}^2\right).$$

Definition 1. 

A $C^{\infty }$ diffeomorphism $f:S^3\to N$ is a contact transformation if

$$\left(d_{x}f\right)H{\left(S^3\right)}_x=H{\left(N\right)}_{f\left(x\right)},x\in S^3.$$

Note that the notion of a contact transformation does not depend upon the particular CR structures one may set on $S^3$ and N, but only on their Levi distributions.

Lemma 4. 

Let $\Theta \in {\mathcal{P}}_{+}\left(N\right)$ be a positively oriented contact form on N and let $f:S^3\to N$ be a $C^{\infty }$ diffeomorphism. The following statements are equivalent:

(i)

f is a contact transformation of $\left(S^3,H\left(S^3\right)\right)$ into $\left(N,H\left(N\right)\right)$.

(ii)

There is a $C^{\infty }$ function

$$\lambda =\lambda \left(f\right)=\lambda \left(f;\theta ,\Theta \right):S^3\to \mathbb{R}\backslash \left\{0\right\}$$

such that $f^{\ast }\Theta =\lambda \theta$.

Proof. 

(i) ⟹ (ii). There exist functions $\lambda ,{\lambda }_1\in C^{\infty }\left(S^3,\mathbb{C}\right)$ such that

$$f^{\ast }\Theta =\lambda \theta +{\lambda }_1{\theta }^1+{\lambda }_{\overline{1}}{\theta }^{\overline{1}}$$

where ${\lambda }_{\overline{1}}=\overline{{\lambda }_1}$. Then for any $x\in S^3$

$${\lambda }_1\left(x\right)={\left(\left(f^{\ast }\Theta \right)T_1\right)}_x={\Theta }_{f\left(x\right)}\left[\left(d_{x}f\right)T_{1,x}\right]=0$$

because of

$$\left(d_{x}f\right)T_{1,x}\in \left(d_{x}f\right)H{\left(S^3\right)}_x{\otimes }_{\mathbb{R}}\mathbb{C}\subset H{\left(N\right)}_{f\left(x\right)}{\otimes }_{\mathbb{R}}\mathbb{C}=\mathrm{Ker}{\left(\Theta \right)}_{f\left(x\right)}{\otimes }_{\mathbb{R}}\mathbb{C}.$$

Then $f^{\ast }\Theta =\lambda \theta$, and $\lambda$ is real valued. To show that $\lambda$ is nowhere vanishing, we argue by contradiction. Let us assume that $\lambda \left(x_0\right)=0$ for some $x_0\in S^3$. Then

$$0=\lambda \left(x_0\right){\theta }_{x_0}={\left(f^{\ast }\Theta \right)}_{x_0}={\Theta }_{f\left(x_0\right)}\circ \left(d_{x_0}f\right)$$

and hence for every $v\in T_{x_0}\left(S^3\right)$

$$\left(d_{x_0}f\right)v\in \mathrm{Ker}{(\Theta )}_{f\left(x_0\right)}=H{\left(N\right)}_{f\left(x_0\right)}⟹\left(d_{x_0}f\right)T_{x_0}\left(S^3\right)\subset H{\left(N\right)}_{f\left(x_0\right)},$$

a contradiction, because

$${dim}_{\mathbb{R}}{T}_{x_0}\left(S^3\right)=3,{dim}_{\mathbb{R}}H{\left(N\right)}_{f\left(x_0\right)}=2,$$

and $d_{x_0}f$ is a $\mathbb{R}$-linear isomorphism.

(ii) ⟹ (i) Let $v\in H{\left(S^3\right)}_x=\mathrm{Ker}{\left(\theta \right)}_x$. Then

$$0=\lambda \left(x\right){\theta }_x\left(v\right)={\left(f^{\ast }\Theta \right)}_x\left(v\right)={\Theta }_{f\left(x\right)}\left[\left(d_{x}f\right)v\right]⟹$$

$$⟹\left(d_{x}f\right)v\in \mathrm{Ker}{(\Theta )}_{f\left(x\right)}=H{\left(N\right)}_{f\left(x\right)}$$

and hence,

$$\left(d_{x}f\right)H{\left(S^3\right)}_x\subset H{\left(N\right)}_{f\left(x\right)}$$

for any $x\in S^3$. □

It should be observed that in the proof of Lemma 4, use was made of the frames $\left\{T_1\right\}\subset C^{\infty }\left(T_{1,0}\left(S^3\right)\right)$ and $\{{\theta }^1,{\theta }^{\overline{1}},\theta \}\subset C^{\infty }\left(T^{\ast }\left(S^3\right)\otimes \mathbb{C}\right)$ and, therefore, of the canonical CR structure $T_{1,0}\left(S^3\right)$ of the sphere. Any other CR structure $\mathcal{H}$ with the same Levi distribution $H\left(S^3\right)=\mathrm{Re}\left\{\mathcal{H}\oplus \overline{\mathcal{H}}\right\}$ would have worked equally well.

Definition 2. 

The function $\lambda =\lambda (f;\theta ,\Theta )$ is called the dilation of f with respect to the contact forms $\theta \in {\mathcal{P}}_{+}\left(S^3\right)$ and $\Theta \in {\mathcal{P}}_{+}\left(N\right)$.

It obeys the following transformation law, with respect to a transformation of the given (positively oriented) pseudohermitian structures.

Lemma 5. 

Let $f:S^3\to N$ be a contact transformation of $\left(S^3,H\left(S^3\right)\right)$ into $(N,H(N\left)\right)$. Let $\widehat{\theta }=e^u\theta$ and $\widehat{\Theta }=e^v\Theta$ with $u\in C^{\infty }\left(S^3,\mathbb{R}\right)$ and $v\in C^{\infty }\left(N,\mathbb{R}\right)$. Then

$$\lambda \left(f;\widehat{\theta },\widehat{\Theta }\right)=exp\left(v\circ f-u\right)\lambda \left(f;\theta ,\Theta \right).$$

(38)

In particular, $\mathrm{sign}\left[\lambda \left(f;\theta ,\Theta \right)\right]\in \{\pm 1\}$ is a CR invariant.

Proof. 

Let us set $\lambda =\lambda \left(f;\theta ,\Theta \right)$ and $\widehat{\lambda }=\lambda \left(f;\widehat{\theta },\widehat{\Theta }\right)$ for the sake of simplicity. Then

$${\left(f^{\ast }\widehat{\Theta }\right)}_x={\widehat{\Theta }}_{f\left(x\right)}\circ \left(d_{x}f\right)=e^{v\left(f\right(x\left)\right)}{\Theta }_{f\left(x\right)}\circ \left(d_{x}f\right)=$$

$$=e^{v\left(f\right(x\left)\right)}{\left(f^{\ast }\Theta \right)}_x=e^{v\left(f\right(x\left)\right)}{\left(\lambda \theta \right)}_x$$

that is

$$f^{\ast }\widehat{\Theta }=e^{v\circ f}\lambda \theta .$$

On the other hand

$$f^{\ast }\widehat{\Theta }=\widehat{\lambda }\widehat{\theta }=\widehat{\lambda }{e}^u\theta$$

yielding $\widehat{\lambda }=e^{v\circ f-u}\lambda$. □

To fix ideas, from now on, we shall work with contact transformations $f:S^3\to N$ of positive dilation, i.e.,

$$\lambda \left(f;\theta ,\Theta \right)>0$$

with respect to some fixed contact form $\Theta \in {\mathcal{P}}_{+}\left(N\right)$. According to Lemma 5, this is a CR-invariant assumption.

Let $\mathcal{H}$ be an arbitrary CR structure on $S^3$ whose Levi distribution is $H\left(S^3\right)$, and let $K>1$ be a constant.

Definition 3. 

A contact transformation $f:S^3\to N$ is called a K-quasiconformal mapping of the pseudohermitian manifold $(S^3,\mathcal{H},\theta )$ into $(N,T_{1,0}\left(N\right),\Theta )$ if

$$\frac{1}{K}{G}_{\theta ,\mathcal{H}}\left(X,X\right)\le \frac{G_{\Theta }^f\left(f_{\ast }X,f_{\ast }Y\right)}{\lambda (f;\theta ,\Theta )}\le K{G}_{\theta ,\mathcal{H}}\left(X,X\right)$$

(39)

for any $X,Y\in H\left(S^3\right)$.

Here, $G_{\theta ,\mathcal{H}}$ is the Levi form of $\left(S^3,\mathcal{H}\right)$ and $G_{\Theta }^f=G_{\Theta }\circ f$. Additionally, $f_{\ast }X$ denotes the $C^{\infty }$ section in the pullback bundle $f^{-1}T\left(N\right)\to S^3$ given by

$$\left(f_{\ast }X\right)\left(x\right)=\left(d_{x}f\right)X_x\in T_{f\left(x\right)}\left(N\right)={\left(f^{-1}TN\right)}_x,x\in S^3.$$

The same symbol $f_{\ast }$ will denote the vector bundle morphism $f_{\ast }:T\left(S^3\right)\to f^{-1}T\left(N\right)$ (descending to a vector bundle morphism $f_{\ast }:H\left(S^3\right)\to f^{-1}H\left(N\right)$, because f is a contact map) determined by the differential $df$. Let

$$J_N:H\left(N\right)\to H\left(N\right),J_N\left(W+\overline{W}\right)=i\left(W-\overline{W}\right),W\in T_{1,0}\left(N\right),$$

be the complex structure along the Levi distribution $H\left(N\right)$. Let us set

$$J_f:H\left(S^3\right)\to H\left(S^3\right),$$

$$J_{f,x}={\left(d_{x}f\right)}^{-1}\circ J_{N,f\left(x\right)}\circ \left(d_{x}f\right),x\in S^3.$$

Then ${\left(J_f\right)}^2=-I$, and hence $J_f$ determines the CR structure

$${\mathcal{H}}_f=\mathrm{Eigen}\left(J_f^{\mathbb{C}};+i\right)\subset H\left(S^3\right)\otimes \mathbb{C}$$

whose Levi distribution is once again $H\left(S^3\right)$. Let $G_f$ be the Levi form of $\left(S^3,{\mathcal{H}}_f\right)$, i.e.,

$$G_{\theta }^f\left(X,Y\right)=\left(d\theta \right)\left(X,J_{f}Y\right),X,Y\in H\left(S^3\right).$$

One has

$$G_{\Theta }^f{\left(f_{\ast }X,f_{\ast }Y\right)}_x=G_{\Theta ,f\left(x\right)}\left(\left(d_{x}f\right)X_x,\left(d_{x}f\right)Y_x\right)=$$

$$={\left(d\Theta \right)}_{f\left(x\right)}\left(\left(d_{x}f\right)X_x,J_{N,f\left(x\right)}\left(d_{x}f\right)Y_x\right)={(d\Theta )}_{f\left(x\right)}\left(\left(d_{x}f\right)X_x,\left(d_{x}f\right)J_{f,x}{Y}_x\right)=$$

$$=\left(f^{\ast }d\Theta \right){(X,J_{f}Y)}_x=\left(d{f}^{\ast }\Theta \right){(X,J_{f}Y)}_x=\left(d\left(\lambda \theta \right)\right){(X,J_{f}Y)}_x=$$

$$=\left(d\lambda \wedge \theta +\lambda d\theta \right)(X,J_{f}Y{)}_x=\lambda \left(x\right)\left(d\theta \right){(X,J_{f}Y)}_x$$

that is,

$$G_{\Theta }^f\left(f_{\ast }X,f_{\ast }Y\right)=\lambda G_f(X,Y)$$

(40)

for any $X,Y\in H\left(S^3\right)$. Consequently the K-quasiconformal requirement (39) may be rephrased as

$$\frac{1}{K}{G}_{\theta ,\mathcal{H}}\left(X,X\right)\le G_f\left(X,X\right)\le K{G}_{\theta ,\mathcal{H}}\left(X,X\right).$$

(41)

Lemma 6. 

Let $f:S^3\to N$ be a contact transformation of $\left(S^3,H\left(S^3\right)\right)$ into $\left(N,H\left(N\right)\right)$. Let $\mathcal{H}$ be a CR structure on $S^3$ such that $H\left(S^3\right)=\mathrm{Re}\left\{\mathcal{H}\oplus \overline{\mathcal{H}}\right\}$ and $\theta \in {\mathcal{P}}_{+}\left(S^3,\mathcal{H}\right)$. If $x\in S^3$ and $w\in {\overline{\mathcal{H}}}_x$ with $w\ne 0$ then

$$\left(d_{x}f\right)w\notin T_{f\left(x\right)}{\left(N\right)}_{f\left(x\right)}$$

that is

$$T_{1,0}{\left(N\right)}_{f\left(x\right)}\cap \left(d_{x}f\right){\overline{\mathcal{H}}}_x=\left(0\right).$$

(42)

Proof. 

We argue by contradiction, i.e., we assume that

$$\left(d_{x}f\right)w\in T_{1,0}{\left(N\right)}_{f\left(x\right)}$$

for some $w\in {\overline{\mathcal{H}}}_x\subset H{\left(S^3\right)}_x{\otimes }_{\mathbb{R}}\mathbb{C}$ with $w\ne 0$. Then (as $T_{1,0}\left(N\right)$ is nondegenerate and $\Theta$ is positively oriented)

$$0<G_{\Theta ,f\left(x\right)}\left(\left(d_{x}f\right)w,\overline{\left(d_{x}f\right)w}\right)=$$

$$={\left(d\Theta \right)}_{f\left(x\right)}\left(\left(d_{x}f\right)w,J_{N,f\left(x\right)}\overline{\left(d_{x}f\right)w}\right)=$$

[as $\overline{\left(d_{x}f\right)w}\in T_{0,1}{\left(N\right)}_{f\left(x\right)}$ and $d_{x}f$ is real]

$$=-i{\left(d\Theta \right)}_{f\left(x\right)}\left(\left(d_{x}f\right)w,\left(d_{x}f\right)\overline{w}\right)=-i{\left(d{f}^{\ast }\Theta \right)}_x(w,\overline{w})=$$

$$=-i{\left(d\left(\lambda \theta \right)\right)}_x(w,\overline{w})=i{\left(d\lambda \wedge \theta +\lambda d\theta \right)}_x\left(\overline{w},w\right)=$$

$$=i\lambda \left(x\right){\left(d\theta \right)}_x\left(\overline{w},w\right)=-\lambda \left(x\right){\left(d\theta \right)}_x\left(\overline{w},J_x^{\mathcal{H}}w\right)=$$

$$=-\lambda \left(x\right)G_{\theta ,\mathcal{H},x}\left(\overline{w},w\right)<0,$$

a contradiction. □

Here, we assumed that the canonical contact form (7) is positively oriented relative to $(S^3,\mathcal{H})$. Otherwise, one merely replaces $\theta$ by $-\theta$ to start with.

The contents of (42) are that, solely as a consequence of $f:S^3\to N$ being a contact transformation of positive dilation $\lambda \left(f\right)>0$,

$$\left(f_{\ast }\overline{\mathcal{H}}\right)\cap f^{-1}{T}_{1,0}\left(N\right)=\left(0\right)$$

for every CR structure $\mathcal{H}$ on $S^3$ whose Levi distribution is $H\left(S^3\right)$.

Let $f:S^3\to N$ and $\mathcal{H}$ be as in Lemma 6. Next, let

$$\left\{L\right\}\subset C^{\infty }\left(U,\mathcal{H}\right),\left\{T_1^N\right\}\subset C^{\infty }\left(V,T_{1,0}\left(N\right)\right),$$

be local frames in $\mathcal{H}$ and $T_{1,0}\left(N\right)$ respectively, defined on the open subsets $U\subset S^3$ and $V\subset N$ such that $U=f^{-1}\left(V\right)$. For every $x\in U$

$$\left(d_{x}f\right){\overline{L}}_x=f_{\overline{1}}^1\left(x;\mathcal{H}\right)T_{1,f\left(x\right)}^N+f_{\overline{1}}^{\overline{1}}\left(x;\mathcal{H}\right)T_{\overline{1},f\left(x\right)}^N$$

for some functions

$$f_{\overline{1}}^1\left(·;\mathcal{H}\right),f_{\overline{1}}^{\overline{1}}\left(·;\mathcal{H}\right)\in C^{\infty }\left(U,\mathbb{C}\right).$$

The adopted notation emphasizes the dependence of the coefficients $f_{\overline{1}}^1$ and $f_{\overline{1}}^{\overline{1}}$ on the CR structure $\mathcal{H}$. Occasionally, if there is no danger of confusion, we drop $\mathcal{H}$ and write merely

$$f_{\overline{1}}^1=f_{\overline{1}}^1\left(·;\mathcal{H}\right),f_{\overline{1}}^{\overline{1}}=f_{\overline{1}}^{\overline{1}}\left(·;\mathcal{H}\right).$$

Lemma 7. 

One has

$$f_{\overline{1}}^{\overline{1}}\left(x;\mathcal{H}\right)\ne 0$$

(43)

for any $x\in U$.

Proof. 

We argue by contradiction, i.e., we assume that $f_{\overline{1}}^{\overline{1}}\left(x_0\right)=0$ for some $x_0\in U$. Then

$$\left(d_{x_0}f\right){\overline{L}}_{x_0}=f_{\overline{1}}^1\left(x_0\right)T_{1,f\left(x_0\right)}^N\in T_{1,0}{\left(N\right)}_{f\left(x_0\right)}$$

and ${\overline{L}}_{x_0}\ne 0$, in contradiction with Lemma 6. □

We adopt the temporary notation

$${\widehat{\mathcal{H}}}_f=\left\{Z\in H\left(S^3\right)\otimes \mathbb{C}:f_{\ast }Z\in f^{-1}{T}_{1,0}\left(N\right)\right\}.$$

(44)

Then

$${\widehat{\mathcal{H}}}_f\cap \overline{\mathcal{H}}=\left(0\right)$$

for any CR structure $\mathcal{H}$ on $S^3$ as in Lemma 6.

Lemma 8. 

Let $f:S^3\to N$ be a contact transformation of positive dilation $\lambda \left(f\right)>0$. For every CR structure $\mathcal{H}$ on $S^3$ such that

$$H\left(S^3\right)=\mathrm{Re}\left\{\mathcal{H}\oplus \overline{\mathcal{H}}\right\},\theta \in {\mathcal{P}}_{+}\left(S^3,\mathcal{H}\right),$$

(45)

there is a field

$$\mu =\mu \left(f,\mathcal{H}\right):\mathcal{H}\to \mathcal{H}$$

of $\mathbb{C}$-anti-linear maps such that

$${\widehat{\mathcal{H}}}_f=\left\{Z-\overline{\mu Z}:Z\in \mathcal{H}\right\}.$$

(46)

Proof. 

Let us start with $W\in {\widehat{\mathcal{H}}}_f$ represented as

$$W=A^{1}L+B^{\overline{1}}\overline{L}$$

with respect to the local frame $\{L,\overline{L}\}$ of $H\left(S^3\right)\otimes \mathbb{C}$. Then

$$f^{-1}{T}_{1,0}\left(N\right)\ni f_{\ast }W=\left(A^1{f}_1^1+B^{\overline{1}}{f}_{\overline{1}}^1\right)T_1^N+\left(A^1{f}_1^{\overline{1}}+B^{\overline{1}}{f}_{\overline{1}}^{\overline{1}}\right)T_{\overline{1}}^N$$

yielding

$$B^{\overline{1}}=-\frac{f_1^{\overline{1}}}{f_{\overline{1}}^{\overline{1}}}{A}^1.$$

Therefore

$$W=A^1\left(L-\frac{f_1^{\overline{1}}}{f_{\overline{1}}^{\overline{1}}}\overline{L}\right),$$

i.e., ${\widehat{\mathcal{H}}}_f$ is (locally, on U) the span of $\left\{L-\left(f_1^{\overline{1}}/f_{\overline{1}}^{\overline{1}}\right)\overline{L}\right\}$. □

Let $x\in S^3$ be an arbitrary point and let us choose local frames $\left\{L\right\}$ and $\left\{T_1^N\right\}$ of the CR structures $\mathcal{H}$ and $T_{1,0}\left(N\right)$, defined on open neighborhoods of the points x and $f\left(x\right)$

$$x\in U\subset S^3,f\left(x\right)\in V\subset N,U=f^{-1}\left(V\right).$$

Our rather pedantic approach to the construction of ${\mu }_x$ (see below) is devised to emphasize that the resulting $\mu$ is globally defined. Indeed we set by definition

$${\mu }_x:{\mathcal{H}}_x\to {\mathcal{H}}_x,{\mu }_x{L}_x=\frac{f_{\overline{1}}^1\left(x;\mathcal{H}\right)}{f_1^1\left(x;\mathcal{H}\right)}{L}_x,$$

(47)

followed by the $\mathbb{C}$-anti-linear extension to the whole of ${\mathcal{H}}_x$. The definition of ${\mu }_x$ does not depend upon the choice of local frames about x and $f\left(x\right)$. Indeed, let us consider local frames

$$\{L^{\prime }\}\subset C^{\infty }\left(U^{\prime },\mathcal{H}\right),\{{T_1^{\prime }}^N\}\subset C^{\infty }\left(V^{\prime },T_{1,0}\left(N\right)\right),$$

$$x\in U^{\prime }\subset S^3,f\left(x\right)\in V^{\prime }\subset N,U^{\prime }=f^{-1}\left(V^{\prime }\right).$$

Then

$$L^{\prime }=u_1^{1}L\mathrm{on}U\cap U^{\prime },{T_1^{\prime }}^N=v_1^1{T}_1^N\mathrm{on}V\cap V^{\prime },$$

for some $C^{\infty }$ functions $u_1^1:U\cap U^{\prime }\to \mathbb{C}$ and $v_1^1:V\cap V^{\prime }\to \mathbb{C}$. A comparison of the representations

$$f_{\ast }L=f_1^1{T}_1^N+f_1^{\overline{1}}{T}_{\overline{1}}^N,f_{\ast }{L}^{\prime }={f^{\prime }}_1^1{T_1^{\prime }}^N+{f^{\prime }}_1^{\overline{1}}{T_{\overline{1}}^{\prime }}^N,$$

yields

$${f^{\prime }}_1^1{v}_1^1=u_1^1{f}_1^1,{f^{\prime }}_1^{\overline{1}}{v}_{\overline{1}}^{\overline{1}}=u_1^1{f}_1^{\overline{1}},$$

(48)

where $v_{\overline{1}}^{\overline{1}}=\overline{v_1^1}$. Let ${\mathcal{H}}_A$ denote the portion of $\mathcal{H}$ over the open set $A\subset S^3$. If

$$\mu :{\mathcal{H}}_U\to {\mathcal{H}}_U,{\mu }^{\prime }:{\mathcal{H}}_{U^{\prime }}\to {\mathcal{H}}_{U^{\prime }},$$

$$\mu L=\frac{f_{\overline{1}}^1}{f_1^1}L,{\mu }^{\prime }{L}^{\prime }=\frac{{f^{\prime }}_{\overline{1}}^1}{{f^{\prime }}_1^1}{L}^{\prime },$$

what one needs to check is that ${\left.\mu \right|}_{U\cap U^{\prime }}={\left.{\mu }^{\prime }\right|}_{U\cap U^{\prime }}$. This is but a standard calculation relying on (48).

Summing up, we built a family of vector bundle morphisms

$$\mu \left(f,\mathcal{H}\right):\mathcal{H}\to \mathcal{H}$$

(49)

associated to the contact transformation $f:S^3\to N$ with $\lambda \left(f\right)>0$, such that ${\widehat{\mathcal{H}}}_f$ is represented by (46). Let $\mathrm{CR}\left[H\left(S^3\right)\right]$ be the set of all CR structures $\mathcal{H}$ on $S^3$ obeying to (45). The family of morphisms (49) is then parametrized by $\mathcal{H}\in \mathrm{CR}\left[H\left(S^3\right)\right]$.

Definition 4. 

Each $\mu (f,\mathcal{H})$ is referred to as the complex dilation of f with respect to the CR structure $\mathcal{H}$.

We previously mentioned that ${\widehat{\mathcal{H}}}_f$ is but a temporary name for the bundle on the right-hand side of (44). Indeed, one has

Lemma 9. 

${\widehat{\mathcal{H}}}_f={\mathcal{H}}_f$.

Proof. 

If $J_N^f=J_N\circ f$, then

$${\mathcal{H}}_f=\mathrm{Eigen}\left[{\left(J_f\right)}^{\mathbb{C}};+i\right]=\left\{Z\in H\left(S^3\right)\otimes \mathbb{C}:J_{f}Z=iZ\right\}=$$

$$=\left\{Z\in H\left(S^3\right)\otimes \mathbb{C}:J_N^f\left(f_{\ast }Z\right)=i{f}_{\ast }Z\right\}=$$

$$=\left\{Z\in H\left(S^3\right)\otimes \mathbb{C}:f_{\ast }Z\in f^{-1}\mathrm{Eigen}\left[{\left(J_N\right)}^{\mathbb{C}};+i\right]\right\}=$$

$$=\left\{Z\in H\left(S^3\right)\otimes \mathbb{C}:f_{\ast }Z\in f^{-1}{T}_{1,0}\left(N\right)\right\}={\widehat{\mathcal{H}}}_f.$$

□

By a result of H. Rossi (cf. [1]), the CR manifold $\left(S^3,\mathcal{H}\left(t\right)\right)$ is not globally embeddable in ${\mathbb{C}}^2$, for any $0<\left|t\right|<1$. Hence, for every nondegenerate CR hypersurface $N\subset {\mathbb{C}}^2$, there is no CR isomorphism $f:\left(S^3,\mathcal{H}\left(t\right)\right)\to N$, except of course for $t=0$ (when one may consider $N=S^3$ and $f=1_{S^3}$). We propose the following weaker version of the global CR embedding problem.

Problem 1. 

Given a strictly pseudoconvex CR manifold M of CR dimension n, find (i) a real hypersurface $N\subset {\mathbb{C}}^{n+1}$ whose induced CR structure $T_{1,0}\left(N\right)$ is strictly pseudoconvex, (ii) a constant $K>1$, and (iii) a K-quasiconformal map $f:M\to N$.

Our treatment of the question in Problem 1 is confined to H. Rossi’s nonembeddable examples $\left(S^3,\mathcal{H}\left(t\right)\right)$. Precisely, we shall discuss the following.

Problem 2. 

Find (i) a function $K:(-1,1)\to (1,+\infty )$, (ii) a family ${\left\{N_t\right\}}_{0<\left|t\right|<1}$ of nondegenerate real hypersurfaces $N_t\subset {\mathbb{C}}^2$, and (iii) a family ${\left\{f_t\right\}}_{0<\left|t\right|<1}$ of $K\left(t\right)$-quasiconformal maps $f_t:\left(S^3,\mathcal{H}\left(t\right)\right)\to N_t$.

2.4. Beltrami’s Equation

Let $N\subset {\mathbb{C}}^2$ be a nondegenerate real hypersurface, and let $f=\left(f^1,f^2\right):S^3\to N$ be a contact transformation of $\left(S^3,H\left(S^3\right)\right)$ into $\left(N,H\left(N\right)\right)$ with $\lambda \left(f\right)>0$. By Lemmas 8 and 9

$$L_t-\overline{{\mu }_f\left(t\right)L_t}\in {\mathcal{H}}_f,\left|t\right|<1,$$

where we have set

$${\mu }_f\left(t\right)=\mu \left[f,\mathcal{H}\left(t\right)\right]:\mathcal{H}\left(t\right)\to \mathcal{H}\left(t\right).$$

Hence (by the very definition of ${\widehat{\mathcal{H}}}_f$)

$$f^{-1}{T}_{1,0}\left(N\right)\ni f_{\ast }\left[L_t-\overline{{\mu }_f\left(t\right)L_t}\right]=f_{\ast }\left[L_t-{\mu }_1^{\overline{1}}\left(t\right){\overline{L}}_t\right]$$

(50)

where the functions ${\mu }_1^{\overline{1}}\left(t\right):S^3\to \mathbb{C}$ are given by

$${\mu }_f\left(t\right)L_t={\mu }_{\overline{1}}^1\left(t\right)L_t,{\mu }_{\overline{1}}^1\left(t\right)=\frac{f_{\overline{1}}^1\left(t\right)}{f_1^1\left(t\right)},{\mu }_1^{\overline{1}}\left(t\right)=\overline{{\mu }_{\overline{1}}^1\left(t\right)},$$

$$f_B^A\left(t\right)=f_B^A\left[·;\mathcal{H}\left(t\right)\right],A,B\in \left\{1,\overline{1}\right\}.$$

Lemma 10. 

Let $f=\left(f^1,f^2\right):S^3\to N$ be a contact transformation of $\left(S^3,H\left(S^3\right)\right)$ into $\left(N,H\left(N\right)\right)$ with $\lambda \left(f\right)>0$. The components $f^j:S^3\to \mathbb{C}$ satisfy Beltrami’s equations

$${\overline{L}}_t\left(f^j\right)={\mu }_{\overline{1}}^1\left(t\right)L_t\left(f^j\right),j\in \{1,2\},\left|t\right|<1.$$

(51)

Proof. 

One has (by (50))

$$f^{-1}{T}^{1,0}\left({\mathbb{C}}^2\right)\supset f^{-1}{T}_{1,0}\left(N\right)\ni L_t\left(f^j\right)\frac{\partial }{\partial {\zeta }^j}+L_t\left(\overline{f^j}\right)\frac{\partial }{\partial {\overline{\zeta }}^j}+$$

$$-{\mu }_1^{\overline{1}}\left(t\right)\left\{{\overline{L}}_t\left(f^j\right)\frac{\partial }{\partial {\zeta }^j}+{\overline{L}}_t\left(\overline{f^j}\right)\frac{\partial }{\partial {\overline{\zeta }}^j}\right\}$$

so that

$$L_t\left(\overline{f^j}\right)-{\mu }_1^{\overline{1}}\left(t\right){\overline{L}}_t\left(\overline{f^j}\right)=0$$

or (by taking complex conjugates)

$${\overline{L}}_t\left(f^j\right)={\mu }_{\overline{1}}^1\left(t\right)L_t\left(f^j\right)$$

which is (51). □

Lemma 11. 

Let $\mu =\mu \left[f,T_{1,0}\left(S^3\right)\right]:T_{1,0}\left(S^3\right)\to T_{1,0}\left(S^3\right)$ be the complex dilation of $f:S^3\to N$ relative to the canonical CR structure $\mathcal{H}\left(0\right)=T_{1,0}\left(S^3\right)$. If $\mu T_1={\mu }_{\overline{1}}^1{T}_1$ then

$${\mu }_{\overline{1}}^1\left(t\right)=\frac{{\mu }_{\overline{1}}^1+t}{1+t{\mu }_{\overline{1}}^1}$$

(52)

for every $\left|t\right|<1$. In particular, the coefficients of the complex dilation ${\mu }_f\left(t\right)$ depend smoothly on the parameter t.

Proof. 

As ${\overline{L}}_0=T_{\overline{1}}$

$$\left(d_{x}f\right)T_{\overline{1},x}=f_{\overline{1}}^1\left(x,0\right)T_{1,f\left(x\right)}^N+f_{\overline{1}}^{\overline{1}}\left(x,0\right)T_{\overline{1},f\left(x\right)}^N$$

hence

$$\left(d_{x}f\right){\overline{L}}_{t,x}=\left(d_{x}f\right)T_{1,x}+t\left(d_{x}f\right)T_{\overline{1},x}=$$

$$=f_1^1(x,0)T_{1,f\left(x\right)}^N+f_1^{\overline{1}}(x,0)T_{\overline{1},f\left(x\right)}^N+$$

$$+t\left[f_{\overline{1}}^1(x,0)T_{1,f\left(x\right)}^N+f_{\overline{1}}^{\overline{1}}(x,0)T_{\overline{1},f\left(x\right)}^N\right]=$$

$$=\left[f_1^1(x,0)+t{f}_{\overline{1}}^1(x,0)\right]T_{1,f\left(x\right)}^N+\left[f_1^{\overline{1}}(x,0)+t{f}_{\overline{1}}^{\overline{1}}(x,0)\right]T_{\overline{1},f\left(x\right)}^N$$

yielding

$$f_{\overline{1}}^1(x,t)=f_1^1(x,0)+t{f}_{\overline{1}}^1(x,0),f_{\overline{1}}^{\overline{1}}(x,t)=f_1^{\overline{1}}(x,0)+t{f}_{\overline{1}}^{\overline{1}}(x,0).$$

(53)

Let us set

$$f_B^A\left(x\right)=f_B^A(x,0),A,B\in \left\{1,\overline{1}\right\}.$$

According to the definition (47), the coefficients of the complex dilation $\mu ={\mu }_f\left(0\right)$ are given by

$$\mu T_1={\mu }_{\overline{1}}^1\left(·,0\right)T_1,{\mu }_{\overline{1}}^1\left(·,0\right)=\frac{f_{\overline{1}}^1}{f_1^1}.$$

Next, let us set $t=0$ into (51) to obtain

$$T_{\overline{1}}\left(f^j\right)={\mu }_{\overline{1}}^1\left(·,0\right)T_1\left(f^j\right).$$

(54)

Let us set

$${\mu }_B^A\left(x\right)={\mu }_B^A\left(x,0\right),A,B\in \left\{1,\overline{1}\right\}.$$

Then

$${\mu }_{\overline{1}}^1(x,t)=\frac{f_{\overline{1}}^1(x,t)}{f_1^1(x,t)}=$$

[by (53) and (54)]

$$=\frac{f_{\overline{1}}^1(x,0)+t{f}_1^1(x,0)}{f_1^1(x,0)+t{f}_{\overline{1}}^1(x,0)}=\frac{{\mu }_{\overline{1}}^1\left(x\right)+t}{1+t{\mu }_{\overline{1}}^1\left(x\right)}.$$

□

Corollary 1. 

The components $f^j$ of a contact transformation $f:S^3\to N\subset {\mathbb{C}}^2$ under the assumptions of Lemma 10 satisfy the Beltrami equation

$${\overline{L}}_t\left(f^j\right)=\frac{{\mu }_{\overline{1}}^1+t}{1+t{\mu }_{\overline{1}}^1}{L}_t\left(f^j\right)$$

for any $j\in \{1,2\}$.

Let $\left(N,T_{1,0}\left(N\right)\right)$ be a nondegenerate 3-dimensional CR manifold and let $\Theta \in {\mathcal{P}}_{+}\left(N\right)$ be a positively oriented contact form. Let $f:S^3\to N$ be a contact transformation with $\lambda \left(f\right)=\lambda (f;\theta ,\Theta )>0$. Let $\mathcal{H}\in \mathrm{CR}\left[H\left(S^3\right)\right]$ and let ${\mu }_f=\mu \left(f,\mathcal{H}\right):\mathcal{H}\to \mathcal{H}$ be the complex dilation of f.

Definition 5. 

The pointwise norm of ${\mu }_f$ is the function $\parallel {\mu }_f\parallel :S^3\to [0,+\infty )$ defined by

$$\parallel {\mu }_f\parallel \left(x\right)={\left[\underset{0\ne Z\in {\mathcal{H}}_x}{sup}\frac{G_{\theta ,x}\left({\mu }_{f,x}Z,\overline{{\mu }_{f,x}Z}\right)}{G_{\theta ,x}\left(Z,\overline{Z}\right)}\right]}^{1/2},x\in S^3.$$

We shall need the following

Theorem 3. 

Let $\mathcal{H}\in \mathrm{CR}\left[H\left(S^3\right)\right]$ and let $f:S^3\to N$ be a contact transformation with $\lambda \left(f\right)>0$. The following statements are equivalent:

(i)

There is $K>1$ such that f is K-quasiconformal.

(ii)

There is $K>1$ such that

$$\parallel {\mu }_f\parallel \le \frac{K-1}{K+1}.$$

(55)

Theorem 3 is stated in [3], p. 61, with $\left(S^3,\mathcal{H}\right)$ replaced by an arbitrary strictly pseudoconvex manifold M, yet the proof is confined to the case where $M=N={\mathbb{H}}_n$ (the Heisenberg group). We give (by following the ideas in [3], pp. 63–65) a proof of the statement as it applies to Rossi’s spheres, and refer to Theorem 3 as the Koranyi–Reimann characterization theorem.

Proof of Theorem 3. 

Let $x_0\in S^3$ and let us choose an open neighborhood $V\subset N$ of $f\left(x_0\right)$ and local orthonormal frames

$$\left\{Z_1\right\}\subset C^{\infty }\left(U,\mathcal{H}\right),\left\{T_1^N\right\}\subset C^{\infty }\left(V,T_{1,0}\left(N\right)\right),U=f^{-1}\left(V\right),$$

$$G_{\theta }\left(Z_1,Z_{\overline{1}}\right)=1,G_{\Theta }\left(T_1^N,T_{\overline{1}}^N\right)=1.$$

Next, let us set

$$E_1=Z_1+Z_{\overline{1}},E_2=J{E}_1=i\left(Z_1-Z_{\overline{1}}\right),$$

$$E_1^N=T_1^N+T_{\overline{1}}^N,E_2^N=J^N{E}_1^N=i\left(T_1^N-T_{\overline{1}}^N\right),$$

so that $\left\{E_a:a\in \{1,2\}\right\}$ and $\left\{E_a^N:a\in \{1,2\}\right\}$ are respectively local frames of $H\left(S^3\right)$ and $H\left(N\right)$. Then

$$f_{\ast }{E}_b=F_b^a{\left(E_a^N\right)}^f$$

for some $C^{\infty }$ functions $F_b^a:U\to \mathbb{R}$ such that $det\left[F_b^a\left(x\right)\right]\ne 0$ for any $x\in U$. Let us consider

$$g:U\to \mathrm{GL}\left(2,\mathbb{R}\right),g=\frac{1}{\sqrt{\lambda \left(f\right)}}\left[\begin{array}{cc}{F}_1^1& F_2^1\\ F_1^2& F_2^2\end{array}\right].$$

We shall need the symplectic group

$$\mathrm{Sp}\left(2,\mathbb{R}\right)=\left\{a\in \mathrm{GL}(2,\mathbb{R}):a^{\tau }{J}_{0}a=J_0\right\},J_0=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right].$$

Lemma 12. 

g is $\mathrm{Sp}\left(2,\mathbb{R}\right)$-valued.

Proof. 

One has

$$\left(d\theta \right)(E_1,E_2)=-2i\left(d\theta \right)(Z_1,Z_{\overline{1}})=2G_{\theta }(Z_1,Z_{\overline{1}})=2$$

and similarly

$$\left(d\Theta \right)(E_1^N,E_2^N)=2.$$

Then [by $f^{\ast }\Theta =\lambda \left(f\right)\theta$ and $\mathrm{Ker}\left(\theta \right)=H\left(S^3\right)$]

$${\left(d\Theta \right)}^f\left(f_{\ast }{E}_1,f_{\ast }{E}_2\right)=\left(d{f}^{\ast }\Theta \right)\left(E_1,E_2\right)=\lambda \left(f\right)\left(d\theta \right)(E_1,E_2)=2\lambda \left(f\right).$$

On the other hand

$${\left(d\Theta \right)}^f\left(f_{\ast }{E}_1,f_{\ast }{E}_2\right)=\left(F_1^1{F}_2^2-F_1^2{F}_2^1\right)(d\Theta )(E_1^N,E_2^N)\circ f=$$

$$=2\left(F_1^1{F}_2^2-F_1^2{F}_2^1\right)=2\lambda \left(f\right)det\left(g\right).$$

It follows that $det\left(g\right)=1$. □

Let us set

$$K=\mathrm{Sp}\left(2,\mathbb{R}\right)\cap \mathrm{O}\left(2\right),A^{+}=\left\{\left(\begin{array}{cc}{e}^s& 0\\ 0& e^{-s}\end{array}\right):s\ge 0\right\},$$

$$j:\mathrm{GL}\left(1,\mathbb{C}\right)=\mathbb{C}\backslash \left\{0\right\}\to \mathrm{GL}\left(2,\mathbb{R}\right),j(x+iy)=\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right).$$

Lemma 13. 

$K=\mathrm{O}\left(2\right)\cap j\left[\mathrm{U}\left(1\right)\right]$.

The proof is straightforward and therefore omitted. Here $\mathrm{U}\left(n\right)=\left\{a\in \mathrm{GL}(n,\mathbb{C}):{\overline{a}}^{\tau }a=I_n\right\}$ so that $\mathrm{U}\left(1\right)=\{a\in \mathbb{C}:|a|=1\}$. We shall need the Cartan decomposition of $\mathrm{Sp}(2,\mathbb{R})$

$$\mathrm{Sp}(2,\mathbb{R})=K{A}^{+}K.$$

By Lemma 12, there exist functions $k,k^{\prime }:U\to K$ and $a:U\to A^{+}$ such that

$$g=ka{k}^{\prime }$$

on U, i.e., there exist $x,y,u,v:U\to \mathbb{R}$ and $s:U\to [0,+\infty )$ such that

$$x^2+y^2=1,u^2+v^2=1,$$

$$g=\left(\begin{array}{cc}x& y\\ -y& x\end{array}\right)\left(\begin{array}{cc}{e}^s& 0\\ 0& e^{-s}\end{array}\right)\left(\begin{array}{cc}u& v\\ -v& u\end{array}\right)=$$

(56)

$$=\left(\begin{array}{cc}xu{e}^s-yv{e}^{-s}& xv{e}^s+yu{e}^{-s}\\ & \\ -yu{e}^s-xv{e}^{-s}& -yv{e}^s+xu{e}^{-s}\end{array}\right)$$

on U. Next

$$Z_1=\frac{1}{2}\left(E_1-i{E}_2\right),Z_{\overline{1}}=\frac{1}{2}\left(E_1+i{E}_2\right),$$

$$T_1^N=\frac{1}{2}\left(E_1^N-i{E}_2^N\right),T_{\overline{1}}^N=\frac{1}{2}\left(E_1^N+i{E}_2^N\right),$$

$$f_{\ast }{Z}_1=f_1^1{\left(T_1^N\right)}^f+f_1^{\overline{1}}{\left(T_{\overline{1}}^N\right)}^f,f_{\ast }{Z}_{\overline{1}}=f_{\overline{1}}^1{\left(T_1^N\right)}^f+f_{\overline{1}}^{\overline{1}}{\left(T_{\overline{1}}^N\right)}^f,$$

yield

$$f_{\ast }{Z}_1=\frac{1}{2}\left[F_1^1+F_2^2+i\left(F_1^2-F_2^1\right)\right]{\left(T_1^N\right)}^f+$$

$$+\frac{1}{2}\left[F_1^1-F_2^2-i\left(F_1^2+F_2^1\right)\right]{\left(T_{\overline{1}}^N\right)}^f$$

and if $g=\left(\begin{array}{cc}{g}_1^1& g_2^1\\ g_1^2& g_2^2\end{array}\right)$ then

$$f_1^1=\frac{1}{2}\left[F_1^1+F_2^2+i\left(F_1^2-F_2^1\right)\right]=$$

(57)

$$=\frac{1}{2}\sqrt{\lambda \left(f\right)}\left(g_1^1+g_2^2\right)+\frac{i}{2}\sqrt{\lambda \left(f\right)}\left(g_1^2-g_2^1\right),$$

$$f_1^{\overline{1}}=\frac{1}{2}\left[F_1^1-F_2^2-i\left(F_1^2+F_2^1\right)\right]=$$

(58)

$$=\frac{1}{2}\sqrt{\lambda \left(f\right)}\left(g_1^1-g_2^2\right)-\frac{i}{2}\sqrt{\lambda \left(f\right)}\left(g_1^2+g_2^1\right).$$

Let us substitute from (56) into (57) and (58) to obtain

$$f_1^1=\sqrt{\lambda \left(f\right)}\left(xu-yv-iyu-ixv\right)\frac{e^s+e^{-s}}{2}=$$

(59)

$$=\sqrt{\lambda \left(f\right)}\left(x-iy\right)\left(u-iv\right)coshs,$$

$$f_1^{\overline{1}}=\sqrt{\lambda \left(f\right)}\left(xu+yv+iyu-ixv\right)\frac{e^s-e^{-s}}{2}=$$

(60)

$$=\sqrt{\lambda \left(f\right)}\left(x+iy\right)\left(u-iv\right)sinhs.$$

Lemma 14. 

$\parallel {\mu }_f\parallel =tanhs$.

Proof. 

We start from ${\mu }_{\overline{1}}^1{f}_1^1=f_{\overline{1}}^1$. Then (by (59) and (60) and their complex conjugates),

$${\mu }_{\overline{1}}^1\left(x-iy\right)\left(u-iv\right)coshs=\left(x-iy\right)\left(u+iv\right)sinhs$$

or

$${\mu }_{\overline{1}}^1={\left(u+iv\right)}^{2}tanhs.$$

(61)

Next, for every $x\in U$ and every $W\in {\mathcal{H}}_x\backslash \left\{0_x\right\}$ one has $W=h{Z}_{1,x}$ for some $h\in \mathbb{C}\backslash \left\{0\right\}$ and then

$$\frac{G_{\theta x}\left({\mu }_{f,x}W,\overline{{\mu }_{f,x}W}\right)}{G_{\theta ,x}\left(W,\overline{W}\right)}=$$

$$=\frac{{(u-iv)}^2{(u+iv)}^2{(tanhs)}^2\overline{h}h}{h\overline{h}}={\left[tanhs\left(x\right)\right]}^2$$

and hence, $\parallel {\mu }_f\parallel \left(x\right)=tanhs\left(x\right)$. □

At this point, we may attack the final part of the proof of Theorem 3. We start from $\parallel {\mu }_f\parallel =tanhs$ (cf. Lemma 14) so that

$$\frac{1+\parallel {\mu }_f\parallel }{1-\parallel {\mu }_f\parallel }=\frac{1+tanhs}{1-tanhs}=\frac{e^s}{e^{-s}}=e^{2s}.$$

Recall that both k and $k^{\prime }$ are $\mathrm{O}\left(2\right)$-valued. Then for any $x\in U$ and any $\mathbf{x}\in {\mathbb{R}}^2$

$$\underset{\left|\mathbf{x}\right|=1}{sup}|g\left(x\right)\mathbf{x}|=\underset{\left|\mathbf{x}\right|=1}{sup}|k\left(x\right)a\left(x\right)k^{\prime }\left(x\right)\mathbf{x}|=$$

$$=\underset{\left|\mathbf{x}\right|=1}{sup}|a\left(x\right)k^{\prime }\left(x\right)\mathbf{x}|=\underset{\left|\mathbf{y}\right|=1}{sup}|a\left(x\right)\mathbf{y}|=e^s$$

where $\mathbf{y}=k^{\prime }\left(x\right)\mathbf{x}$. Similarly

$$\underset{\left|\mathbf{x}\right|=1}{inf}|g\left(x\right)\mathbf{x}|=e^{-s}.$$

It follows that

$$\frac{1+\parallel {\mu }_f\parallel \left(x\right)}{1-\parallel {\mu }_f\parallel \left(x\right)}=\frac{{sup}_{\left|\mathbf{x}\right|=1}\left|g\left(x\right)\mathbf{x}\right|}{{inf}_{\left|\mathbf{x}\right|=1}\left|g\left(x\right)\mathbf{x}\right|}.$$

(62)

Proof of (i) ⟹ (ii). 

If f is K-quasiconformal for some $K>1$, then for every $x\in S^3$ and every $X\in H{\left(S^3\right)}_x$

$$\frac{\lambda {\left(f\right)}_x}{K}{G}_{\theta ,x}\left(X,X\right)\le G_{\Theta ,f\left(x\right)}\left(\left(d_{x}f\right)X,\left(d_{x}f\right)X\right)\le \lambda {\left(f\right)}_{x}K{G}_{\theta ,x}\left(X,X\right)$$

or

$$\frac{1}{K}|\mathbf{x}{|}^2\le |g\left(x\right)\mathbf{x}{|}^2\le K|\mathbf{x}{|}^2$$

where

$$\mathbf{x}=\left(x^1,x^2\right)\in {\mathbb{R}}^2,X=x^1{E}_{1,x}+x^2{E}_{2,x}\in H{\left(S^3\right)}_x.$$

Consequently,

$$\frac{1}{\sqrt{K}}\le \underset{\left|\mathbf{x}\right|=1}{inf}|g\left(x\right)\mathbf{x}|,\underset{\left|\mathbf{x}\right|=1}{sup}|g\left(x\right)\mathbf{x}|\le \sqrt{K},$$

so that [by (62)]

$$\frac{1+\parallel {\mu }_f\parallel }{1-\parallel {\mu }_f\parallel }\le K$$

or

$$\parallel {\mu }_f\parallel \le \frac{K-1}{K+1}.$$

□

Proof of (ii) ⟹ (i). 

If there is $K>1$ such that (55) holds, then

$$\frac{e^{s\left(x\right)}}{e^{-s\left(x\right)}}=\frac{{sup}_{\left|\mathbf{x}\right|=1}\left|g\left(x\right)\mathbf{x}\right|}{{inf}_{\left|\mathbf{x}\right|=1}\left|g\left(x\right)\mathbf{x}\right|}=\frac{1+\parallel {\mu }_f\parallel \left(x\right)}{1-\parallel {\mu }_f\parallel \left(x\right)}\le K$$

so that $e^{s\left(x\right)}\le K{e}^{-s\left(x\right)}$. Let $x^1{E}_{1,x}+x^2{E}_{2,x}\in H{\left(S^3\right)}_x$ be a unit vector and let us set $\mathbf{x}=\left(x^1,x^2\right)$. Then

$$e^{-s\left(x\right)}\le |g\left(x\right)\mathbf{x}{|}^2\le e^{s\left(x\right)}\le K{e}^{-s\left(x\right)}\le K.$$

Similarly

$$\frac{1}{K}\le |g\left(x\right)\mathbf{x}{|}^2.$$

□

□

2.5. Quasiconformality to the Standard Sphere

An interesting particular case of the CR embedding problem was considered by E. Barletta and S. Dragomir (cf. [10]) who asked which strictly pseudoconvex CR manifolds M, of CR dimension n, can be globally embedded as the standard sphere $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ with the ordinary CR structure $T_{1,0}\left(S^{2n+1}\right)$ induced by the complex structure of ${\mathbb{C}}^{n+1}$. Their findings are that the Pontrjagin forms of the Fefferman metric $F_{\theta }\in \mathrm{Lor}\left[C\left(M\right)\right]$ are CR invariants of M (and when a certain Pontrjagin form P vanishes (i.e., $P=0$), the corresponding transgression class $\left[T\left(P\right)\right]$ is a CR invariant, as well) and among those CR invariants, one pinpoints obstructions to the posed question (i.e., whether M and $S^{2n+1}$ are CR equivalent).

A weaker version of E. Barletta and S. Dragomir’s problem (cf. op. cit.) consistent with the formulation of our Problem 1, is to ask which strictly pseudoconvex CR manifolds M, of CR dimension n, are K-quasiconformally equivalent to the standard sphere $S^{2n+1}$. As with Problem 1, the question can be asked—and it is especially meaningful to ask—when M fails to be globally embeddable. In the spirit of the present paper, we confine the question to the case of 3-dimensional (i.e., $n=1$) CR manifolds and then to the particular case of Rossi’s spheres

$$\left(M,\mathcal{H}\right)\in \left\{\left(S^3,\mathcal{H}\left(t\right)\right):\left|t\right|<1\right\}.$$

Problem 3. 

Find a function $K:(-1,1)\to (1,+\infty )$ and a family ${\left\{f_t\right\}}_{\left|t\right|<1}$ of $K\left(t\right)$-quasiconformal maps $f_t:S^3\to S^3$ of the Rossi sphere $\left(S^3,\mathcal{H}\left(t\right)\right)$ onto the standard sphere $\left(S^3,T_{1,0}\left(S^3\right)\right)$.

Of course $f_0=1_{S^3}:S^3\to S^3$ is a CR equivalence of $\left(S^3,\mathcal{H}\left(0\right)\right)$ and itself $\left(S^3,T_{1,0}\left(S^3\right)\right)$. Yet given a constant $K>1$ and a value of the parameter $0<\left|t\right|<1$, the identity mapping $1_{S^3}$ is not K-quasiconformal in general, and the pair $(K,t)$ is subject to constraints.

Theorem 4. 

Let $K>1$ and $0<\left|t\right|<1$ such that $f=1_{S^3}$ is a K-quasiconformal map of $\left(S^3,\mathcal{H}\left(t\right)\right)$ onto $\left(S^3,T_{1,0}\left(S^3\right)\right)$. Then

$$K\ge \frac{1+\left|t\right|}{1-\left|t\right|}.$$

(63)

Proof. 

Note that $f=1_{S^3}$ is a contact transformation $f:S^3\to S^3$ with $\lambda \left(f\right)=\lambda (f;\theta ,\theta )=1$. Let us consider the (globally defined) frames of $H\left(S^3\right)$

$$E_1=T_1+T_{\overline{1}},E_2=i\left(T_1-T_{\overline{1}}\right),$$

$$E_1^t=L_t+{\overline{L}}_t=(1+t)E_1,E_2^t=i\left(L_t-{\overline{L}}_t\right)=(1-t)E_2.$$

The complex structures $J^t:H\left(S^3\right)\to H\left(S^3\right)$ and $J=J^0:H\left(S^3\right)\to H\left(S^3\right)$ (determined by the CR structures $\mathcal{H}\left(t\right)$ and $T_{1,0}\left(S^3\right)$) are related by

$$J^t{E}_1=\frac{1}{1+t}{J}^t{E}_1^t=\frac{1}{1+t}{E}_2^t=\frac{1-t}{1+t}{E}_2=\frac{1-t}{1+t}J{E}_1$$

and similarly,

$$J^t{E}_2=\frac{1+t}{1-t}J{E}_2.$$

Recall that

$$\left(d\theta \right)(E_1,E_2)=2G_{\theta }\left(T_1,T_{\overline{1}}\right)=1.$$

Then, for any $X=X^1{E}_1+X^2{E}_2\in H\left(S^3\right)$

$$G_{\theta }^t\left(X,X\right)=\left(d\theta \right)\left(X,J^{t}X\right)=\frac{1-t}{1+t}{\left(X^1\right)}^2+\frac{1+t}{1-t}{\left(X^2\right)}^2$$

so that the Levi forms of the pseudohermitian manifolds $\left(S^3,\mathcal{H}\left(t\right),\theta \right)$ and $\left(S^3,T_{1,0}\left(S^3\right),\theta \right)$ are related by

$$G_{\theta }^t(X,X)=\frac{1}{1-t^2}{G}_{\theta }(X,X)+\frac{t}{1-t^2}\left[(t-2){\left(X^1\right)}^2+(t+2){\left(X^2\right)}^2\right]=$$

$$=\frac{1-t}{1+t}{G}_{\theta }(X,X)+\frac{4t}{1-t^2}{\left(X^2\right)}^2=\frac{1+t}{1-t}{G}_{\theta }(X,X)-\frac{4t}{1-t^2}{\left(X^1\right)}^2.$$

To establish the lower bound (63) on K, we distinguish two cases, as (I) $t>0$ or (II) $t<0$. In the first case the K-quasiconformality of $f=1_{S^3}$

$$\frac{1}{K}{G}_{\theta }(X,X)\le G_{\theta }^t(X,X)\le K{G}_{\theta }(X,X)$$

(64)

for $X=E_2$ yields

$$K{G}_{\theta }(E_2,E_2)\ge G_{\theta }^t(E_2,E_2)=\frac{1+t}{1-t}{G}_{\theta }(E_2,E_2)$$

hence,

$$K\ge \frac{1+t}{1-t}.$$

In the second case, let us set $X=E_1$ in (64) so that

$$K{G}_{\theta }(E_1,E_1)\ge G_{\theta }^t(E_1,E_1)=\frac{1-t}{1+t}{G}_{\theta }(E_1,E_1)$$

hence

$$K\ge \frac{1-t}{1+t}.$$

□

The bound (63) is consistent with A. Korányi and H. Reimann’s theorem. Indeed, if ${\mu }_f\left(t\right):\mathcal{H}\left(t\right)\to \mathcal{H}\left(t\right)$ is the complex dilation of $f=1_{S^3}$, then $\parallel {\mu }_f\parallel =\left|t\right|$ and (63) is a corollary of (55) in Theorem 3.

2.6. Fefferman’s Metrics

Let $\left(N,T_{1,0}\left(N\right)\right)$ be a nondegenerate 3-dimensional CR manifold, and let $\Theta \in {\mathcal{P}}_{+}\left(N\right)$. Let $\left\{T_1^N\right\}$ be a local frame of $T_{1,0}\left(N\right)$, defined on the open set $V\subset N$. Let $T^N\in \mathfrak{X}\left(N\right)$ be the Reeb vector field of $(N,\Theta )$. Let $\left\{{\Theta }^1\right\}$ be the corresponding adapted coframe, i.e., ${\Theta }^1\left(T_1^N\right)=1$, ${\Theta }^1\left(T_{\overline{1}}^N\right)=0$ and ${\Theta }^1\left(T^N\right)=0$. Let $f:S^3\to N$ be a contact transformation with $\lambda \left(f\right)=\lambda (f;\theta ,\Theta )>0$. Here, $\theta$ is the canonical pseudohermitian structure on $S^3$ (given by (7)). Let $f^{\ast }:{\Omega }^p\left(N\right)\to {\Omega }^p\left(S^3\right)$ be the pullback by f of differential p-forms on N, $p\in \{1,2,3\}$. Then,

$$f^{\ast }\Theta =\lambda \left(f\right)\theta ,f^{\ast }{\Theta }^1=f_1^1{\theta }^1+f_{\overline{1}}^1{\theta }^{\overline{1}}+f_0^1\theta .$$

Next, we consider the canonical circle bundles

$$\begin{array}{cc}{S}^1\to & C\left(S^3,\mathcal{H}\left(t\right)\right)\\ & \\ & \downarrow {\pi }^t\\ & \\ & S^3\end{array}\begin{array}{cc}{S}^1\to & C\left(S^3,{\mathcal{H}}_f\right)\\ & \\ & \downarrow {\pi }_f\\ & \\ & S^3\end{array}\begin{array}{cc}{S}^1\to & C\left(N,T_{1,0}\left(N\right)\right)\\ & \\ & \downarrow {\pi }^N\\ & \\ & N\end{array}$$

(so that ${\pi }^0=\pi$ (cf. our Section 2.1.5)) and the Fefferman metrics

$$F_{\theta }^t=F\left(\mathcal{H}\left(t\right),\theta \right)\in \mathrm{Lor}\left[C\left(S^3,\mathcal{H}\left(t\right)\right)\right],$$

$$F_f=F\left({\mathcal{H}}_f,\theta \right)\in \mathrm{Lor}\left[C\left(S^3,{\mathcal{H}}_f\right)\right],$$

$$F_{\Theta }\in \mathrm{Lor}\left[C\left(N,T_{1,0}\left(M\right)\right)\right],$$

(so that $F_{\theta }^0=F_{\theta }$). We also write briefly $C\left(N\right)=C\left(N,T_{1,0}\left(M\right)\right)$. The principal bundle $S^1\to C\left(S^3,\mathcal{H}\left(t\right)\right)\to S^3$ is described in Section 2. In addition, every $c\in C{\left(N\right)}_p$ with $p\in V$ may be represented as

$$c=\left[\Lambda {\left(\Theta \wedge {\Theta }^1\right)}_p\right],\Lambda \in \mathbb{C}\backslash \left\{0\right\}.$$

To describe $S^1\to C\left(S^3,{\mathcal{H}}_f\right)\to S^3$, we recall that, given a frame $\left\{Z_1\right\}\subset C^{\infty }(U,\mathcal{H})$, the CR structure ${\mathcal{H}}_f$ is the span of

$$L_f=Z_1-{\mu }_1^{\overline{1}}{Z}_{\overline{1}}\in C^{\infty }\left({\mathcal{H}}_f\right),{\mu }_1^{\overline{1}}=\frac{f_1^{\overline{1}}}{f_{\overline{1}}^{\overline{1}}},$$

$$f_B^A=f_B^A\left(·,\mathcal{H}\right),\mathcal{H}\in \mathrm{CR}\left[H\left(S^3\right)\right],A,B\in \{1,\overline{1}\}.$$

Let $\left\{{\theta }^1\right\}$ and $\left\{{\theta }_f^1\right\}$ be the adapted coframes determined by

$${\theta }^1\left(Z_1\right)=1,{\theta }^1\left(Z_{\overline{1}}\right)=0,{\theta }^1\left(T\right)=0,$$

$${\theta }_f^1\left(L_f\right)=1,{\theta }_f^1\left({\overline{L}}_f\right)=0,{\theta }_f^1\left(T\right)=0.$$

Then

$${\theta }_f^1=\frac{1}{1-|{\mu }_{\overline{1}}^1{|}^2}\left({\theta }^1+{\mu }_{\overline{1}}^1{\theta }^{\overline{1}}\right)$$

and every $c\in C{\left(S^3,{\mathcal{H}}_f\right)}_x$ may be (locally) represented as

$$c=\left[\alpha {\left(\theta \wedge {\theta }_f^1\right)}_x\right]=\left[\alpha {\left(\theta \wedge {\theta }^1+{\mu }_{\overline{1}}^1{\theta }^{\overline{1}}\right)}_x\right],\alpha \in \mathbb{C}\backslash \left\{0\right\}.$$

Every $(2,0)$-form on N is locally represented as $\Omega =\Lambda \theta \wedge {\Theta }^1$ for some $\Lambda \in C^{\infty }\left(V,\mathbb{C}\right)$. Then [by $f_{\overline{1}}^1={\mu }_{\overline{1}}^1{f}_1^1$]

$$f^{\ast }\Omega =f_1^1{\Lambda }^f\lambda \left(f\right)\theta \wedge {\theta }_f^1$$

where ${\Lambda }^f=\Lambda \circ f$.

Proposition 1. 

Let $f:S^3\to N$ be a contact transformation with $\lambda \left(f\right)>0$ and let $\mathcal{H}\in \mathrm{CR}\left[H\left(S^3\right)\right]$ be a CR structure on $S^3$ whose Levi distribution is $H\left(S^3\right)$. The pullback $f^{\ast }:{\Omega }^2\left(N\right)\to {\Omega }^2\left(S^3\right)$ induces a $C^{\infty }$ diffeomorphism

$$C\left(f\right):C\left(N\right)\to C\left(S^3,{\mathcal{H}}_f\right),$$

$$C\left(f\right)\left(C\right)=\left[f_1^1\left(x\right)\Lambda {\left(\theta \wedge {\theta }^1+{\mu }_{\overline{1}}^1\theta \wedge {\theta }^{\overline{1}}\right)}_x\right],$$

(65)

for every $C\in C{\left(N\right)}_{f\left(x\right)}$ locally represented as

$$C=\left[\Lambda {\left(\Theta \wedge {\Theta }^1\right)}_x\right],\Lambda \in \mathbb{C},x\in U.$$

Proof. 

Let $y\in N$ and let $V\subset N$ be an open neighborhood of y, the domain of a (local) frame $\left\{T_1^N\right\}\subset C^{\infty }\left(V,T_{1,0}\left(N\right)\right)$. Let $C\in C{\left(N\right)}_y$ and let us set $x=f^{-1}\left(y\right)\in U=f^{-1}\left(V\right)$. Then $C=\left[\Lambda {(\Theta \wedge {\Theta }^1)}_y\right]$ for some $\Lambda \in \mathbb{C}\backslash \left\{0\right\}$ and we set

$$C\left(f\right)\left(C\right)=\left[\Lambda {\left\{f^{\ast }\left(\Theta \wedge {\Theta }^1\right)\right\}}_x\right]$$

thus yielding (65). The definition of $C\left(f\right)\left(C\right)$ does not depend upon the choice of local frame $\left\{T_1^N\right\}$ about $y=f\left(x\right)$. □

The investigation of the metric properties of $C\left(f\right)$ [in particular, the calculation of $C{\left(f\right)}^{\ast }{F}_f-F_{\theta }^t$ for $\mathcal{H}\in \left\{\mathcal{H}\left(t\right):\left|t\right|<1\right\}$] is an open problem.

3. Sobolev Solutions to Beltrami’s Equation

The purpose of this section is to address the problem of solving the Beltrami equations

$${\overline{L}}_t\left(g\right)=\mu \left(·,t\right)L_t\left(g\right),\left|t\right|<1,$$

(66)

under appropriate assumptions on a given family of functions $\mu \left(·,t\right):S^3\to \mathbb{C}$, $\left|t\right|<1$. To solve (66), we follow the approach by A. Koranyi and H. Reimann (cf. [3], pp. 69–74). There, one looks for weak solutions, in a Folland–Stein space, to the Beltrami equation

$$\overline{V}\left(f\right)=\mu V\left(f\right),V\equiv \frac{\partial }{\partial \zeta }+i\overline{\zeta }\frac{\partial }{\partial \tau },$$

on the Heisenberg group ${\mathbb{H}}_1=\mathbb{C}\times \mathbb{R}$, for a given function $\mu :{\mathbb{H}}_1\to \mathbb{C}$ such that ${\parallel \mu \parallel }_{\infty }<1$. Our problem (66) is formulated on the sphere $S^3$, rather than the Heisenberg group ${\mathbb{H}}_1$. Of course the sphere minus a point and the Heisenberg group may be identified by the Cayley transform, and we profit from certain ideas by C-Y. Hsiao and P-L. Yung (cf. [4]) to transpose (66) on ${\mathbb{H}}_1$. Equation (66) may also be written as

$$\overline{Z}\left(g\right)=\frac{\mu \left(·,t\right)-t}{1-t\mu \left(·,t\right)}Z\left(g\right)$$

(67)

where $Z=T_1=\overline{w}\partial /\partial z-\overline{z}\partial /\partial w$. By a change of dependent variable $f=g\circ H^{-1}$ or $g=f\circ H$, Equation (67) goes over to

$$\overline{u}\overline{V}\left(f\right)=\frac{\lambda \left(·,t\right)-t}{1-t\lambda \left(·,t\right)}uV\left(f\right),$$

(68)

$$\lambda (x,t)=\mu \left(H^{-1}\left(x\right),t\right),x\in {\mathbb{H}}_1,\left|t\right|<1.$$

Here, $H={\psi }^{-1}\circ \mathcal{C}:S^3\backslash \left\{(0,-1)\right\}\to {\mathbb{H}}_1$ is the $C^{\infty }$ diffeomorphism in Section 2.1.6. Equation (68) is central to the present section, and it is our purpose to solve it by an iterative argument relying on Banach’s fixed-point theorem.

Let $\mathcal{S}\left({\mathbb{H}}_1\right)$ be the Schwartz class, consisting of all functions $\phi \in C^{\infty }\left({\mathbb{R}}^3\right)$ such that $p_{\alpha ,\beta }\left(\phi \right)<\infty$ for any $\alpha ,\beta \in {\mathbb{Z}}_{+}^3$. Here $\{p_{\alpha ,\beta }:\alpha ,\beta \in {\mathbb{Z}}_{+}^3\}$ is the separating family of semi-norms on $C^{\infty }\left({\mathbb{R}}^3\right)$ given by

$$p_{\alpha ,\beta }\left(\phi \right)=\underset{\mathbf{x}\in {\mathbb{R}}^3}{sup}\left|{\mathbf{x}}^{\alpha }{D}^{\beta }\phi \left(\mathbf{x}\right)\right|.$$

If $g\in \mathcal{S}\left({\mathbb{H}}_1\right)$, then a necessary condition for solving

$$\overline{V}\left(f\right)=g$$

(69)

(the inhomogeneous tangential Cauchy–Riemann equation on ${\mathbb{H}}_1$) is that $g\ast \overline{S}=0$ (i.e., g must be orthogonal to the kernel of V) where

$$\overline{S}\left(\zeta ,\tau \right)=\frac{1}{{\pi }^2{\left({\left|\zeta \right|}^2+i\tau \right)}^2},$$

$$\left(g\ast \overline{S}\right)\left(x\right)={\int }_{{\mathbb{H}}_1}\overline{S}\left(y^{-1}x\right)g\left(y\right)dy.$$

The canonical solution to (69) (i.e., the solution orthogonal to the kernel of $\overline{V}$) is $f=g\ast k$, where

$$k\left(\zeta ,\tau \right)=\frac{1}{{\pi }^2}\frac{\overline{\zeta }}{\left(\tau +i|\zeta {|}^2\right)\left(\tau -i|\zeta {|}^2\right)}.$$

Cf. P.C. Greiner, J.J. Kohn and E.M. Stein [7]. Let us set

$$b\left(\zeta ,\tau \right)=\left(Vk\right)(\zeta ,\tau )=\frac{2i}{{\pi }^2}\frac{{\overline{\zeta }}^2}{\left(\tau +i|\zeta {|}^2\right)\left(\tau -i|\zeta {|}^2\right)}$$

so that

$$f=g\ast k⟹V\left(f\right)=g\ast V\left(k\right)=g\ast b=\overline{V}\left(f\right)\ast b.$$

The kernel $b\left(\zeta ,\tau \right)$ is homogeneous [with respect to the parabolic dilations ${\delta }_s\left(\zeta ,\tau \right)=\left(s\zeta ,s^2\tau \right)$ on the Heisenberg group ${\mathbb{H}}_1$ (with $s>0$)] of degree $-4$ and

$${\int }_{{\Sigma }^2}b\left(\zeta ,\tau \right)d\sigma =0.$$

Here ${\Sigma }^2={\Sigma }^2\left(1\right)$, and

$${\Sigma }^2\left(r\right)=\{x\in {\mathbb{H}}_1:\left|x\right|=r\},$$

$$\left|x\right|={\left({\left|z\right|}^4+t^2\right)}^{1/4},x=\left(\zeta ,\tau \right)\in {\mathbb{H}}_1,$$

is the Heisenberg sphere of radius $r>0$. Therefore (by a result of A. Korányi and S. Vági [11]) for every $1<p<\infty$ the convolution operator $B\left(g\right)=g\ast b$ extends from $\mathcal{S}\left({\mathbb{H}}_1\right)$ to a bounded operator on $L^p\left({\mathbb{H}}_1,{\theta }_0\right)$. Additionally, $k\left(\zeta ,\tau \right)$ is homogeneous of degree $-3$ so that the convolution operator $K\left(g\right)=g\ast k$ extends from $\mathcal{S}\left({\mathbb{H}}_1\right)$ to a bounded operator

$$K:L^p\left({\mathbb{H}}_1,{\theta }_0\right)\to L^q\left({\mathbb{H}}_1,{\theta }_0\right),\frac{1}{q}=\frac{1}{p}-\frac{1}{4},1<p<q<\infty .$$

Cf. G.B. Folland and E.M. Stein [12]. Let $W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$ be the Folland–Stein space of all $L^2$ functions on ${\mathbb{H}}_1$ admitting weak E-derivatives. Let $h\in C^{\infty }\left({\mathbb{H}}_1\right)$ such that $\overline{V}\left(h\right)=0$. Let us look for a solution $f\in L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ to the Beltrami Equation (68) such that $f-h\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$. To this end, we set

$$g=\overline{V}\left(f-h\right).$$

Note that if f were $C^1$, then we would have $\overline{V}\left(f\right)=g$ and

$$V\left(f\right)=V(f-h)+V\left(h\right)=B\left(g\right)+V\left(h\right).$$

At this point, we substitute $\overline{V}\left(f\right)$ and $V\left(f\right)$ into Equation (68), respectively, by g and $B\left(g\right)+V\left(h\right)$ so as to obtain

$$\overline{u}g=\frac{\lambda \left(·,t\right)-t}{1-t\lambda \left(·,t\right)}u\left[B\left(g\right)+V\left(h\right)\right].$$

(70)

Solving for g in (70) is equivalent to seeking a fixed point of

$$F_t\left(g\right)=\frac{\lambda \left(·,t\right)-t}{1-t\lambda \left(·,t\right)}\left[B\left(g\right)+V\left(h\right)\right]{\left(\frac{u}{\left|u\right|}\right)}^2.$$

Let us set

$$\alpha (x,t)=\frac{\lambda \left(x,t\right)-t}{1-t\lambda \left(x,t\right)}{\left[\frac{u\left(x\right)}{\left|u\right(x\left)\right|}\right]}^2,x\in {\mathbb{H}}_1,\left|t\right|<1,$$

and consider the recurrent sequence

$$g_0=0,g_{\nu +1}=F_t\left(g_{\nu }\right),\nu \ge 0.$$

Then

$$g_{\nu +1}=\sum _{k=0}^{\nu }{\left[\alpha (·,t)B\right]}^k\left(\alpha (·,t)V\left(h\right)\right),\nu \ge 0.$$

Then a formal solution to (70) is

$$g=\sum _{\nu =0}^{\infty }{\left[\alpha (·,t)B\right]}^{\nu }\left(\alpha (·,t)V\left(h\right)\right).$$

(71)

The series (71) converges in $L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ provided that

$$F_t:L^2\left({\mathbb{H}}_1,{\theta }_0\right)\to L^2\left({\mathbb{H}}_1,{\theta }_0\right),\left|t\right|<1,$$

are contractions. From now on, we assume that ${\left\{\mu \left(·,t\right)\right\}}_{\left|t\right|<1}$ is a smooth 1-parameter family of measurable functions $\mu \left(·,t\right):S^3\to \mathbb{C}$ of compact support

$$\mathrm{Supp}\left[\mu \left(·,t\right)\right]\subset S^3\backslash \left\{(0,-1)\right\},\left|t\right|<1,$$

such that

$$\parallel \mu \left(·,t\right){\parallel }_{\infty }={\mathrm{ess}\sup }_{p\in S^3}\left|\mu \left(p,t\right)\right|<\frac{1-\left|t\right|}{1+\left|t\right|}.$$

The choice of the upper bound on the essential supremum of $\left|\mu \left(·,t\right)\right|$ will be explained in a moment. As a consequence of our choice, the function $\lambda \left(·,t\right)$ has compact support $\mathrm{Supp}\left[\lambda \left(·,t\right)\right]\subset {\mathbb{H}}_1$ and

$$\parallel \lambda \left(·,t\right){\parallel }_{\infty }<\frac{1-\left|t\right|}{1+\left|t\right|}.$$

(72)

Then $\alpha \left(·,t\right)$ has compact support $\mathrm{Supp}\alpha \left(·,t\right)\subset {\mathbb{H}}_1$ and

$$\parallel \alpha \left(·,t\right){\parallel }_{\infty }=inf\left\{C>0:\left|\alpha (x,t)\right|\le C\mathrm{a}.\mathrm{e}.x\in {\mathbb{H}}_1\right\}=$$

$$=inf\left\{C>0:\left|\frac{\lambda \left(x,t\right)-t}{1-t\lambda \left(x,t\right)}\right|\le C\mathrm{a}.\mathrm{e}.x\in {\mathbb{H}}_1\right\}$$

and hence (72) yields

Lemma 15. 

$$\parallel \alpha \left(·,t\right){\parallel }_{\infty }<1$$

(73)

for every $\left|t\right|<1$.

Proof. 

To prove (73), we ought to choose $0<C_0<1$ such that

$$\left|\frac{\lambda \left(x,t\right)-t}{1-t\lambda \left(x,t\right)}\right|\le C_0\mathrm{a}.\mathrm{e}.x\in {\mathbb{H}}_1.$$

Yet

$$\left|\frac{\lambda \left(x,t\right)-t}{1-t\lambda \left(x,t\right)}\right|\le \frac{\left|\lambda \left(x,t\right)\right|+\left|t\right|}{1-\left|t\right|\left|\lambda \left(x,t\right)\right|}$$

so it suffices to choose $0<C_0<1$ such that

$$\frac{\left|\lambda \left(x,t\right)\right|+\left|t\right|}{1-\left|t\right|\left|\lambda \left(x,t\right)\right|}\le C_0\mathrm{a}.\mathrm{e}.x\in {\mathbb{H}}_1$$

or

$$|\lambda \left(x,t\right)|\le \frac{C_0-\left|t\right|}{1+\left|t\right|C_0}\mathrm{a}.\mathrm{e}.x\in {\mathbb{H}}_1.$$

Therefore, one ought to choose $0<C_0<1$ such that

$$\parallel \lambda \left(·,t\right){\parallel }_{\infty }\le \frac{C_0-\left|t\right|}{1+\left|t\right|C_0}⟺C_0\ge \frac{\parallel \lambda \left(·,t\right){\parallel }_{\infty }+\left|t\right|}{1-\left|t\right|\parallel \lambda \left(·,t\right){\parallel }_{\infty }}$$

which is possible only provided that

$$\frac{\parallel \lambda \left(·,t\right){\parallel }_{\infty }+\left|t\right|}{1-\left|t\right|\parallel \lambda \left(·,t\right){\parallel }_{\infty }}<1$$

or equivalently

$$\parallel \lambda \left(·,t\right){\parallel }_{\infty }<\frac{1-\left|t\right|}{1+\left|t\right|}$$

which is (72). □

As $\alpha \left(·,t\right)\in L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and $\mathrm{Supp}\left[\alpha \left(·,t\right)\right]$ is compact, it must be that $\alpha \left(·,t\right)V\left(h\right)\in L^2\left({\mathbb{H}}_1,{\theta }_0\right)$. Then $F_t$ is a map of $L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ into $L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and for any $g,v\in L^2\left({\mathbb{H}}_1,{\theta }_0\right)$

$$\parallel F_t\left(g+v\right)-F_t\left(g\right){\parallel }_{L^p\left({\mathbb{H}}_1\right)}=\parallel \alpha (·,t)B\left(v\right){\parallel }_{L^p\left({\mathbb{H}}_1\right)}\le \parallel \alpha (·,t)B\parallel \parallel v{\parallel }_{L^p\left({\mathbb{H}}_1\right)}$$

so that $F_t$ is a contraction provided that

$$\parallel \alpha (·,t)B\parallel <1.$$

(74)

If this is the case, the series (71) converges in $L^2\left({\mathbb{H}}_1,{\theta }_0\right)$. Moreover, if the sum g of the series (71) satisfies the integrability condition

$$g\ast \overline{S}=0$$

(75)

then solving for f in $\overline{V}\left(f-h\right)=g$ gives the solution f to the Beltrami Equation (68)

$$f=g\ast k+h,f-h\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right).$$

The property $f-h\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$ of the solution describes its holomorphic behavior at ∞. The operator norm in (75) is

$$\parallel \alpha (·,t)B\parallel =sup\left\{\frac{\parallel \alpha (·,t)B\left(g\right){\parallel }_{L^2({\mathbb{H}}_1,{\theta }_0)}}{\parallel g{\parallel }_{L^2({\mathbb{H}}_1,{\theta }_0)}}:\begin{array}{c}g\in L^2\left({\mathbb{H}}_1,{\theta }_0\right),\\ g\ne 0\end{array}\right\}.$$

To compute the operator norm (and prove (75)), we need to represent B as a multiplier on the Fourier transform. For every $\lambda \in \mathbb{R}\backslash \left\{0\right\}$ we consider the space

$${\mathfrak{H}}_{\lambda }=L^{2}H\left(\mathbb{C},{\gamma }_{\lambda }\right)=\mathcal{O}\left(\mathbb{C}\right)\cap L^2\left(\mathbb{C},{\gamma }_{\lambda }\right)$$

of all holomorphic functions $\varphi :\mathbb{C}\to \mathbb{C}$ such that

$$\parallel \varphi {\parallel }_{{\gamma }_{\lambda }}={\left(\frac{\left|\lambda \right|}{\pi }{\int }_{\mathbb{C}}|\varphi \left(z\right){|}^2{\gamma }_{\lambda }\left(z\right)dm\left(z\right)\right)}^{1/2}<\infty ,$$

$${\gamma }_{\lambda }\left(z\right)=exp\left(-{\left|\lambda \right|\left|z\right|}^2\right),$$

where m is the Lebesgue measure on ${\mathbb{R}}^2$. Then ${\mathfrak{H}}_{\lambda }$ is a Hilbert space with the scalar product

$${⟨\varphi ,\psi ⟩}_{{\mathfrak{H}}_{\lambda }}=\frac{\left|\lambda \right|}{\pi }{\int }_{\mathbb{C}}\varphi \left(z\right)\overline{\psi \left(z\right)}{\gamma }_{\lambda }\left(z\right)dm\left(z\right).$$

The Bargmann representation of the Heisenberg group ${\mathbb{H}}_1$ is the unitary representation of ${\mathbb{H}}_1$ on ${\mathfrak{H}}_{\lambda }$ given by

$$T_{\lambda }:{\mathbb{H}}_1\to {\mathrm{End}}_{\mathbb{C}}\left({\mathfrak{H}}_{\lambda }\right),$$

$$\left[T_{\lambda }(\zeta ,\tau )\varphi \right]\left(z\right)=\left\{\begin{array}{cc}exp\left({\displaystyle -\frac{\lambda }{2}\left(i\tau +{\left|\zeta \right|}^2\right)-\lambda \overline{\zeta }z}\right)\varphi (z+\zeta )\hfill & \mathrm{if}\lambda >0,\hfill \\ \\ exp\left({\displaystyle -\frac{\lambda }{2}\left(i\tau -{\left|\zeta \right|}^2\right)+\lambda \overline{\zeta }z}\right)\varphi (z+\zeta )\hfill & \mathrm{if}\lambda <0,\hfill \end{array}\right.$$

$$(\zeta ,\tau )\in {\mathbb{H}}_1,\varphi \in {\mathfrak{H}}_{\lambda },z\in \mathbb{C}.$$

Lemma 16. 

$T_{\lambda }\left(\zeta ,\tau \right)=T_{-\lambda }\left(\overline{\zeta },-\tau \right)$.

Let ${\mathfrak{h}}_1$ be the Lie algebra of ${\mathbb{H}}_1$. The same symbol $T_{\lambda }$ will denote the induced representation of the Lie algebra ${\mathfrak{h}}_1$ on ${\mathfrak{H}}_{\lambda }$

$$T_{\lambda }:{\mathfrak{h}}_1\to {\mathrm{End}}_{\mathbb{C}}\left({\mathfrak{H}}_{\lambda }\right),T_{\lambda }\left(A\right)=\left(d_0{T}_{\lambda }\right)A_0,A\in {\mathfrak{h}}_1.$$

The Lewy operator $\overline{V}=\partial /\partial \overline{\zeta }-i\zeta \partial /\partial \tau$ and the Reeb vector field $\partial /\partial \tau$ are known to be left invariant. Hence, ${\mathfrak{h}}_1{\otimes }_{\mathbb{R}}\mathbb{C}$ is the span of $\{V,\overline{V},\partial /\partial \tau \}$.

Lemma 17. 

(i)

If $\lambda >0$ then

$$T_{\lambda }\left(V\right)=\frac{\partial }{\partial z},T_{\lambda }\left(\overline{V}\right)=-\lambda z,T_{\lambda }\left(\frac{\partial }{\partial \tau }\right)=-\frac{i\lambda }{\zeta }.$$

(ii)

If $\lambda <0$ then

$$T_{\lambda }\left(V\right)=\lambda z,T_{\lambda }\left(\overline{V}\right)=\frac{\partial }{\partial z},T_{\lambda }\left(\frac{\partial }{\partial \tau }\right)=-\frac{i\lambda }{\zeta }.$$

(iii)

$T_{\lambda }:{\mathfrak{h}}_1\to {\mathrm{End}}_{\mathbb{C}}\left({\mathfrak{H}}_{\lambda }\right)$ is a unitary representation.

The Fourier transform at $\lambda$ of a function $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$ is the operator

$$T_{\lambda }\left(f\right):{\mathfrak{H}}_{\lambda }\to {\mathfrak{H}}_{\lambda },$$

$$\left[T_{\lambda }\left(f\right)\varphi \right]\left(z\right)={\int }_{{\mathbb{H}}_1}f(\zeta ,\tau )\left[T_{\lambda }(\zeta ,\tau )\varphi \right]\left(z\right)d\xi d\eta d\tau ,$$

$$\varphi \in {\mathfrak{H}}_{\lambda },z\in \mathbb{C}.$$

Here $\zeta =\xi +i\eta$ are the real and imaginary parts of $\zeta$. We recall that a bounded linear operator $A:{\mathfrak{H}}_{\lambda }\to {\mathfrak{H}}_{\lambda }$ is an operator of trace class if

$$\parallel A{\parallel }_1=\mathrm{Tr}|A|:=\sum _{n=1}^{\infty }{⟨{\left(A^{\ast }A\right)}^{1/2}{\varphi }_n,{\varphi }_n⟩}_{\lambda }<\infty$$

for some complete orthonormal system $\{{\varphi }_n:n\ge 1\}\subset {\mathfrak{H}}_{\lambda }$ (and thus for all). If this is the case, then the trace of A

$$\mathrm{Tr}A:=\sum _{\nu =1}^{\infty }{⟨A{\varphi }_n,{\varphi }_n⟩}_{\lambda }$$

is an absolutely convergent series, and its sum is independent of the choice of a complete orthonormal system in ${\mathfrak{H}}_{\lambda }$.

Lemma 18. 

The Fourier transform $T_{\lambda }\left(f\right):{\mathfrak{H}}_{\lambda }\to {\mathfrak{H}}_{\lambda }$ of every $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$ is an operator of trace class.

The norm of $T_{\lambda }\left(f\right)$ (the trace norm) is defined by

$$\parallel T_{\lambda }\left(f\right){\parallel }^2=\mathrm{Tr}\left\{T_{\lambda }^{\ast }\left(f\right)T_{\lambda }\left(f\right)\right\}$$

where $T_{\lambda }^{\ast }\left(f\right)=T_{\lambda }{\left(f\right)}^{\ast }$ (the adjoint of $T_{\lambda }\left(f\right)$).

Lemma 19. 

Let $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$.

(i)

The inversion formula for the Fourier transform is

$$f(\zeta ,\tau )=\frac{1}{4{\pi }^2}{\int }_{\mathbb{R}}\mathrm{Tr}\left\{T_{\lambda }^{\ast }(\zeta ,\tau )T_{\lambda }\left(f\right)\right\}\left|\lambda \right|d\lambda .$$

(76)

(ii)

The Plancherel formula for the Fourier transform is

$${\parallel f\parallel }^2=\frac{1}{4{\pi }^2}{\int }_{\mathbb{R}}\parallel T_{\lambda }\left(f\right){\parallel }^2\left|\lambda \right|d\lambda$$

(77)

where $\parallel f\parallel$ is the $L^2$ norm of f.

Cf. J. Faraut [13]. On the basis of the formulas (76) and (77), the Fourier transform $T_{\lambda }\left(f\right)$ may be extended from functions of Schwartz class $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$ to square integrable functions $f\in L^2({\mathbb{H}}_1,{\theta }_0)$.

Lemma 20. 

The Fourier transform of the convolution product

$$(f\ast g)\left(y\right)={\int }_{{\mathbb{H}}_1}f\left(x\right)g\left(x^{-1}y\right)dx,f,g\in \mathcal{S}\left({\mathbb{H}}_1\right),$$

is given by

$$T_{\lambda }(f\ast g)=T_{\lambda }\left(f\right)T_{\lambda }\left(g\right).$$

Lemma 21. 

The system $\{{\varphi }_n:n\ge 1\}\subset {\mathfrak{H}}_{\lambda }$ given by

$${\varphi }_n\left(z\right)=\sqrt{\frac{|\lambda {|}^n}{n!}}{z}^n,z\in \mathbb{C},n\in \mathbb{N},$$

is a complete orthonormal system in ${\mathfrak{H}}_{\lambda }$.

Let

$$t_{n,m}^{\lambda }(\zeta ,\tau )={⟨T_{\lambda }(\zeta ,\tau ){\varphi }_m,{\varphi }_n⟩}_{\lambda }$$

be the Fourier coefficients of the operator $T_{\lambda }(\zeta ,\tau )$ with respect to ${\left\{{\varphi }_n\right\}}_{n\ge 1}$. This is an infinite matrix given by the following.

Lemma 22. 

(i)

If $\lambda >0$ and $m\ge n$, then

$$t_{n,m}^{\lambda }(\zeta ,\tau )=$$

$$=\sqrt{\frac{n!}{m!}}{\left(\sqrt{\lambda }\zeta \right)}^{m-n}exp\left(-\frac{i\lambda \tau }{2}\right)exp\left(-\frac{{\lambda \left|\zeta \right|}^2}{2}\right)L_n^{m-n}\left({\lambda \left|\zeta \right|}^2\right).$$

(ii)

If $\lambda >0$ and $m<n$, then

$$t_{n,m}^{\lambda }(\zeta ,\tau )=\overline{t_{m,n}^{\lambda }(-\zeta ,-\tau )}=$$

$$=\sqrt{\frac{m!}{n!}}{\left(-\sqrt{\lambda }\overline{\zeta }\right)}^{n-m}exp\left(-\frac{i\lambda \tau }{2}\right)exp\left(-\frac{{\lambda \left|\zeta \right|}^2}{2}\right)L_m^{n-m}\left({\lambda \left|\zeta \right|}^2\right).$$

(iii)

If $\lambda <0$, then

$$t_{n,m}^{\lambda }(\zeta ,\tau )=\overline{t_{n,m}^{\left|\lambda \right|}(\zeta ,\tau )}.$$

Cf. A. Korányi and H.M. Reimann [3], pp. 70–71. Here,

$$L_n^{\alpha }\left(x\right)=\sum _{k=0}^n\frac{{(-1)}^k}{k!(n-k)!}\frac{\Gamma (n+\alpha +1)}{\Gamma (k+\alpha +1)}{x}^k,\alpha >-1,x\ge 0,$$

are the Laguerre polynomials. From now on, the Fourier transform of a function $f\in \mathcal{S}\left({\mathbb{H}}_1\right)$ will be represented as an infinite matrix

$$\widehat{f}\left(\lambda \right)={\left[{\widehat{f}}_{m,n}^{\lambda }\right]}_{m,n\in \mathbb{N}},{\widehat{f}}_{m,n}^{\lambda }={⟨T_{\lambda }\left(f\right){\varphi }_n,{\varphi }_m⟩}_{{\gamma }_{\lambda }},$$

so that

$$T_{\lambda }\left(f\right){\varphi }_m=\sum _{n=1}^{\infty }{\widehat{f}}_{n,m}^{\lambda }{\varphi }_n.$$

Lemma 23. 

The Fourier transform $\widehat{b}\left(\lambda \right)=\left[{\widehat{b}}_{m,n}^{\lambda }\right]$ of

$$b\left(\zeta ,\tau \right)=Vk\left(\zeta ,\tau \right)=\frac{2i}{{\pi }^2}\frac{{\overline{\zeta }}^2}{\left(\tau +i|\zeta {|}^2\right)\left(\tau -i|\zeta {|}^2\right)}$$

is given by

$${\widehat{b}}_{m,n}^{\lambda }=\left\{\begin{array}{cc}{\displaystyle -{\delta }_{m+2,n}\sqrt{\frac{m+1}{m}}}\hfill & \mathrm{if}\lambda >0,\hfill \\ \\ {\displaystyle {\delta }_{m-2,n}\sqrt{\frac{m-2}{m-1}}}\hfill & \mathrm{if}\lambda <0.\hfill \end{array}\right.$$

(78)

Let us consider the subspaces $L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)\subset L^2\left({\mathbb{H}}_1,{\theta }_0\right)$ defined by

$$L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)=\left\{f\in L^2\left({\mathbb{H}}_1,{\theta }_0\right):\widehat{f}\left(\lambda \right)=0\mathrm{a}.\mathrm{e}.\lambda >0\right\},$$

$$L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)=L^2\left({\mathbb{H}}_1,{\theta }_0\right)\ominus L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right).$$

Lemma 24. 

$B{L}_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)\subset L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)$.

Next, for every $k\in \mathbb{Z}$ let us set

$$U^k=\left\{f\in L^2\left({\mathbb{H}}_1,{\theta }_0\right):f\left(\zeta e^{i\phi },\tau \right)=e^{ik\phi }f\left(\zeta ,\tau \right)\right\}.$$

Lemma 25. 

$L^2\left({\mathbb{H}}_1,{\theta }_0\right)={⨁}_{k\in \mathbb{Z}}{U}^k$.

Lemma 26. 

(i)

If $f\in U^k$ then

$$m-n\ne +k⟹{\widehat{f}}_{m,n}^{\lambda }=0\mathrm{for}\mathrm{a}.\mathrm{e}.\lambda >0,$$

$$m-n\ne -k⟹{\widehat{f}}_{m,n}^{\lambda }=0\mathrm{for}\mathrm{a}.\mathrm{e}.\lambda <0.$$

(ii)

$B{U}^k\subset U^{k+2}$.

Next let us consider the complete orthogonal sum

$$D_j={\widehat{⨁}}_{k\le j}{U}^k.$$

Lemma 27. 

(i)

The complete orthogonal sums ${\left\{D_j\right\}}_{j\in \mathbb{Z}}$ satisfy the following multiplication law

$$f\in D_j\mathrm{and}\alpha \left(·,t\right)\in D_m\cap L^{\infty }\left({\mathbb{H}}_1,{\theta }_0\right)⟹$$

$$⟹f·\alpha (·,t)\in D_{j+m}.$$

(ii)

$L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)$ are multiplication invariant, i.e.,

$$f\in L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)\mathrm{and}\alpha \left(·,t\right)\in L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right)\cap L^{\infty }\left({\mathbb{H}}_1,{\theta }_0\right)⟹$$

$$⟹f·\alpha \left(·,t\right)\in L_{\pm }^2\left({\mathbb{H}}_1,{\theta }_0\right).$$

Theorem 5. 

Let $h\in {\mathrm{CR}}^{\infty }\left({\mathbb{H}}_1\right)$ be a CR function [i.e., $\overline{V}\left(h\right)=0$] and let us assume that $\alpha \left(·,t\right)\in L^{\infty }\left({\mathbb{H}}_1\right)$. Let us assume that one of the following conditions is satisfied

(1)

$\alpha \left(·,t\right),\alpha \left(·,t\right)V\left(h\right)\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and ${\displaystyle \parallel \alpha \left(·,t\right){\parallel }_{\infty }<\frac{1}{\sqrt{2}}}$.

(2)

$\alpha \left(·,t\right)\in D_{-2}$, $\alpha \left(·,t\right)V\left(h\right)\in D_{-1}$ and ${\displaystyle \parallel \alpha \left(·,t\right){\parallel }_{\infty }<\frac{1}{\sqrt{2}}}$.

(3)

$\alpha \left(·,t\right)\in D_{-2}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$, $\alpha \left(·,t\right)V\left(h\right)\in D_{-1}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and $\parallel \alpha \left(·,t\right){\parallel }_{\infty }<1$.

Then the Beltrami equation

$${\displaystyle \overline{u}\overline{V}\left(f\right)=\frac{\lambda \left(·,t\right)-t}{1-t\lambda \left(·,t\right)}uV\left(f\right)}$$

has a unique solution $f_t$ such that $f_t-h\in W_E^{1,2}\left({\mathbb{H}}_1,{\theta }_0\right)$.

Proof. 

We ought to show that the series

$$g=\sum _{\nu =0}^{\infty }{\left[\alpha (·,t)B\right]}^{\nu }\left(\alpha (·,t)V\left(h\right)\right)$$

converges in $L^2\left({\mathbb{H}}_1,{\theta }_0\right)$, and its sum g satisfies the integrability condition $g\ast \overline{S}=0$. For every $f\in L^2({\mathbb{H}}_1,{\theta }_0)$, its Fourier transform at $\lambda$ is

$$T_{\lambda }\left(f\right){\varphi }_n=\sum _{m=1}^{\infty }{\widehat{f}}_{m,n}^{\lambda }{\varphi }_m$$

and hence, its trace norm is

$$\parallel T_{\lambda }\left(f\right){\parallel }^2=\mathrm{Tr}\left\{T_{\lambda }^{\ast }\left(f\right)T_{\lambda }\left(f\right)\right\}=\sum _{n=1}^{\infty }{⟨T_{\lambda }^{\ast }\left(f\right)T_{\lambda }\left(f\right){\varphi }_n,{\varphi }_n⟩}_{{\gamma }_{\lambda }}=$$

$$=\sum _{n=1}^{\infty }\parallel T_{\lambda }\left(f\right){\varphi }_n{\parallel }^2=\sum _{n,m=1}^{\infty }|{\widehat{f}}_{m,n}^{\lambda }{|}^2.$$

Then

$$T_{\lambda }\left(Bf\right){\varphi }_n=T_{\lambda }\left(f\ast b\right)=T_{\lambda }\left(f\right)T_{\lambda }\left(b\right){\varphi }_n=\sum _{m,k=1}^{\infty }{\widehat{b}}_{m,n}^{\lambda }{\widehat{f}}_{k,m}^{\lambda }{\varphi }_k$$

i.e.,

$${\widehat{\left(Bf\right)}}_{k,n}^{\lambda }=\sum _m{\widehat{b}}_{m,n}^{\lambda }{\widehat{f}}_{k,m}^{\lambda }=$$

[by (78) in Lemma 23]

$$=\sum _m{\widehat{f}}_{k,m}^{\lambda }\left\{\begin{array}{cc}{\displaystyle -{\delta }_{m+2,n}\sqrt{\frac{m+1}{m}}}\hfill & \mathrm{if}\lambda >0\hfill \\ \\ {\displaystyle {\delta }_{m-2,n}\sqrt{\frac{m-2}{m-1}}}\hfill & \mathrm{if}\lambda <0\hfill \end{array}\right.=\left\{\begin{array}{cc}{\displaystyle -\sqrt{\frac{n-1}{n-2}}{\widehat{f}}_{k,n-2}^{\lambda }}\hfill & \mathrm{if}\lambda >0\hfill \\ \\ {\displaystyle \sqrt{\frac{n}{n+1}}{\widehat{f}}_{k,n+2}^{\lambda }}\hfill & \mathrm{if}\lambda <0\hfill \end{array}\right.$$

so that

$$\parallel T_{\lambda }\left(Bf\right){\parallel }^2=\sum _{n,m}\left\{\begin{array}{cc}{\displaystyle \frac{n-1}{n-2}{\left|{\widehat{f}}_{m,n-2}^{\lambda }\right|}^2}\hfill & \mathrm{if}\lambda >0\hfill \\ \\ {\displaystyle \frac{n}{n+1}{\left|{\widehat{f}}_{m,n+2}^{\lambda }\right|}^2}\hfill & \mathrm{if}\lambda <0\hfill \end{array}\right.=\sum _{k,m}\left\{\begin{array}{cc}{\displaystyle \frac{k+1}{k}{\left|{\widehat{f}}_{m,k}^{\lambda }\right|}^2}\hfill & \mathrm{if}\lambda >0\hfill \\ \\ {\displaystyle \frac{k-2}{k-1}{\left|{\widehat{f}}_{m,k}^{\lambda }\right|}^2}\hfill & \mathrm{if}\lambda <0\hfill \end{array}\right.$$

and hence [by $(k-2)/(k-1)<1$ and $(k+1)/k<2$]

$$\parallel T_{\lambda }\left(Bf\right)\parallel \le \sqrt{2}\parallel T_{\lambda }\left(f\right)\parallel$$

and then

$$\parallel Bf\parallel \le \sqrt{2}\parallel f\parallel ,f\in L^2({\mathbb{H}}_1,{\theta }_0).$$

(79)

Similarly, if $f\in L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ then

$$\parallel Bf{\parallel }^2=\frac{1}{4{\pi }^2}{\int }_{\mathbb{R}}\parallel T_{\lambda }\left(Bf\right)\parallel \left|\lambda \right|d\lambda =$$

[as $\widehat{f}\left(\lambda \right)=0$ for a.e. $\lambda >0$]

$$=\frac{1}{4{\pi }^2}{\int }_{-\infty }^0\sum _{n,m}\frac{n-2}{n-1}{\left|{\widehat{f}}_{m,n}^{\lambda }\right|}^2\left|\lambda \right|d\lambda$$

yielding

$$\parallel Bf\parallel \le \parallel f\parallel ,f\in L_{-}^2({\mathbb{H}}_1,{\theta }_0).$$

(80)

Let us examine now the three assumptions in Theorem 5. By (79) and ${\displaystyle \parallel \alpha \left(,t\right){\parallel }_{\infty }<\frac{1}{\sqrt{2}}}$, it follows that the operator norm of

$$\alpha (·,t)B:L^2({\mathbb{H}}_1,{\theta }_0)\to L^2({\mathbb{H}}_1,{\theta }_0)$$

is $\parallel \alpha (·,t)B\parallel <1$, and hence

$$\sum _{\nu =0}^{\infty }{\left[\alpha (·,t)B\right]}^{\nu }\left[\alpha (·,t)Vh\right]$$

converges to some $g_t\in L^2({\mathbb{H}}_1,{\theta }_0)$.

(1)

As $\alpha (·,t)\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and $\alpha (·,t)Vh\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ and [by Lemma 24]

$$B\left[L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)\right]\subset L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$$

it follows that $g_t\in L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right)$.

(2)

As

$$D_k={\widehat{⨁}}_{\ell \le k}{U}^{\ell },D_{k+2}={\widehat{⨁}}_{\ell \le k+2}{U}^{\ell }={\widehat{⨁}}_{\ell \le k}{U}^{\ell +2},$$

$$B\left(U^{\ell }\right)\subset U^{\ell +2}[\mathrm{by}\left(\mathrm{ii}\right)\mathrm{in}\mathrm{Lemma}26]$$

one has

$$B\left(D_k\right)\subset D_{k+2}.$$

(81)

It should be observed that (81) is independent of any of the assumptions in Theorem 5. If $\alpha \left(·,t\right)\in D_{-2}={{\displaystyle \widehat{⨁}}}_{\ell \le -2}{U}^{\ell }$, then (by (81)

$$\alpha \left(·,t\right)B{D}_k\subset \alpha \left(·,t\right)·D_{k+2}\subset$$

(by (i) in Lemma 27 with $j=k+2$ and $m=-2$)

$$\subset D_k$$

so that $D_k$ is invariant by $\alpha \left(·,t\right)B$. Moreover $\alpha \left(·,t\right)Vh\in D_{-1}$ yields $g_t\in D_{-1}$.

(3)

If $\alpha \left(·,t\right)\in D_{-2}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)$ then $\alpha \left(·,t\right)\in D_{-2}$ was already shown to imply $\alpha \left(·,t\right)B{D}_k\subset D_k$ (here useful for $k=-1$). On the other hand (by $B{L}_{-}^2({\mathbb{H}}_1,{\theta }_0)\subset L_{-}^2({\mathbb{H}}_1,{\theta }_0)$),

$$\alpha \left(·,t\right)B\left\{D_{-1}\cap L_{-}^2({\mathbb{H}}_1,{\theta }_0)\right\}\subset D_{-1}\cap \left\{\alpha \left(·,t\right)·L_{-}^2({\mathbb{H}}_1,{\theta }_0)\right\}\subset$$

(by $\alpha \left(·,t\right)\in L_{-}^2({\mathbb{H}}_1,{\theta }_0)$)

$$\subset D_{-1}\cap L_{-}^2({\mathbb{H}}_1,{\theta }_0)$$

yielding $g_t\in D_{-1}\cap L_{-}^2({\mathbb{H}}_1,{\theta }_0)$. Summing up, under the assumptions (1)–(3) in Theorem 5, the function $g_t$ belongs to one of the spaces

$$L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right),D_{-1},D_{-1}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right).$$

The proof of Theorem 5 may be completed by applying the following lemma:

Lemma 28. 

Let $S\in \left\{L_{+}^2\left({\mathbb{H}}_1,{\theta }_0\right),D_{-1},D_{-1}\cap L_{-}^2\left({\mathbb{H}}_1,{\theta }_0\right)\right\}$. For every $g\in S$, one has $\widehat{g}\left(\lambda \right)\widehat{\overline{S}}\left(\lambda \right)=0$. Equivalently, each $g\in S$ satisfies the integrability condition $g\ast \overline{S}=0$.

□

4. Conclusions and Open Problems

Sobolev-type solutions to the Beltrami equation

$$\overline{V}\left(f\right)=\mu V\left(f\right)$$

(82)

on the Heisenberg group ${\mathbb{H}}_1$ were first produced by A. Korányi and H.M. Reimann [3], relying on work by P.C. Greiner, J.J. Kohn and E.M. Stein [7], on the solution to the Lewy equation $\overline{V}\left(f\right)=g$. We consider the Beltrami equations associated to the non-embeddable CR structures $\mathcal{H}\left(t\right)$, $\left|t\right|<1$, on $S^3$ as discovered by H. Rossi [1], and transplant said equations on ${\mathbb{H}}_1$ by using the CR diffeomorphism $H:U=S^3\backslash \left\{(0,-1)\right\}\approx {\mathbb{H}}_1$ (associated with the Cayley map). This gives a 1-parameter family of first order PDEs (with variable coefficients) on ${\mathbb{H}}_1$, similar to Korányi and Reimann’s Beltrami Equation (82), which may be simultaneously treated by an outgrowth of Korányi and Reimann’s techniques (borrowed from [7] for the part of complex analysis, and from J. Faraut [13] for the part of harmonic analysis). It is an open problem whether the same CR diffeomorphism H may be used to transplant Fourier calculus from ${\mathbb{H}}_1$ to the open set $U\subset S^3$ (and whether the resulting tools are effective in a direct study of Equations (3)). We expect the resulting local harmonic analysis on $S^3$ to be similar to that proposed by R.S. Strichartz [14]. Cf. also [15].

The success in [10] to discover obstructions to CR equivalence of a strictly pseudoconvex real hypersurface $M\subset {\mathbb{C}}^{n+1}$ to the sphere $S^{2n+1}$ (such as the first Pontrjagin form of the Fefferman metric) prompts the question of whether (other) characteristic forms of $F_{\theta }^t$ [the Fefferman metric of a Rossi sphere $(S^3,\mathcal{H}\left(t\right))$] may be identified as obstructions to the existence of a K-quasiconformal map $f:(S^3,\mathcal{H}\left(t\right))\to (S^3,\mathcal{H}\left(0\right))$. Our discussion of Fefferman’s metric in Section 2.1.5 and Section 2.6 is only tentative, and a deeper study is relegated to further work.