2.1 Singularities of holomorphic foliations

To start with, recall the definition of singular holomorphic foliation on a complex surface M.

Each open set $\mathbb {U}_i$ is called a flow box. For each $c\in \mathbb {T}_i$ , the Riemann surface $\Phi _i^{-1}\{t=c\}$ in $\mathbb {U}_i$ is called a plaque. Property (2) above ensures that in the intersection of two flow boxes, plaques are mapped to plaques.

A leaf L is a minimal connected subset of M such that if L intersects a plaque, it contains that plaque. A transversal is a Riemann surface immersed in M which is transverse to each leaf of M.

The local theory of singular holomorphic foliations is closely related to holomorphic vector fields. One recalls some basic concepts in $\mathbb {C}^2$ ; see [Reference Brunella5, Reference Fornæss and Sibony11, Reference Nguyên17, Reference Nguyên18].

Suppose the holomorphic vector field $Z=P({\partial }/{\partial z})+Q({\partial }/{\partial w})$ admits a singularity at the origin. Let $\unicode{x3bb} _1$ , $\unicode{x3bb} _2$ be the eigenvalues of the Jacobian matrix $\textstyle (\! \begin {smallmatrix} P_z & P_w \\ Q_z & Q_w \end {smallmatrix} \!)$ at the origin.

In this paper, all singularities are assumed to be non-degenerate. Then the foliation defined by integral curves of Z has an isolated singularity at $0$ . Degenerate singularities are studied in [Reference Brunella5]. Seidenberg’s reduction theorem [Reference Seidenberg21] shows that degenerate singularities can be resolved into non-degenerate ones after finitely many blow-ups.

One can verify that the quotient is unchanged by multiplication of Z by any non-vanishing holomorphic function.

One could consider $\unicode{x3bb} ^{-1}={\unicode{x3bb} _2}/{\unicode{x3bb} _1}$ instead of $\unicode{x3bb} $ , but then $\unicode{x3bb} \notin \mathbb {R}$ if and only if ${\unicode{x3bb} ^{-1}\notin \mathbb {R}}$ . Thus, the notion of hyperbolicity is well defined. Also, being non-hyperbolic, in the Poincaré domain or Siegel domain, is well defined. The complex number $\unicode{x3bb} $ will be called an eigenvalue of Z at the singularity, with an inessential abuse due to this exchange $\unicode{x3bb} \leftrightarrow \unicode{x3bb} ^{-1}$ . The unordered pair $\{\unicode{x3bb} ,\unicode{x3bb} ^{-1}\}$ is invariant under local biholomorphic changes of coordinates.

Consider a holomorphic foliation $(M,E,\mathscr {F}{\kern1.5pt})$ where E is discrete. When one tries to linearize a vector field near an isolated non-degenerate singularity, one has to divide power series coefficients by quantities $m_1+\unicode{x3bb} m_2-1$ and $m_1+\unicode{x3bb} m_2-\unicode{x3bb} $ where $m_1$ , $m_2\in \mathbb {Z}_{\geqslant 0}$ with $m_1+m_2\geqslant 2$ . To ensure convergence, these quantities have to be non-zero and not too close to zero.

These quantities are non-zero if and only if $\unicode{x3bb} \notin \mathbb {Q}_{\neq 1}$ . They do not have $0$ as a limit if and only if $\unicode{x3bb} \notin \mathbb {R}_{\leqslant 0}$ , that is, the singularity is in the Poincaré domain.

We are now ready to state some linearization results in $\mathbb {C}^2$ .

Remark 2.6. The linear part of a singular holomorphic vector field is

$$ \begin{align*} (az+bw)\frac{\partial}{\partial z}+(cz+dw)\frac{\partial}{\partial w} \end{align*} $$

for some $a,b,c,d\in \mathbb {C}$ with $ad-bc\neq 0$ if the singularity is assumed to be non-degenerate. It is non-linearizable if and only if the Jordan normal form of the Jacobian matrix $\textstyle (\! \begin {smallmatrix} a & b\\ c & d \end {smallmatrix} \!)$ has a rank-2 block $\textstyle (\! \begin {smallmatrix} a & 1\\ 0 & a \end {smallmatrix} \!)$ with $a\neq 0$ . In this case $\unicode{x3bb} =1$ , hence Poincaré’s theorem holds. The vector field is holomorphically equivalent to its linear part $(az+w){\partial }/{\partial z}+aw({\partial }/{\partial w})$ , but is not linearizable.

For the resonant case $\unicode{x3bb} \in \mathbb {Q}_{\neq {1}}$ and the degenerate case, one may use the Poincaré–Dulac normal form [Reference Arnold and Ilyashenko2, Ch. 3, §3.2, pp. 54].

In particular, all hyperbolic singularities are linearizable.

To get linearization for $\unicode{x3bb} $ in the Siegel domain, the following result assumes the more advanced Brjuno condition.

Theorem 2.7. (Brjuno [Reference Arnold and Ilyashenko2, Reference Brjuno4])

A singular holomorphic vector field with a non-resonant linear part is holomorphically linearizable if its eigenvalue $\unicode{x3bb} \in \mathbb {R}$ satisfies the condition

$$ \begin{align*} \sum_{n\geqslant 1}\frac{\log q_{n+1}}{q_n}<\infty, \end{align*} $$

where $p_n/q_n$ is the nth approximant of the continued fraction expansion of $\unicode{x3bb} $ .

The golden ratio

$$ \begin{align*} \frac{\sqrt{5}-1}{2}=1+\frac{1}{1+\frac{1}{1+\cdots}} \end{align*} $$

is a Brjuno number. Indeed, any irrational number whose continued fraction expansion ends with a string of 1s

$$ \begin{align*} \alpha=a_0+\frac{1}{a_1+\frac{1}{\cdots}}=[a_0,a_1,\ldots,a_k,1,1,\ldots]\in\mathbb{R}\backslash\mathbb{Q} \quad(a_0\in\mathbb{Z} ,a_1,\ldots, a_k\in\mathbb{N}), \end{align*} $$

is a Brjuno number. The Brjuno numbers are dense in $\mathbb {R}\backslash \mathbb {Q}$ . See [Reference Lee14, Propositions 1.2 and 1.3].

In this paper, all singularities are assumed to be linearizable.

2.2 Directed harmonic currents

Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field $Z=z{\partial }/{\partial z}+\unicode{x3bb} w({\partial }/{\partial w})$ with $\unicode{x3bb} \in \mathbb {R}^*$ . One may assume $0<|\unicode{x3bb} |\leqslant 1$ after switching z and w if necessary. There are always two separatrices $\{z=0\}$ and $\{w=0\}$ . Other leaves can be parametrized as

(1) $$ \begin{align} L_\alpha:=\{(z,w)=\psi_\alpha(\zeta):= (e^{i \zeta},\alpha e^{i \unicode{x3bb} \zeta})=(e^{-v+i u},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})\} \quad (\alpha\neq0), \end{align} $$

where $\zeta =u+iv\in \mathbb {C}$ . The map

$$ \begin{align*} \begin{aligned} \Psi:\mathbb{C}\times\mathbb{C}^*&\longrightarrow\mathbb{C}^2\\ (\zeta,\alpha)&\longmapsto (e^{i \zeta},\alpha e^{i \unicode{x3bb} \zeta}) \end{aligned} \end{align*} $$

is locally biholomorphic. Here $\alpha $ is the coordinate on the transversal and $\zeta $ is the coordinate on leaves. It is not injective since $\Psi (\zeta +2\pi ,\alpha )=\Psi (\zeta ,\alpha e^{2\pi i\unicode{x3bb} })$ .

Two numbers $\alpha $ , $\beta \in \mathbb {C}^*$ are equivalent $\alpha \sim \beta $ if $\beta =e^{2k\pi i \unicode{x3bb} }\alpha $ for some $k\in \mathbb {Z}$ . The following statements are equivalent:

• • $\alpha \sim \beta $ ;

• • $L_\alpha =L_\beta $ ;

• • $\psi _\alpha =\psi _\beta \circ (\text {translation of }2k\pi )$ for some $k\in \mathbb {Z}$ .

Let $\mathscr {C}_{\mathscr {F}}$ (respectively, $\mathscr {C}_{\mathscr {F}}^{1,1}$ ) denote the space of functions (respectively, forms of bidegree $(1,1)$ ) defined on leaves of the foliation which are compactly supported on $M\backslash E$ , leafwise smooth and transversally continuous. A form $\iota \in \mathscr {C}_{\mathscr {F}}^{1,1}$ is said to be positive if its restriction to every plaque is a positive (1,1)-form.

A directed harmonic current T on $\mathscr {F}$ is a continuous linear form on $\mathscr {C}_{\mathscr {F}}^{1,1}$ satisfying the following two conditions:

1. (1) $i\partial \bar {\partial } T=0$ in the weak sense, that is, $T(i\partial \bar {\partial }f)=0$ for all $f\in \mathscr {C}_{\mathscr {F}}$ , where in the expression $i\partial \bar {\partial }f$ one only considers $\partial \bar {\partial }$ along the leaves;

2. (2) T is positive, that is, $T(\iota )\geqslant 0$ for all positive forms $\iota \in \mathscr {C}_{\mathscr {F}}^{1,1}$ .

It is well known (see, for example, [Reference Berndtsson and Sibony3, Reference Dinh, Nguyên and Sibony6, Reference Fornæss and Sibony11]) that a directed harmonic current T on a flow box $\mathbb {U}\cong \mathbb {B}\times \mathbb {T}$ can be locally expressed as

(2) $$ \begin{align} T=\int_{\alpha\in\mathbb{T}} h_\alpha [P_\alpha]\, d\mu(\alpha). \end{align} $$

The $h_\alpha $ are non-negative harmonic functions on the local leaves $P_\alpha $ and $\mu $ is a Borel measure on the transversal $\mathbb {T}$ . If $h_\alpha =0$ at some point on $P_\alpha $ , then by the mean value theorem $h_\alpha \equiv 0$ . For all such $\alpha \in \mathbb {T}$ , we replace $h_\alpha $ by the constant function $1$ and we set $d\mu (\alpha )=0$ . Thus, we get a new expression of T where $h_\alpha>0$ for all $\alpha \in \mathbb {T}$ .

Such an expression is not unique since $T=\int _{\alpha \in \mathbb {T}}(h_\alpha g(\alpha ))[P_\alpha ](({1}/{g(\alpha )})\, d\mu (\alpha ))$ for any measurable positive function $g:\mathbb {T}\rightarrow \mathbb {R}_{>0}$ which is finite and non-zero almost everywhere. The expression is unique after normalization, which means that for each $\alpha \in \mathbb {T}$ one fixes $h_\alpha (z_0,w_0)=1$ at some point $(z_0,w_0)\in P_\alpha $ .

Each harmonic function $h_\alpha $ on the leaf $V_\alpha $ can be pulled back by the parametrization $\Psi $ as the harmonic function

$$ \begin{align*} H_\alpha(u,v):=h_\alpha(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i\unicode{x3bb} u}). \end{align*} $$

The domain of definition for u, v will be precisely described later in this section.

In §1 the notion of periodic current was introduced. Here is an equivalent characterization.

A current T of the form (2) is $dd^c$ -closed on $\mathbb {D}^2\backslash \{0\}$ . But its trivial extension $\tilde {T}$ across the singularity $0$ is not necessarily $dd^c$ -closed on $\mathbb {D}^2$ . It is true when T is compactly supported, for example when T is a localization of a current on a compact manifold, by the following argument (see [Reference Dinh, Nguyên and Sibony6, Lemma 2.5] for details).

Let T be a directed harmonic current on $M\backslash E$ , where M is a compact complex manifold and E is a finite set. The current T can be extended by zero through E in order to obtain the positive current $\tilde T$ on M. Next, we apply the following result.

Here $-T$ is a negative current of bidimension $(1,1)$ on $M\backslash E$ with $dd^c (-T)\geqslant 0$ and E has dimension $0$ . So for the trivial extension $\tilde {T}$ on M one has $dd^c(-\tilde {T})\geqslant 0$ . Moreover, $\tilde {T}$ is compactly supported since M is compact. Thus

$$ \begin{align*} \langle dd^c\tilde{T},1 \rangle = \langle \tilde{T},dd^c1 \rangle = 0. \end{align*} $$

Combining with $dd^c\tilde {T}\leqslant 0$ from the extension theorem, one concludes that $dd^c\tilde {T}=0$ on M. Thus, locally near any singularity, the trivial extension $\tilde {T}$ is $dd^c$ -closed.

Let $\beta :=idz\wedge d\bar {z}+idw\wedge d\bar {w}$ be the standard Kähler form on $\mathbb {C}^2$ . The mass of T on a domain $U\subset \mathbb {D}^2$ is denoted by $\|T\|_U:=\int _U T\wedge \beta $ . In this paper, all currents are assumed to have finite mass on $\mathbb {D}^2$ .

Definition 2.10. (See [Reference Nguyên19, §2.4])

Let T be a directed harmonic current on $(\mathbb {D}^2,\mathscr {F},\{0\})$ . We define the Lelong number by the limit

$$ \begin{align*} \mathscr{L}(T,0)=\limsup\limits_{r\rightarrow0+}\frac{1}{\pi r^2}\|T\|_{r\mathbb{D}^2}\in[0,+\infty]. \end{align*} $$

The limit can be infinite when the trivial extension $\tilde {T}$ across the origin is not $dd^c$ -closed [Reference Nguyên19, Example 2.11]. When $\tilde {T}$ is $dd^c$ -closed, the following theorem ensures the finiteness.

In our case, the function

$$ \begin{align*} r\mapsto \frac{1}{\pi r^2}\|\tilde{T}\|_{r\mathbb{D}^2}=\frac{1}{\pi r^2}\|T\|_{r\mathbb{D}^2} \end{align*} $$

is increasing with $r\in (0, 1]$ . In particular,

$$ \begin{align*} \mathscr{L}(T,0)=\lim\limits_{r\rightarrow0+}\frac{1}{\pi r^2}\|T\|_{r\mathbb{D}^2}\in\bigg[0,\dfrac{1}{\pi}\|T\|_{\mathbb{D}^2}\bigg]. \end{align*} $$

In this paper, the symbols $\lesssim $ and $\gtrsim $ stand for inequalities up to a multiplicative positive constant depending only on $\unicode{x3bb} $ . We write $\approx $ when both inequalities are satisfied.

3.1 Positive case $\unicode{x3bb}>0$

For any $\alpha \in \mathbb {C}^*$ fixed, the leaf $L_\alpha $ is contained in a real three-dimensional Levi flat CR manifoldFootnote ¹ $|w|=|\alpha | |z|^{\unicode{x3bb} }$ , which can be viewed as a curve in $|z|=e^{-v}$ , $|w|=|\alpha | e^{-\unicode{x3bb} v}$ coordinates. The norms $|z|$ and $|w|$ depend only on v. When $v\rightarrow +\infty $ , the point on the leaf tends to the singularity $(0,0)$ described by Figures 5 and 6.

Figure 6

Case $|\alpha |\geqslant 1$ .

If one fixes some $v=-\log r$ , then $|z|=r$ and $|w|=|\alpha | r^\unicode{x3bb} $ is fixed. The set $\mathbb {T}^2_r:=\{(z,w)\in \mathbb {D}^2:|z|=r,|w|=|\alpha | r^\unicode{x3bb} \}$ is a torus and the intersection of the leaf $L_\alpha $ with this torus is a smooth curve $L_{\alpha ,r}:=L_\alpha \cap \mathbb {T}^2_r$ .

When $\unicode{x3bb} \in \mathbb {Q}$ , this curve $L_{\alpha ,r}$ is closed. See Figure 7.

When $\unicode{x3bb} \notin \mathbb {Q}$ , this curve $L_{\alpha ,r}$ is dense on the torus $\mathbb {T}_r^2$ . See Figures 8 and 9.

In this case the two curves $L_{\alpha ,r}$ and $L_{\alpha e^{2\pi i \unicode{x3bb} },r}$ are two different parametrizations of the same image. The dashed curve in Figure 8 is not only the image of $L_{\alpha ,r}$ for $u\in [2\pi ,4\pi )$ but also the image of $L_{\alpha e^{2\pi i \unicode{x3bb} },r}$ for $u\in [0,2\pi )$ . This raises ambiguity while normalizing harmonic functions on a leaf $L_\alpha $ .

Such ambiguity can be resolved once one restricts everything to an open subset $U_\epsilon :=\{(z,w)\in \mathbb {D}^2~|~{\textrm {arg}}(z)\in (0,2\pi -\epsilon ),z\neq 0,w\neq 0\}$ for some fixed $\epsilon \in [0,\pi )$ . Any leaf $L_\alpha $ on $U_\epsilon $ decomposes into a disjoint union of infinitely many components:

$$ \begin{align*} L_\alpha\cap U_\epsilon=\bigcup\limits_{k\in\mathbb{Z}}\!\bigg\{(e^{-v+iu},\alpha e^{2k\pi i\unicode{x3bb}} e^{-\unicode{x3bb} v+i\unicode{x3bb} u})~|~u\kern1.2pt{\in}\kern1.2pt(0,2\pi-\epsilon),v\kern1.2pt{>}\kern1.2pt\frac{\log^+|\alpha|}{\unicode{x3bb}}\bigg\}. \end{align*} $$

For example, in Figure 10, the curve and the dashed curve are two distinct components of $L_{1,1}\cup U_{\epsilon}$ .

Figure 7

A closed curve on a torus.

Figure 8

Two loops.

Figure 9

Twenty loops.

Figure 10

Two components of $L_{1,1}\cup U_{\epsilon}$ .

Such a parametrization is yet not unique. For example, for any $k_0\in \mathbb {Z}$ one can parametrize

$$ \begin{align*} L_\alpha\cap U_\epsilon=\bigcup\limits_{k\in\mathbb{Z}}\!\bigg\{\!(e^{-v+iu},\alpha e^{2k\pi i\unicode{x3bb}} e^{-\unicode{x3bb} v+i\unicode{x3bb} u})\kern1.2pt{|}\kern1.2pt u\kern1.2pt{\in}\kern1.2pt(2k_0\pi,2k_0\pi+2\pi-\epsilon),v>\frac{\log^+|\alpha|}{\unicode{x3bb}}\bigg\}\kern-1pt. \end{align*} $$

The parametrization is unique once one fixes $k_0$ , for example, $k_0=0$ . I remark for the time being that all other choices of $k_0$ will be used for analysing non-periodic currents in §5.2.

3.2 Resolving ambiguity in the irrational case

Let $\unicode{x3bb} \notin \mathbb {Q}$ . Let T be a harmonic current directed by $\mathscr {F}$ . Then $T|_{P_\alpha }$ has the form $h_\alpha (z,w)[P_\alpha ]$ . One may assume that $h_\alpha $ is nowhere 0 for every $\alpha $ . Let

$$ \begin{align*} H_\alpha(u+iv):=h_\alpha\circ \psi_\alpha\bigg(u+iv+i\frac{\log^+|\alpha|}{\unicode{x3bb}}\bigg). \end{align*} $$

This is a positive harmonic function for $\mu $ -almost all $\alpha \in \mathbb {C}^*$ defined in a neighbourhood of the upper half-plane $\mathbb {H}=\{(u+iv)\in \mathbb {C}~|~v>0\}$ , determined by the Poisson integral formula

$$ \begin{align*} H_\alpha(u+iv)=\frac{1}{\pi} \int_{y\in\mathbb{R}}H_\alpha(y) \frac{v} {v^2+(y-u)^2} dy+C_\alpha v. \end{align*} $$

One can normalize $H_\alpha $ by setting $H_\alpha (0)=1$ . But by doing so one may normalize data over the same leaf for multiple times. Indeed, any pair of equivalent numbers $\alpha \sim \beta $ in $\mathbb {C}^*$ , $\beta =\alpha e^{2k\pi i \unicode{x3bb} }$ , may provide us with two different normalizations $H_{\alpha }$ and $H_{\beta }$ on the same leaf $L_{\alpha }=L_{\beta }$ . A major task is to find formulas for the mass and the Lelong number independent by the choice of normalization.

The ambiguity is described by the following proposition.

Proposition 3.2. If $\beta =\alpha e^{2k \pi i \unicode{x3bb} }$ for some $k\in \mathbb {Z}$ , then the two normalized positive harmonic functions $H_\alpha $ and $H_\beta $ satisfy

$$ \begin{align*} H_\alpha(u+iv) = H_\alpha(2k\pi) H_\beta(u-2k\pi+iv). \end{align*} $$

In other words, they differ by a translation and a multiplication by a non-zero constant.

Proof. When $|\alpha |<1$ , by definition

$$ \begin{align*} H_\alpha(u+iv)=h_\alpha(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u}), \quad H_\alpha(0)=h_\alpha(1,\alpha). \end{align*} $$

Thus, the normalized harmonic function is

$$ \begin{align*} H_\alpha(u+iv)=\frac{h_\alpha(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})}{h_\alpha(1,\alpha)}, \end{align*} $$

and for the same reason

$$ \begin{align*}H_\beta(u+iv)=\frac{h_\beta(e^{-v+iu},\beta e^{-\unicode{x3bb} v+i \unicode{x3bb} u})}{h_\beta(1,\beta)}. \end{align*} $$

The two functions $h_\alpha $ and $h_\beta $ are the positive harmonic coefficient of T on the same leaf $L_\alpha =L_\beta $ , hence they differ up to multiplication by a positive constant $C>0$ :

$$ \begin{align*} \begin{aligned} h_\alpha(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})&=C\cdot h_\beta(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})\\ &=C\cdot h_\beta(e^{-v+iu},\beta e^{-2k \pi i \unicode{x3bb}} e^{-\unicode{x3bb} v+i \unicode{x3bb} u})\\ &=C\cdot h_\beta(e^{-v+i(u-2k \pi)},\beta e^{-\unicode{x3bb} v+i \unicode{x3bb} (u-2k \pi)}). \end{aligned} \end{align*} $$

Thus,

$$ \begin{align*} \begin{aligned} H_\alpha(u+iv)&=\frac{h_\alpha(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})}{h_\alpha(1,\alpha)}=\frac{C\cdot h_\beta(e^{-v+i(u-2k \pi)},\beta e^{-\unicode{x3bb} v+i \unicode{x3bb} (u-2k \pi)})}{C\cdot h_\beta(1,\alpha)}\\ &=\frac{h_\beta(e^{-v+i(u-2k \pi)},\beta e^{-\unicode{x3bb} v+i \unicode{x3bb} (u-2k \pi)})}{h_\beta(1,\beta)}\cdot \frac{h_\beta(1,\beta)}{h_\beta(1,\alpha)}\\ &=H_\beta(u-2k \pi+iv)\cdot \frac{h_\beta(1,\beta)}{h_\beta(1,\alpha)}. \end{aligned} \end{align*} $$

When $u=2k \pi $ and $v=0$ one has $H_\alpha (2k \pi )={h_\beta (1,\beta )}/{h_\beta (1,\alpha )}$ . Thus, one gets the equality. The proof for the case $|\alpha |>1$ is similar.

Take the open subset $U:=\{(z,w)\in \mathbb {D}^2~|~z\notin \mathbb {R}_{\geqslant 0},w\neq 0\}$ . See Figures 11 and 12.

Figure 11

Domain U in coordinates $(z,w)$ .

Figure 12

Domain U in coordinates $(u,v)$ .

Any leaf $L_\alpha $ in U is a disjoint union of infinitely many components. Once $\alpha $ is fixed, there is a one-to-one correspondence between these components and strips in Figure 12.

$$ \begin{align*} L_\alpha\cap U=\bigcup\limits_{k\in\mathbb{Z}}\tilde{L}_{\alpha e^{2k\pi i\unicode{x3bb}}}:=\bigcup\limits_{k\in\mathbb{Z}}\!\bigg\{(e^{-v+iu},\alpha e^{2k\pi i\unicode{x3bb}} e^{-\unicode{x3bb} v+i\unicode{x3bb} u})\kern1.2pt{|}\kern1.2pt u\kern1.2pt{\in}\kern1.2pt(0,2\pi),v\kern1.2pt{>}\kern1.2pt\frac{\log^+|\alpha|}{\unicode{x3bb}}\bigg\}. \end{align*} $$

Normalizing $H_{\alpha e^{2k\pi i\unicode{x3bb} }}$ on $\tilde {L}_{\alpha e^{2k\pi i\unicode{x3bb} }}$ avoids ambiguity. Thus, the mass

$$ \begin{align*} \begin{aligned} \|T\|_{U}&=\int_{(z,w)\in U}T\wedge i\partial\bar{\partial}(|z|^2+|w|^2)\\ &=\int_{\alpha\in\mathbb{C}^*}\int_{v>{\log^+|\alpha|}/{\unicode{x3bb}}}\int_{u=0}^{2\pi}H_{\alpha}(u+iv) 2(e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,du\, dv\, d\mu(\alpha)\\ &=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^{2}\, du\, dv\, d\mu(\alpha) \end{aligned} \end{align*} $$

for some positive measure $\mu $ on $\mathbb {C}^*$ . Here, $\|\psi _\alpha '\|^2$ is the jacobian coming from the $(1,1)$ -form $i\partial \bar {\partial }(|z|^2+|w|^2)$ on $L_\alpha $ after a change of coordinates and a translation on v:

(3) $$ \begin{align} \|\psi_\alpha'\|^2=\left\{ \begin{aligned} &2(e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) & (|\alpha|< 1),\\ &2(|\alpha|^{-{2}/{\unicode{x3bb}}}e^{-2v}+\unicode{x3bb}^2 e^{-2\unicode{x3bb} v}) & (|\alpha|\geqslant 1). \end{aligned} \right. \end{align} $$

Since H is harmonic in a neighbourhood of $\mathbb {H}$ , it is continuous in $\mathbb {H}$ . So

$$ \begin{align*} \begin{aligned} \|T\|_{U}&=\lim\limits_{\epsilon\rightarrow 0+}\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=0}^{2\pi+\epsilon}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2 \,du\, dv\, d\mu(\alpha)\\ &=\lim\limits_{\epsilon\rightarrow0+}\|T\|_{\bigcup\limits_{k\in\mathbb{Z}}\tilde{L}_{\alpha e^{2k \pi i\unicode{x3bb}}}}\\ &=\|T\|_{\mathbb{D}^2}. \end{aligned} \end{align*} $$

Thus, we can express the mass by a formula independent of the choice of normalization

$$ \begin{align*} \|T\|_{\mathbb{D}^2}=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2 \,du\, dv\, d\mu(\alpha). \end{align*} $$

Lemma 3.3. For each $k_0\in \mathbb {Z}$ fixed,

(4) $$ \begin{align} \|T\|_{\mathbb{D}^2}=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=2k_0\pi}^{2k_0\pi+2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2 \,du \,dv \,d\mu(\alpha). \end{align} $$

Proof. The disjoint union $L_\alpha \cap U=\bigcup \nolimits _{k\in \mathbb {Z}}\tilde {L}_{\alpha e^{2k\pi i\unicode{x3bb} }}$ can be parametrized in many other ways. For instance,

$$ \begin{align*} L_\alpha\cap U=\bigcup\limits_{k\in\mathbb{Z}}\bigg\{(e^{-v+iu},\alpha e^{2k\pi i\unicode{x3bb}} e^{-\unicode{x3bb} v+i\unicode{x3bb} u})~|~u\in(2k_0\pi,2k_0\pi+2\pi),v>\dfrac{\log^+|\alpha|}{\unicode{x3bb}}\bigg\}. \end{align*} $$

By the same argument as above one concludes.

3.3 Negative case $\unicode{x3bb} <0$

As in the positive case, for any $\alpha \in \mathbb {C}^*$ fixed, the leaf $L_\alpha $ is contained in a real three-dimensional analytic Levi-flat CR manifold $|w|=|\alpha | |z|^{\unicode{x3bb} }$ , which can be viewed as a curve in $|z|,|w|$ coordinates. The norms $|z|$ and $|w|$ depend only on v.

The difference is that in the negative case, no leaf $L_\alpha $ tends to the singularity $(0,0)$ . For r sufficiently small, the leaf $L_\alpha $ is outside of $r \mathbb {D}^2$ . See Figure 13.

Figure 13

Case $\unicode{x3bb} <0$ .

Like the positive case $\unicode{x3bb}>0$ , when one fixes $|z|=r$ for some $r\in (0,1)$ , $|w|=|\alpha | |z|^\unicode{x3bb} $ is uniquely determined and the real two-dimensional leaf $L_\alpha $ becomes a real 1-dimensional curve $L_{\alpha ,r}:=L_\alpha \cap \mathbb {T}^2_r$ on the torus $\mathbb {T}^2_r:=\{(z,w)\in \mathbb {D}^2~|~|z|=r,|w|=|\alpha | r^\unicode{x3bb} \}$ . It is a closed curve if $\unicode{x3bb} \in \mathbb {Q}$ , and a dense curve on $\mathbb {T}^2_r$ if $\unicode{x3bb} \notin \mathbb {Q}$ .

Let T be a harmonic current directed by $\mathscr {F}$ . Then $T|_{P_\alpha }$ has the form $h_\alpha (z,w)[P_\alpha ]$ . Let $H_\alpha :=h_\alpha \circ \psi _\alpha (u+iv)$ . It is a positive harmonic function for $\mu $ -almost all $\alpha \in \mathbb {D}^*$ defined on a neighbourhood of a horizontal strip $\{(u,v)\in \mathbb {R}^2~|~0<v<{\log |\alpha |}/{\unicode{x3bb} }\}$ .

As in the case $\unicode{x3bb}>0$ , one only calculates the mass on an open subset $U:=\{(z,w)\in \mathbb {D}^2~|~z\notin \mathbb {R}_{\geqslant 0},w\neq 0\}$ . For each $\alpha \in \mathbb {D}^*$ one normalizes $H_\alpha $ by setting $H_\alpha (0)=1$ to fix the expression $T:=\int h_\alpha [P_\alpha ]\, d\mu (\alpha )$ . Similarly to Lemma 3.3, for each $k_0\in \mathbb {Z}$ fixed,

$$ \begin{align*} \begin{aligned} \|T\|_{\mathbb{D}^2} &=\int_{0<|\alpha|<1}\!\int_{v=0}^{{\log|\alpha|}/{\unicode{x3bb}}}\!\int_{u=2 k_0 \pi}^{2 k_0 \pi+2 \pi}\!\kern-1.2pt H_\alpha(u+iv) 2 (e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,du \,dv\, d\mu(\alpha),\\ \mathscr{L}(T,0)&=\lim\limits_{r\rightarrow 0+}\frac{1}{r^2}\|T\|_{r\mathbb{D}^2}\\ & =\lim\limits_{r\rightarrow 0+}\frac{1}{r^2}\int_{0<|\alpha|<r^{1-\unicode{x3bb}}}\int_{v=-\log r}^{({\log|\alpha|-\log r})/{\unicode{x3bb}}}\int_{u=2 k_0 \pi}^{2 k_0 \pi+2 \pi} \\&\qquad\qquad\qquad\ \kern1pt\qquad\qquad H_\alpha(u+iv) 2 (e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,du \,dv \,d\mu(\alpha). \end{aligned} \end{align*} $$

These formulas will be calculated in later sections.

5.1 Periodic currents, still a Fourier series

Periodic currents behave similarly to currents in the rational case $\unicode{x3bb} \in \mathbb {Q}$ . Suppose $H_\alpha $ is periodic, that is, there is some $b\in \mathbb {Z}_{\geqslant 1}$ such that $H_\alpha (u+iv)=H_\alpha (u+2\pi b+iv)$ for any $u+iv\in \mathbb {H}$ . Periodic harmonic functions are characterized as in (5) of Lemma 4.1.

According to Lemma 3.3, the mass is

$$ \begin{align*} \|T\|_{\mathbb{D}^2}=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=2k_0\pi}^{2k_0\pi+2\pi}H_\alpha(u+iv)\|\psi_\alpha'\|^2\, du\wedge dv \,d\mu(\alpha), \end{align*} $$

for any $k_0\in \mathbb {Z}$ , in particular for $k_0=0,1,\ldots ,b-1$ . Thus, we may calculate

$$ \begin{align*} b\|T\|_{\mathbb{D}^2}&=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=0}^{2\pi b}H_\alpha(u+iv)\|\psi_\alpha'\|^2 \,du\wedge dv \,d\mu(\alpha)\\ \|T\|_{\mathbb{D}^2}&=\frac{1}{b}\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=0}^{2\pi b}H_\alpha(u+iv)\|\psi_\alpha'\|^2 \,du\wedge dv\, d\mu(\alpha),\\ &=\frac{1}{b}\bigg\{\int_{|\alpha|<1}\int_{v>0}\int_{u=0}^{2\pi b}H_\alpha(u+iv) 2(e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,du\wedge dv \,d\mu(\alpha)\\ &\quad +\int_{|\alpha|\geqslant1}\int_{v>0}\int_{u=0}^{2\pi b}H_\alpha(u+iv) 2(|\alpha|^{-{2}/{\unicode{x3bb}}} e^{-2v}+\unicode{x3bb}^2 e^{-2\unicode{x3bb} v}) \,du\wedge dv \,d\mu(\alpha)\bigg\},\\ &=\frac{2\pi b}{b}\bigg\{\int_{|\alpha|<1}\int_{v>0}(a_0(\alpha)+b_0(\alpha) v) 2(e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,dv \,d\mu(\alpha)\\ & \quad+\int_{|\alpha|\geqslant1}\int_{v>0}(a_0(\alpha)+b_0(\alpha) v) 2(|\alpha|^{-{2}/{\unicode{x3bb}}} e^{-2v}+\unicode{x3bb}^2 e^{-2\unicode{x3bb} v}) \,dv\, d\mu(\alpha)\bigg\},\\ &=2\pi \bigg\{\int_{|\alpha|<1}a_0(\alpha) (1+|\alpha|^2 \unicode{x3bb})\, d\mu(\alpha)+\int_{|\alpha|\geqslant1}a_0(\alpha) (|\alpha|^{-{2}/{\unicode{x3bb}}}+\unicode{x3bb})\, d\mu(\alpha) \nonumber \\&\quad+\int_{|\alpha|<1}b_0(\alpha) \bigg(\frac{1}{2}+\frac{1}{2}|\alpha|^2\bigg)\, d\mu(\alpha)+\int_{|\alpha|\geqslant1}b_0(\alpha) \bigg(\frac{1}{2}+\frac{1}{2}|\alpha|^{-{2}/{\unicode{x3bb}}}\bigg) \,d\mu(\alpha)\bigg\}\\ &\approx \int_{\alpha\in\mathbb{C}^*}a_0(\alpha)\, d\mu(\alpha)+\int_{\alpha\in\mathbb{C}^*}b_0(\alpha) \,d\mu(\alpha), \end{align*} $$

which is the same expression as in the case $\unicode{x3bb} \in \mathbb {Q}_{>0}$ .

Next, the Lelong number is calculated as

$$ \begin{align*} ~&\mathscr{L}(T,0)\\[3pt]&\quad=\lim\limits_{r\rightarrow 0+}\frac{1}{r^2}\|T\|_{r\mathbb{D}^2}\\[3pt] &\quad=\lim\limits_{r\rightarrow 0+}\frac{1}{r^2}2\pi \bigg\{\int_{|\alpha|<r^{1-\unicode{x3bb}}}\int_{v>-\log r}(a_0(\alpha)+b_0(\alpha)v) 2 (e^{-2v}\\[3pt] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,dv\, d\mu(\alpha)\\[3pt] & \qquad+ \int_{r^{1-\unicode{x3bb}}\leqslant |\alpha|<1}\int_{v>({\log|\alpha|-\log r}/{\unicode{x3bb}})}(a_0(\alpha)+b_0(\alpha)v) 2 (e^{-2v}\\[3pt] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v}) \,dv\, d\mu(\alpha)\bigg\}\\[3pt] & \qquad+ \int_{|\alpha|\geqslant 1}\int_{v>{-\log r}/{\unicode{x3bb}}}(a_0(\alpha)+b_0(\alpha)v) 2 (|\alpha|^{-{2}/{\unicode{x3bb}}}e^{-2v}+\unicode{x3bb}^2 e^{-2\unicode{x3bb} v}) \,dv\, d\mu(\alpha)\bigg\}\\[3pt] &\quad=\lim\limits_{r\rightarrow 0+}2\pi \bigg\{\int_{|\alpha|<r^{1-\unicode{x3bb}}}a_0(\alpha) (1+\unicode{x3bb} |\alpha|^2 r^{2\unicode{x3bb}-2})\, d\mu(\alpha)\\[3pt] & \qquad +\int_{|\alpha|\geqslant r^{1-\unicode{x3bb}}}a_0(\alpha) (|\alpha|^{-{2}/{\unicode{x3bb}}} r^{{2}/{\unicode{x3bb}}-2}+\unicode{x3bb})\, d\mu(\alpha)\\[3pt] & \qquad +\int_{|\alpha|< r^{1-\unicode{x3bb}}}b_0(\alpha) \bigg(\frac{1}{2}+\frac{1}{2}|\alpha|^2 r^{2\unicode{x3bb}-2}-\log r-\unicode{x3bb} |\alpha|^2 r^{2\unicode{x3bb}-2} \log r\bigg)\, d\mu(\alpha)\\[3pt] & \qquad +\int_{r^{1-\unicode{x3bb}}\leqslant |\alpha|<1}b_0(\alpha) \bigg(\frac{1}{2}+\frac{1}{2}|\alpha|^{-{2}/{\unicode{x3bb}}} r^{{2}/{\unicode{x3bb}}-2}-\log r-\unicode{x3bb}^{-1} |\alpha|^{-{2}/{\unicode{x3bb}}} r^{2\unicode{x3bb}-2}\log r\\[3pt] & \qquad +\log |\alpha|+\unicode{x3bb}^{-1} |\alpha|^{-{2}/{\unicode{x3bb}}} \log|\alpha| r^{2\unicode{x3bb}-2}\bigg)\, d\mu(\alpha)\\[3pt] & \qquad +\int_{|\alpha|\geqslant 1}b_0(\alpha) \bigg(\frac{1}{2}+\frac{1}{2}|\alpha|^{-{2}/{\unicode{x3bb}}} r^{{2}/{\unicode{x3bb}}-2}-\log r-\unicode{x3bb}^{-1} |\alpha|^{-{2}/{\unicode{x3bb}}} r^{2\unicode{x3bb}-2}\log r\bigg)\, d\mu(\alpha)\bigg\}, \end{align*} $$

exactly the same expression as in the positive rational case with $b=1$ . Using the same argument as in Lemma 4.2, one may assume that $b_0(\alpha )=0$ for $\mu $ -almost all $\alpha \in \mathbb {C}^*$ . One concludes that

$$ \begin{align*} \mathscr{L}(T,0)\approx\int_{\alpha\in\mathbb{C}^*}a_0(\alpha)\, d\mu(\alpha)\approx\|T\|_{\mathbb{D}^2}. \end{align*} $$

The Lelong number is strictly positive, the same as in the case $\unicode{x3bb} \in \mathbb {Q}\cup (0,1)$ .

5.2 Non-periodic current

For periodic currents, one takes an average among b expressions (4) in the previous section. For non-periodic currents, there is no canonical way of normalization. The key technique is to calculate expressions (4) for all $k_0\in \mathbb {Z}$ .

The Lelong number is expressed as

$$ \begin{align*} \begin{aligned} \mathscr{L}(T,0)&=\lim\limits_{r\rightarrow 0+}\frac{1}{r^2}\bigg\{\int_{|\alpha|<r^{1-\unicode{x3bb}}}\int_{v>-\log r}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2du \,dv \,d\mu(\alpha)\\ & \quad +\int_{r^{1-\unicode{x3bb}}\leqslant|\alpha|<1}\int_{v>({\log|\alpha|-\log r})/{\unicode{x3bb}}}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2du \,dv \,d\mu(\alpha)\\ & \quad +\int_{|\alpha|\geqslant1}\int_{v>{-\log r}/{\unicode{x3bb}}}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)\|\psi_\alpha'\|^2du \,dv \,d\mu(\alpha)\bigg\} \end{aligned} \end{align*} $$

Recall the Poisson integral formula after multiplying by a non-zero constant:

$$ \begin{align*} H_\alpha(u+iv)=\frac{1}{\pi} \int_{y\in\mathbb{R}}H_\alpha(y) \frac{v} {v^2+(y-u)^2} dy+{C}_\alpha v. \end{align*} $$

Using the same argument as in Lemma 4.2, one may assume ${C}_\alpha =0$ for all $\alpha \in \mathbb {C}^*$ .

Lemma 5.1. For any $v\geqslant {1}/{\unicode{x3bb} }>1$ and for any $u\in \mathbb {R}$ ,

$$ \begin{align*} \begin{aligned} \frac{{\partial}/{\partial v}(-\frac12({v}/({v^2+(u-y)^2})e^{-2v}))}{{v}/({v^2+(u-y)^2})e^{-2v}}\in\bigg(\frac{1}{2},2\bigg), \\ \frac{{\partial}/{\partial v}(-({1}/{2\unicode{x3bb}})({v}/({v^2+(u-y)^2}))e^{-2\unicode{x3bb} v})}{{v}/({v^2+(u-y)^2})e^{-2\unicode{x3bb} v}}\in\bigg(\frac{1}{2},2\bigg). \end{aligned} \end{align*} $$

Proof. This can be calculated directly:

$$ \begin{align*} \begin{aligned} \frac{\partial}{\partial v}\bigg(-\frac{1}{2}\frac{v}{v^2+(u-y)^2}e^{-2v}\bigg)&=\bigg(\frac{v}{v^2+(u-y)^2}+\bigg(-\frac{1}{2}\bigg)\frac{1}{v^2+(u-y)^2}\\ & \quad +\bigg(-\frac{1}{2}\bigg)\frac{v(-2v)}{(v^2+(u-y)^2)^2}\bigg) e^{-2v}\\ \frac{{\partial}/{\partial v}(-\frac12({v}/({v^2+(u-y)^2}))e^{-2v})}{{v}/({v^2+(u-y)^2})e^{-2v}}&=1+\bigg(-\frac{1}{2}\frac{1}{v}\bigg)+\frac{v}{v^2+(u-y)^2}\\ &\in\bigg(1-\frac{1}{2v},1+\frac{1}{v}\bigg)\subseteq\bigg(\frac{1}{2},2\bigg) \quad (v>1),\\ \frac{\partial}{\partial v}\bigg (-\frac{1}{2\unicode{x3bb}}\frac{v}{v^2+(u-y)^2}e^{-2\unicode{x3bb} v}\bigg)&=\bigg(\frac{v}{v^2+(u-y)^2}+\bigg(-\frac{1}{2\unicode{x3bb}}\bigg)\frac{1}{v^2+(u-y)^2}\\ & \quad +\bigg(-\frac{1}{2\unicode{x3bb}}\bigg)\frac{v(-2v)}{(v^2+(u-y)^2)^2}\bigg) e^{-2\unicode{x3bb} v}\\ \frac{{\partial}/{\partial v}(-({1}/{2\unicode{x3bb}})({v}/({v^2+(u-y)^2}))e^{-2\unicode{x3bb} v})}{{v}/({v^2+(u-y)^2})e^{-2\unicode{x3bb} v}}&=1+\bigg(-\frac{1}{2\unicode{x3bb}}\frac{1}{v}\bigg)+\frac{1}{\unicode{x3bb}}\frac{v}{v^2+(u-y)^2}\\ &\in\bigg(1-\frac{1}{2\unicode{x3bb} v},1+\frac{1}{\unicode{x3bb} v}\bigg)\subseteq\bigg(\frac{1}{2},2\bigg) \!\quad \!\bigg(v\geqslant\frac{1}{\unicode{x3bb}}\bigg). \end{aligned} \end{align*} $$

Corollary 5.2. For any r such that $0<r\leqslant e^{-{1}/{\unicode{x3bb} }}$ ,

$$ \begin{align*} &\frac{1}{r^2}\int_{v>-\log r}H_\alpha(u+iv)\|\psi_\alpha'\|^2 dv\approx H_\alpha(u+(-\log r) i) \quad(0<|\alpha|<r^{1-\unicode{x3bb}}),\\ &\frac{1}{r^2}\int_{v>({\log|\alpha|-\log r})/{\unicode{x3bb}}}H_\alpha(u+iv)\|\psi_\alpha'\|^2 dv\\ &\quad\approx H_\alpha\bigg(u+\bigg(\frac{\log|\alpha|-\log r}{\unicode{x3bb}}\bigg) i\bigg) \quad (r^{1-\unicode{x3bb}}\leqslant|\alpha|<1),\\ &\frac{1}{r^2}\int_{v>({\log|\alpha|-\log r})/{\unicode{x3bb}}}H_\alpha(u+iv)\|\psi_\alpha'\|^2 dv\approx H_\alpha\bigg(u+\bigg(\frac{-\log r}{\unicode{x3bb}}\bigg) i\bigg) \quad (|\alpha|\geqslant 1). \end{align*} $$

Figure 14 explains Corollary 5.2. We remark that Corollary 5.2 is true for $r\in (0,1)$ after a dilation $(z,w)\mapsto (e^{{1}/{2\unicode{x3bb} }}z,e^{{1}/{2\unicode{x3bb} }}w)$ .

Figure 14

${1}/{r^2}$ (The integration on $v>-\log r$ ) $\approx $ (The value at $v=-\log r$ ).

Proof. The assumption $0<r\leqslant e^{-{1}/{\unicode{x3bb} }}$ implies $-\log r\geqslant {1}/{\unicode{x3bb} }$ . Hence, for $v\geqslant -\log r\geqslant {1}/{\unicode{x3bb} }$ , Lemma 5.1 holds.

First, when $0<|\alpha |\leqslant r^{1-\unicode{x3bb} }$ ,

$$ \begin{align*} \begin{aligned} &\int_{v>-\log r}H_\alpha(u+iv)\|\psi_\alpha'\|^2dv\\ &\quad=\frac{1}{\pi}\int_{v>-\log r}\int_{y\in\mathbb{R}}H_\alpha(y)\frac{v}{v^2+(u-y)^2}2 (e^{-2v}+\unicode{x3bb}^2 |\alpha|^2 e^{-2\unicode{x3bb} v})\, dy\, dv\\ &\quad\approx \frac{1}{\pi}\int_{y\in\mathbb{R}}H_\alpha(y)\bigg\{\int_{v>-\log r}\frac{\partial}{\partial v}\bigg(\frac{v}{v^2+(u-y)^2}(-e^{-2v}-\unicode{x3bb} |\alpha|^2 e^{-2\unicode{x3bb} v})\bigg)\, dv\bigg\}\, dy\\ &\quad=\frac{1}{\pi}\int_{y\in\mathbb{R}}H_\alpha(y)\frac{-\log r}{(-\log r)^2+(u-y)^2}(r^2+\unicode{x3bb} |\alpha|^2 r^{2\unicode{x3bb}})\, dy\\ &\quad=H_\alpha(u+(-\log r) i)(r^2+\unicode{x3bb} |\alpha|^2 r^{2\unicode{x3bb}})\\ &\quad\approx r^2 H_\alpha(u+(-\log r) i). \end{aligned} \end{align*} $$

For the same reason, when $r^{1-\unicode{x3bb} }\leqslant |\alpha |<1$ , which implies $({\log |\alpha |-\log r})/{\unicode{x3bb} }\geqslant -\log r\geqslant {1}/{\unicode{x3bb} }$ ,

$$ \begin{align*} &\int_{v>({\log|\alpha|-\log r})/{\unicode{x3bb}}}H_\alpha(u+iv)\|\psi_\alpha'\|^2\, dv\\ &\quad\approx H_\alpha\bigg(u+\bigg(\frac{\log|\alpha|-\log r}{\unicode{x3bb}}\bigg) i\bigg) (|\alpha|^{-{2}/{\unicode{x3bb}}} r^{{2}/{\unicode{x3bb}}}+\unicode{x3bb} r^2)\\ &\quad\approx r^2 H_\alpha\bigg(u+\bigg(\frac{\log|\alpha|-\log r}{\unicode{x3bb}}\bigg) i\bigg). \end{align*} $$

Finally, when $|\alpha |\geqslant 1$ one has ${-\log r}/{\unicode{x3bb} }\geqslant -\log r\geqslant {1}/{\unicode{x3bb} }$ and

$$ \begin{align*} \int_{v>{-\log r}/{\unicode{x3bb}}}H_\alpha(u+iv)\|\psi_\alpha'\|^2dv &\approx H_\alpha\bigg(u+\bigg(\frac{-\log r}{\unicode{x3bb}}\bigg) i\bigg) (|\alpha|^{-{2}/{\unicode{x3bb}}} r^{{2}/{\unicode{x3bb}}}+\unicode{x3bb} r^2)\\ &\approx r^2 H_\alpha\bigg(u+\bigg(\frac{-\log r}{\unicode{x3bb}}\bigg) i\bigg).\\[-3.9pc] \end{align*} $$

Thus,

$$ \begin{align*} \begin{aligned} \mathscr{L}(T,0)&\approx \lim\limits_{r\rightarrow0+}\bigg\{ \int_{|\alpha|<r^{1-\unicode{x3bb}}}\int_{u=0}^{2\pi}H_\alpha(u+(-\log r) i) \,du \,d\mu(\alpha)\\ & \quad +\int_{r^{1-\unicode{x3bb}}\leqslant|\alpha|<1}\int_{u=0}^{2\pi}H_\alpha\bigg(u+\bigg(\frac{\log|\alpha|-\log r}{\unicode{x3bb}}\bigg) i\bigg) \,du \,d\mu(\alpha)\\ & \quad +\int_{|\alpha|\geqslant 1}\int_{u=0}^{2\pi}H_\alpha\bigg(u+\bigg(\frac{-\log r}{\unicode{x3bb}}\bigg) i\bigg) \,du \,d\mu(\alpha)\bigg\}, \end{aligned} \end{align*} $$

by inequalities (6) and (7) in the previous subsection. All terms are positive, so the order of taking the limit and integration can change:

$$ \begin{align*} \begin{aligned} \mathscr{L}(T,0)&\approx\lim\limits_{v\rightarrow+\infty}\int_{\alpha\in\mathbb{C}^*}\int_{u=0}^{2\pi}H_\alpha(u+iv) \,du \,d\mu(\alpha)\\ &=\lim\limits_{k\rightarrow+\infty}\int_{\alpha\in\mathbb{C}^*}\int_{u=0}^{2\pi}\int_{y\in\mathbb{R}}H_\alpha(y)\frac{2k\pi}{(2k\pi)^2+(u-y)^2} \,dy\, du\, d\mu(\alpha). \end{aligned} \end{align*} $$

Fix some $k\in \mathbb {Z}$ , $k\geqslant 2$ . Define intervals $I_N$ for all $N\in \mathbb {Z}$ as follows:

$$ \begin{align*} I_0&=[-2k\pi+2\pi,2k\pi),\\ I_N&= \left\{ \begin{aligned} &[2k N\pi,2k(N+1)\pi) & (N>0),\\ &[2k(N-1)\pi +2\pi,2k N\pi+2\pi) & (N<0). \end{aligned} \right. \end{align*} $$

Thus, $\mathbb {R}=\bigcup \nolimits _{N\in \mathbb {Z}}I_N$ is a disjoint union.

Lemma 5.3. For any $u\in (0,2\pi )$ , one has

$$ \begin{align*} \frac{2k\pi}{(2k\pi)^2+(u-y)^2}\geqslant \frac{1}{1+(N+1)^2}\frac{1}{2k\pi} \quad(y\in I_N). \end{align*} $$

Thus,

$$ \begin{align*} \begin{aligned} \mathscr{L}(T,0)&\approx\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^*}\int_{u=0}^{2\pi}\int_{y\in I_N}H_\alpha(y)\frac{2k\pi}{(2k\pi)^2+(u-y)^2} \,dy\, du\, d\mu(\alpha)\\ &\geqslant\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^*}\int_{y\in I_N}\int_{u=0}^{2\pi}H_\alpha(y)\frac{1}{1+(N+1)^2}\frac{1}{2k\pi} \,du\, dy\, d\mu(\alpha)\\ &=\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^*}\int_{y\in I_N}H_\alpha(y)\frac{1}{1+(N+1)^2}\frac{1}{k} \,dy \,d\mu(\alpha). \end{aligned} \end{align*} $$

By Lemma 3.3 and Corollary 5.2 after a dilation,

$$ \begin{align*} \|T\|_{\mathbb{D}^2}&=\int_{\alpha\in\mathbb{C}^*}\int_{v>0}\int_{u=2k_0\pi}^{2k_0\pi+2\pi}H_\alpha(u+iv) \|\psi_\alpha'\|^2 \,du\wedge dv \,d\mu(\alpha)\quad{(k_0\in\mathbb{Z})}\\ &\approx\int_{\alpha\in\mathbb{C}^*}\int_{\alpha\in\mathbb{C}^*}\int_{y=2k_0\pi}^{2k_0\pi+2\pi}H_\alpha(y) \,dy\, d\mu(\alpha) \end{align*} $$

is the integral of y on any interval of length $2\pi $ . Since $I_0$ has length $(2k-1) 2\pi $ and $I_N$ has length $2k\pi $ for $N\neq 0$ ,

$$ \begin{align*} \int_{\alpha\in\mathbb{C}^*}\int_{y\in I_0}H_\alpha(y) \,dy\, d\mu(\alpha)&\approx(2k-1) \|T\|_{\mathbb{D}^2}\\ &\geqslant k \|T\|_{\mathbb{D}^2},\\ \int_{\alpha\in\mathbb{C}^*}\int_{y\in I_N}H_\alpha(y) \,dy\, d\mu(\alpha)&\approx k \|T\|_{\mathbb{D}^2} \quad(N\neq 0). \end{align*} $$

Thus,

$$ \begin{align*} \mathscr{L}(T,0)\gtrsim\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\frac{1}{1+(N+1)^2} \|T\|_{\mathbb{D}^2}\approx \|T\|_{\mathbb{D}^2} \end{align*} $$

is non-zero.