1 Introduction
The dynamical properties of singular holomorphic foliations have recently drawn a great deal of attention; see the discussions in [Reference Dinh and Sibony9, Reference Fornæss and Sibony11, Reference Fornæss, Sibony and Wold13, Reference Nguyên15, Reference Nguyên17, Reference Nguyên18]. Let us mention one of the remarkable results which establishes the unique ergodicity for general singular holomorphic foliations on compact Kähler surfaces.
Theorem 1.1. (Dinh, Nguyên and Sibony [Reference Dinh, Nguyên and Sibony7])
Let $\mathscr {F}$ be a holomorphic foliation with only hyperbolic singularities in a compact Kähler surface $(X,\omega )$ . Assume that $\mathscr {F}$ admits no directed positive closed current. Then there exists a unique positive $dd^c$ -closed current T of mass $1$ directed by $\mathscr {F}$ .
The first version was stated for $X=\mathbb {P}^2$ and proved by Fornæss and Sibony [Reference Fornæss and Sibony12]. Later Dinh and Sibony proved the unique ergodicity for foliations in $\mathbb {P}^2$ with an invariant curve [Reference Dinh and Sibony8]. So one may expect to describe recurrence properties of leaves by studying the density distribution of directed harmonic currents. One has the following result about leaves.
Theorem 1.2. (Fornæss and Sibony [Reference Fornæss and Sibony12])
Let $(X,\mathscr {F},E)$ be a holomorphic foliation on a compact complex surface X with singular set E. Assume that:
-
(1) there is no invariant analytic curve;
-
(2) all the singularities are hyperbolic;
-
(3) there is no non-constant holomorphic map $\mathbb {C}\rightarrow X$ such that out of E the image of $\mathbb {C}$ is locally contained in a leaf.
Then every harmonic current T directed by $\mathscr {F}$ gives no mass to each single leaf.
A practical way to measure the density of harmonic currents is to use the notion of Lelong number introduced by Skoda [Reference Skoda22]. Indeed Theorem 1.2 above is equivalent to the statement that the Lelong number of T vanishes everywhere outside E. Another result holds near hyperbolic singularities.
Theorem 1.3. (Nguyên [Reference Nguyên16])
Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field $Z(z,w)=z ({\partial }/{\partial z})+\unicode{x3bb} w ({\partial }/{\partial w}),$ where $\unicode{x3bb} \in \mathbb {C}\backslash \mathbb {R}$ , that is to say, $0$ is a hyperbolic singularity. Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Then the Lelong number of T at $0$ vanishes.
Next, Nguyên applies this result to prove the existence of Lyapunov exponents for singular holomorphic foliations on compact projective surfaces [Reference Nguyên20]. Very recently he has proved in [Reference Nguyên19] that for every $n\geqslant 2,$ the Lelong numbers of any directed harmonic current which gives no mass to invariant hyperplanes vanishes near weakly hyperbolic singularities in $\mathbb {C}^n.$ This result is optimal; see [Reference Dinh and Wu10]. The mass-distribution problem would be completed once we could understand the behaviour of harmonic currents near non-hyperbolic non-degenerate singularities, and near degenerate singularities.
The present paper answers (partly) the problem in the non-hyperbolic linearizable singularity case. Here is our first main result.
Theorem 1.4. Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field $Z(z,w)=z ({\partial }/{\partial z})+\unicode{x3bb} w ({\partial }/{\partial w})$ , where $\unicode{x3bb} \in \mathbb {R}^*$ . Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Assume $T\neq 0$ . Then the Lelong number of T at $0$ :
-
• is strictly positive and could be infinite if $\unicode{x3bb}>0$ ;
-
• vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ .
For the foliation concerned $(\mathbb {D}^2,\mathscr {F},\{0\})$ , a local leaf $P_\alpha $ , with $\alpha \in \mathbb {C}^*$ , can be parametrized by $(z,w)=(e^{-v+iu},\alpha e^{-\unicode{x3bb} v+i\unicode{x3bb} u})$ , with $u,v\in \mathbb {R}$ . See the parametrization (1) for details. The monodromy group around the singularity is generated by $(z,w)\mapsto (z,e^{2\pi i\unicode{x3bb} }w)$ . It is a cyclic group of finite order when $\unicode{x3bb} \in \mathbb {Q}^*$ , of infinite order when $\unicode{x3bb} \notin \mathbb {Q}$ .
We are now ready to introduce the notion of periodic current, an essential tool in this paper. A directed harmonic current T is called periodic if it is invariant under some cofinite subgroup of the monodromy group, that is, under the action of $(z,w)\mapsto (z,e^{2k\pi i \unicode{x3bb} }w)$ for some $k\in \mathbb {Z}_{>0}$ .
Observe that if $\unicode{x3bb} =({a}/{b})\in \mathbb {Q}^*$ with $a\in \mathbb {Z}^*$ , $b\in \mathbb {Z}_{>0}$ , then any directed harmonic current is invariant under the action of $(z,w)\mapsto (z,e^{2b\pi i \unicode{x3bb} }w)$ , hence is periodic. But when $\unicode{x3bb} \notin \mathbb {Q}^*$ , the periodicity is a non-trivial assumption. It does not follow from the ergodicity of irrational rotation because the current is only continuous on leaf parameters $(u,v)$ for each fixed $\alpha $ . It may not be continuous in variables $(z,w)$ .
We are in a position to state our second main result.
Theorem 1.5. Using the same notation as above, the Lelong number of T at the singularity is $0$ when $\unicode{x3bb} <0$ and the current is periodic, in particular, when $\unicode{x3bb} \in \mathbb {Q}_{<0}$ .
It remains open to determine the possible Lelong number values of non-periodic T when $\unicode{x3bb} <0$ is irrational.
Section 2 reviews the definition of singular holomorphic foliations, directed harmonic currents, the mass and the Lelong number. Section 3 describes the topology of leaves near linearizable non-hyperbolic singularities, resolves the ambiguity of normalizing harmonic functions on the leaves and provides practical formulas for the mass and the Lelong number. Section 4 calculates the Lelong number when $\unicode{x3bb} \in \mathbb {Q}_{>0}$ . Section 5 calculates the Lelong number when $\unicode{x3bb} \in \mathbb {R}_{>0}\backslash \mathbb {Q}$ , with an analysis on Poisson integrals of non-periodic currents. Section 6 calculates the Lelong number when $\unicode{x3bb} <0$ , assuming that the currents are periodic.
2 Background
2.1 Singularities of holomorphic foliations
To start with, recall the definition of singular holomorphic foliation on a complex surface M.
Definition 2.1. Let $E\subset M$ be some closed subset, possibly empty, such that $\overline {M\backslash E}=M$ . A singular holomorphic foliation $(M,E,\mathscr {F}{\kern1.5pt})$ consists of a holomorphic atlas $\{(\mathbb {U}_i,\Phi _i)\}_{i\in I}$ on $M\backslash E$ which satisfies the following conditions.
-
(1) For each $i\in I$ , $\Phi _i: \mathbb {U}_i\rightarrow \mathbb {B}_i\times \mathbb {T}_i$ is a biholomorphism, where $\mathbb {B}_i$ and $\mathbb {T}_i$ are domains in $\mathbb {C}$ .
-
(2) For each pair $(\mathbb {U}_i,\Phi _i)$ and $(\mathbb {U}_j,\Phi _j)$ with $\mathbb {U}_i\cap \mathbb {U}_j\neq \emptyset $ , the transition map
$$ \begin{align*} \Phi_{ij}:=\Phi_i\circ\Phi_j^{-1}:\Phi_j(\mathbb{U}_i\cap\mathbb{U}_j)\rightarrow\Phi_i(\mathbb{U}_i\cap\mathbb{U}_j) \end{align*} $$has the form$$ \begin{align*} \Phi_{ij}(b,t)=(\Omega(b,t),\Lambda(t)), \end{align*} $$where $(b,t)$ are the coordinates on $\mathbb {B}_j\times \mathbb {T}_j$ , and the functions $\Omega $ , $\Lambda $ are holomorphic, with $\Lambda $ independent of b.
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].
Definition 2.2. Let $Z=P(z,w){\partial }/{\partial z}+Q(z,w){\partial }/{\partial w}$ be a holomorphic vector field defined in a neighbourhood $\mathbb {U}$ of $(0,0)\in \mathbb {C}^2$ . One says that Z is:
-
(1) singular at $(0,0)$ if $P(0,0)=Q(0,0)=0$ ;
-
(2) linear if it can be written as
$$ \begin{align*} Z=\unicode{x3bb}_1 z\frac{\partial}{\partial z}+\unicode{x3bb}_2 w \frac{\partial}{\partial w} \end{align*} $$where $\unicode{x3bb} _1$ , $\unicode{x3bb} _2\in \mathbb {C}$ are not simultaneously zero; -
(3) linearizable if it is linear after a biholomorphic change of coordinates.
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.
Definition 2.3. The singularity is non-degenerate if both $\unicode{x3bb} _1$ , $\unicode{x3bb} _2$ are non-zero. This condition is biholomorphically invariant.
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.
Definition 2.4. A singularity of Z is hyperbolic if the quotient $\unicode{x3bb} :=({\unicode{x3bb} _1}/{\unicode{x3bb} _2})\in \mathbb {C}\backslash \mathbb {R}$ . It is non-hyperbolic if $\unicode{x3bb} \in \mathbb {R}^*$ . It is in the Poincaré domain if $\unicode{x3bb} \in \mathbb {C}\backslash \mathbb {R}_{\leqslant 0}$ . It is in the Siegel domain if $\unicode{x3bb} \in \mathbb {R}_{<0}$ .
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 $\mathbb {C}^2$ .
Theorem 2.5. (Poincaré; see [Reference Arnold and Ilyashenko2, Ch. 4, §1.2, pp. 72])
A singular holomorphic vector field in $\mathbb {C}^2$ is holomorphically equivalent to its linear part if its eigenvalue $\unicode{x3bb} \in (\mathbb {C}\backslash \mathbb {R}_{\leqslant 0})\backslash \mathbb {Q}_{\neq 1}$ .
Remark 2.6. The linear part of a singular holomorphic vector field is
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
where $p_n/q_n$ is the nth approximant of the continued fraction expansion of $\unicode{x3bb} $ .
The golden ratio
is a Brjuno number. Indeed, any irrational number whose continued fraction expansion ends with a string of 1s
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
where $\zeta =u+iv\in \mathbb {C}$ . The map
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) $i\partial \bar {\partial } T=0$ 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) 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
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
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.
Proposition 2.8. A directed harmonic current T is periodic if and only if there exists some $k\in \mathbb {Z}_{>0}$ such that $H_\alpha (u+2 k\pi ,v)=H_\alpha (u,v)$ for all $u,v$ and for $\mu $ -almost all $\alpha $ .
Proof. By definition T is invariant under $(z,w)\mapsto (z,e^{2k\pi i \unicode{x3bb} }w)$ for some $k\in \mathbb {Z}_{>0}$ , which is equivalent to $H_\alpha (u+2 k\pi ,v)=H_\alpha (u,v)$ for all $u,v$ and $\mu $ -almost all $\alpha $ .
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.
Theorem 2.9. (Alessandrini and Bassanelli [Reference Alessandrini and Bassanelli1, Theorem 5.6])
Let $\Omega $ be an open subset of $\mathbb {C}^n$ and Y an analytic subset of $\Omega $ of dimension less than p. Suppose T is a negative current of bidimension $(p,p)$ on $\Omega \backslash Y$ such that $dd^c T\geqslant 0$ . Then the following assertions hold.
-
(1) The mass of T near Y is locally finite. In particular, T admits a trivial extension by $0$ across Y, denoted by $\tilde {T}$ .
-
(2) $dd^c\tilde {T}\geqslant 0$ on $\Omega $ .
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
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
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.
Theorem 2.11. (Skoda [Reference Skoda22])
Let T be a positive $dd^c$ -closed $(1,1)$ -current in $\mathbb {D}^2$ . Then the function $r\mapsto {1}/{\pi r^2}\|T\|_{r\mathbb {D}^2}$ is increasing with $r\in (0, 1]$ .
In our case, the function
is increasing with $r\in (0, 1]$ . In particular,
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 Parametrization of leaves
Recall the parametrization of an arbitrary leaf $L_\alpha $ :
To calculate the mass $\|T\|_{\mathbb {D}^2}$ and the Lelong number $\mathscr {L}(T,0)$ , we shall study $\Psi ^{-1}(r \mathbb {D}^2)$ for $r\in (0,1]$ . Define $P_\alpha :=L_\alpha \cap \mathbb {D}^2$ and $P_\alpha ^{(r)}:=L_\alpha \cap r \mathbb {D}^2$ . Define $\log ^+(x):=\max \{0,\log (x)\}$ for $x>0$ .
Lemma 3.1. The range of $(u,v)$ for a point $(z,w)\in P_\alpha $ and $P_{\alpha }^{(r)}$ is an upper half-plane when $\unicode{x3bb}>0$ , or a horizontal strip when $\unicode{x3bb} <0$ . More precisely:
-
(1) when $\unicode{x3bb}>0$ ,
$$ \begin{align*} (z,w)\in P_\alpha & \Longleftrightarrow\, v>\frac{\log^+|\alpha|}{\unicode{x3bb}},\\ (z,w)\in P_\alpha^{(r)} & \Longleftrightarrow \left\{ \begin{aligned} &v>\frac{\log|\alpha|-\log r}{\unicode{x3bb}} & (|\alpha|\geqslant r^{1-\unicode{x3bb}}),\\ &v>-\log r & (|\alpha|<r^{1-\unicode{x3bb}}); \end{aligned} \right. \end{align*} $$ -
(2) when $\unicode{x3bb} <0$ , $P_\alpha =\emptyset $ for $|\alpha |\geqslant 1$ , $P_\alpha ^{(r)}=\emptyset $ for $|\alpha |\geqslant r^{1-\unicode{x3bb} }$ and for the other $\alpha $ ,
$$ \begin{align*} (z,w)\in P_\alpha & \Longleftrightarrow\, 0<v<\frac{\log|\alpha|}{\unicode{x3bb}},\\ (z,w)\in P_\alpha^{(r)} & \Longleftrightarrow -\log r<v<\frac{\log|\alpha|-\log r}{\unicode{x3bb}}. \end{align*} $$
Proof. Recall that $(z,w)=(e^{-v+i u},\alpha e^{-\unicode{x3bb} v+i \unicode{x3bb} u})$ on $L_{\alpha }$ . So for any $r\in (0,1]$ , $(z,w)\in P_\alpha ^{(r)}$ if and only if both $|z|=e^{-v}<r$ and $|w|=|\alpha | e^{-\unicode{x3bb} v}<r$ .
When $\unicode{x3bb}>0$ one has $v>-\log r$ and $v>({\log |\alpha |-\log r})/{\unicode{x3bb} }$ . In particular, for $r=1$ , one has $v>0$ and $v>{\log |\alpha |}/{\unicode{x3bb} }$ .
When $\unicode{x3bb} <0$ one has $-\log r<v<({\log |\alpha |-\log r})/{\unicode{x3bb} }$ . In particular, for $r=1$ , one has $0<v<{\log |\alpha |}/{\unicode{x3bb} }$ . If there is no solution for v then $P_{\alpha }^{(r)}=\emptyset $ .
When $\unicode{x3bb}>0$ , the range of v is unbounded for each fixed $\alpha \in \mathbb {C}^*$ . See Figures 1 and 2.
When $\unicode{x3bb} <0$ , the range of v is bounded for each fixed $\alpha $ . See Figures 3 and 4.
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.
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:
For example, in Figure 10, the curve and the dashed curve are two distinct 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
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
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
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
In other words, they differ by a translation and a multiplication by a non-zero constant.
Proof. When $|\alpha |<1$ , by definition
Thus, the normalized harmonic function is
and for the same reason
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 $C>0$ :
Thus,
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.
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.
Normalizing $H_{\alpha e^{2k\pi i\unicode{x3bb} }}$ on $\tilde {L}_{\alpha e^{2k\pi i\unicode{x3bb} }}$ avoids ambiguity. Thus, the mass
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:
Since H is harmonic in a neighbourhood of $\mathbb {H}$ , it is continuous in $\mathbb {H}$ . So
Thus, we can express the mass by a formula independent of the choice of normalization
Lemma 3.3. For each $k_0\in \mathbb {Z}$ fixed,
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,
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.
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,
These formulas will be calculated in later sections.
4 Positive rational case: $\unicode{x3bb} =({a}/{b})\in \mathbb {Q}$ , $\unicode{x3bb} \in (0,1]$
Write $\unicode{x3bb} ={a}/{b}$ where $a,b\in \mathbb {Z}_{\geqslant 1}$ are coprime. Then in $\mathbb {D}^2$ , for any $\alpha \in \mathbb {C}^*$ , the union $L_\alpha \cup \{0\}$ is the algebraic curve $\{w^b=\alpha ^b z^a\}\cap \mathbb {D}^2$ . In other words, every leaf is a separatrix. In this section it will be shown that any directed harmonic current T has non-zero Lelong number.
The parametrization map $\psi _\alpha (\zeta ):=(e^{i\zeta },\alpha e^{i\unicode{x3bb} \zeta })$ is now periodic: $\psi _\alpha (\zeta +2\pi b)=\psi _\alpha (\zeta )$ . Let T be a directed harmonic current. Then $T|_{P_\alpha }$ has the form $h_\alpha (z,w)[P_\alpha ]$ . Let
This is a positive harmonic function for $\mu $ -almost all $\alpha \in \mathbb {C}^*$ defined in a neighbourhood of the upper half-plane $\mathbb {H}:=\{(u+iv)\in \mathbb {C}~|~v>0\}$ . Moreover, it is periodic: $H_\alpha (u+iv)=H_\alpha (u+2\pi b+iv)$ . Periodic harmonic functions can be characterized by the following lemma.
Lemma 4.1. Let $F(u,v)$ be a harmonic function in a neighbourhood of $\mathbb {H}$ . If $F(u,v)=F(u+2\pi b,v)$ for all $(u,v)\in \mathbb {H}$ , then
for some $a_k$ , $b_k\in \mathbb {R}$ . Moreover, if $F|_{\mathbb {H}}\geqslant 0$ , then $a_0,b_0\geqslant 0$ .
Proof. By periodicity
for some functions $A_k(v)$ , $B_k(v)$ . They are smooth since F is harmonic. Moreover,
Thus,
Hence,
for some $a_k$ , $a_{-k}$ , $b_k$ , $b_{-k}\in \mathbb {R}$ . One obtains the equality.
If $F|_{\mathbb {H}}\geqslant 0$ , then for any $v\geqslant 0$ ,
Thus, $a_0,b_0\geqslant 0.$
For $\alpha ,\beta \in \mathbb {C}^*$ , the two maps $\psi _\alpha $ and $\psi _\beta $ parametrize the same leaf $L_\alpha =L_\beta $ if and only if $\beta =\alpha e^{2\pi i ({k}/{b})}$ for some $k\in \mathbb {Z}$ , that is $\alpha $ and $\beta $ differ from multiplying a bth root of unity. Thus, a transversal can be chosen as the sector $\mathbb {S}:=\{\alpha \in \mathbb {C}^*~|~{\textrm {arg}}(\alpha )\in [0,{2\pi }/{b})\}$ . One fixes a normalization by setting $H_\alpha (0)=h_\alpha \circ \psi _\alpha (i({\log ^+|\alpha |}/{\unicode{x3bb} }))=1$ .
The mass of the current T is
In particular, one calculates the $(1,1)$ -form $i\partial \bar {\partial }(|z|^2+|w|^2)$ on $L_\alpha $ , where $z=e^{-v+iu},w=\alpha e^{-\unicode{x3bb} v+i\unicode{x3bb} u}$ , using
whence
Thus,
By Lemma 4.1,
where $a_0(\alpha )$ , $b_0(\alpha )$ are positive for $\mu $ -almost all $\alpha $ . Thus,
The Lelong number can now be calculated as follows:
First one analyses the $a_0(\alpha )$ part. When $|\alpha |<r^{1-\unicode{x3bb} }$ ,
is uniformly bounded with respect to $\alpha $ and r. When $|\alpha |\geqslant r^{1-\unicode{x3bb} }$
is also uniformly bounded with respect to $\alpha $ and r. Thus,
Next one analyses the $b_0(\alpha )$ part.
Lemma 4.2. The Lelong number of T at $0$ is finite only if $b_0(\alpha )=0$ for $\mu $ -almost all $\alpha \in \mathbb {S}$ .
Proof. Suppose not, that is, $\int _{\alpha \in \mathbb {S}}b_0(\alpha )\, d\mu (\alpha )=B_0>0$ . Then
contradicting the finiteness of the Lelong number stated in Theorem 2.11.
Thus, one may assume $b_0(\alpha )=0$ for $\mu $ -almost all $\alpha \in \mathbb {S}$ . Then the Lelong number
is strictly positive.
5 Positive irrational case $\unicode{x3bb} \notin \mathbb {Q}$ , $\unicode{x3bb} \in (0,1)$
Now $\{z=0\}$ and $\{w=0\}$ are the only two separatrices in $\mathbb {D}^2$ . For each fixed $\alpha \in \mathbb {C}^*$ , the map $\psi _\alpha (\zeta )=(e^{i \zeta },\alpha e^{i \unicode{x3bb} \zeta })$ is injective since $\unicode{x3bb} \notin \mathbb {Q}$ .
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
for any $k_0\in \mathbb {Z}$ , in particular for $k_0=0,1,\ldots ,b-1$ . Thus, we may calculate
which is the same expression as in the case $\unicode{x3bb} \in \mathbb {Q}_{>0}$ .
Next, the Lelong number is calculated as
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
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
Recall the Poisson integral formula after multiplying by a non-zero constant:
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 $u\in \mathbb {R}$ ,
Proof. This can be calculated directly:
Corollary 5.2. For any r such that $0<r\leqslant e^{-{1}/{\unicode{x3bb} }}$ ,
Figure 14 explains Corollary 5.2. We remark that Corollary 5.2 is true for $r\in (0,1)$ after a dilation $(z,w)\mapsto (e^{{1}/{2\unicode{x3bb} }}z,e^{{1}/{2\unicode{x3bb} }}w)$ .
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 $0<|\alpha |\leqslant r^{1-\unicode{x3bb} }$ ,
For the same reason, when $r^{1-\unicode{x3bb} }\leqslant |\alpha |<1$ , which implies $({\log |\alpha |-\log r})/{\unicode{x3bb} }\geqslant -\log r\geqslant {1}/{\unicode{x3bb} }$ ,
Finally, when $|\alpha |\geqslant 1$ one has ${-\log r}/{\unicode{x3bb} }\geqslant -\log r\geqslant {1}/{\unicode{x3bb} }$ and
Thus,
by inequalities (6) and (7) in the previous subsection. All terms are positive, so the order of taking the limit and integration can change:
Fix some $k\in \mathbb {Z}$ , $k\geqslant 2$ . Define intervals $I_N$ for all $N\in \mathbb {Z}$ as follows:
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
Proof. Elementary.
Thus,
By Lemma 3.3 and Corollary 5.2 after a dilation,
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 $N\neq 0$ ,
Thus,
is non-zero.
6 Periodic currents in the negative case $\unicode{x3bb} <0$
Now we treat the case $\unicode{x3bb} <0$ . We assume the currents are periodic. Recall that when $\unicode{x3bb} \in \mathbb {Q}$ all directed currents are periodic. So such currents include all currents for $\unicode{x3bb} \in \mathbb {Q}_{<0}$ .
Recall the formulas of the mass and of the Lelong number obtained in §3.3, for each $k_0\in \mathbb {Z}$ fixed:
We now prove Theorem 1.5. Suppose that there exists some $b\in \mathbb {Z}_{\leqslant 1}$ such that $H_{\alpha }(u+iv)=H_{\alpha }(u+2\pi b+iv)$ for all $\alpha \in \mathbb {D}^*$ and all $(u,v)$ in a neighbourhood of the strip $\{(u+iv)\in \mathbb {C}~|~u\in \mathbb {R},v\in [0,{\log |\alpha |}/{\unicode{x3bb} }]\}$ . One proves the following result.
Lemma 6.1. Let $F(u,v)$ be a positive harmonic function on a neighbourhood of the horizontal strip $\{(u+iv)\in \mathbb {C}~|~u\in \mathbb {R},v\in [0,C]\}$ for some $C>0$ . Suppose $F(u,v)=F(u+2\pi b,v)$ on this strip. Then
for some $a_k,b_k\in \mathbb {R}$ with $a_0\geqslant 0$ and $b_0\geqslant 0$ .
Proof. The proof is almost the same as that of Lemma 4.1. Using Fourier series and calculating the Laplacian, one concludes that
for some $a_k,b_k,p,q\in \mathbb {R}$ . For any $v\in [0,C]$ , $F(u,v)\geqslant 0$ implies
Thus, $p\geqslant 0$ and $q\geqslant -C^{-1} p$ . One may write $p+q v=p (1-C^{-1} v)+(q+C^{-1} p) v$ with $p=:a_0\geqslant 0$ and $q+C^{-1} p=:b_0\geqslant 0$ .
For periodic currents one may assume
for some $a_k(\alpha ),b_k(\alpha )\in \mathbb {R}$ with $a_0(\alpha )\geqslant 0$ and $b_0(\alpha )\geqslant 0$ . According to Lemma 3.3, for any $k_0\in \mathbb {Z}$ , use the Jacobian (3):
Next, using $0=\int _0^{2\pi b}\cos ({ku}/{b})du$ for $k\neq 0$ and the same for $\sin ({ku}/{b})$ , let us calculate the average among $k_0=0,1,\ldots ,b-1$ for the mass
and for the Lelong number
We introduce the two functions of $r\in (0,1]$ given by elementary integrals,
to describe the contributions from the $a_0(\alpha )$ part and from the $b_0(\alpha )$ part. Here we recall that every positive linear function of v on is a sum of and $b_0(\alpha)\,v$ with $a_0(\alpha),b_0(\alpha)\geqslant 0$ . The two summands correspond to the dotted line and the dashed line in Figure 15.
Then we can express
Observe that
Fix any $\alpha \in \mathbb {D}^*$ ; by definition $r^2I_a(r)$ and $r^2I_b(r)$ are increasing for $r\in (0,1]$ , since the interval of integration $(-\log r,({\log |\alpha |-\log r})/{\unicode{x3bb} })$ is expanding and the function integrated is positive. In particular, for any $r\in (0,1]$ ,
It is more subtle to talk about monotonicity of $I_a(r)$ and $I_b(r)$ . We expect upper bounds of $I_a(r)/I_a(1)$ and $I_b(r)/I_b(1)$ for $r\in (0,1]$ which are independent of $\alpha $ , that is, depend only on $\unicode{x3bb} $ .
Lemma 6.2. For any $r\in (0,1)$ and any $\alpha \in \mathbb {C}$ with $0<|\alpha |<r^{1-\unicode{x3bb} }<1$ , one has
Proof. Differentiation gives
It suffices to show that $({d}/{dr})I_a(r)>0$ when $r\in (0,1)$ and $0<|\alpha |<r^{1-\unicode{x3bb} }$ .
Introduce the new variable $t:={|\alpha |}/{r^{1-\unicode{x3bb} }}\in (0,1)$ . In the big parentheses, replace $|\alpha |$ by $t r^{1-\unicode{x3bb} }$ and $\log |\alpha |$ by $\log (t)+(1-\unicode{x3bb} )\log (r)$ :
since $\unicode{x3bb} \in [-1,0)$ implies $t^{2+{2}/{\unicode{x3bb} }}\geqslant 1$ .
It is not true that $I_b(r)$ is increasing on $(0,1]$ , but on a smaller half-neighbourhood of $0$ , independent of $\alpha $ , it is increasing. This suffices to give an upper bound for $I_b(r)/I_b(1)$ .
Lemma 6.3. For any $r\in (0,e^{{1}/{2 \unicode{x3bb} (1-\unicode{x3bb} )}})$ and any $\alpha \in \mathbb {C}$ with $0<|\alpha |<r^{1-\unicode{x3bb} }<1$ , one has
Proof. Differentiation gives
It suffices to show that ${d}/{dr}I_b(r)>0$ when $0<r<e^{{1}/{2 \unicode{x3bb} (1-\unicode{x3bb} )}}$ and $0<|\alpha |<r^{1-\unicode{x3bb} }$ .
Again, introduce the variable $t:={|\alpha |}/{r^{1-\unicode{x3bb} }}\in (0,1)$ and replace $\alpha $ and $\log |\alpha |$ in the parentheses:
End of proof of Theorem 1.5.
From the foregoing, the Lelong number is zero:
since $\|T\|_{\mathbb {D}^2}=2 \pi \int _{0<|\alpha |<1}(a_0(\alpha ) I_a(1)+b_0(\alpha ) I_b(1))\, d\mu (\alpha )$ is finite.
Acknowledgements
The author thanks Joël Merker and an anonymous referee for valuable suggestions which help to improve the presentation.