A remark on a weighted version of Suita conjecture for higher derivatives

Abstract

In this article, we consider the set of points for the holding of the equality in a weighted version of Suita conjecture for higher derivatives, and give relations between the set and the integer valued points of a class of harmonic functions (maybe multi-valued). For planar domains bounded by finite analytic closed curves, we give relations between the set and Dirichlet problem.

Appendix

Let $\Omega$ be an open Riemann surface, which admits a nontrivial Green function $G_{\Omega }$. It is known that $G_{\Omega }(z,z')$ is a harmonic function with respect to $z\in \Omega \backslash \{z'\}$ when $z'\in \Omega$ is fixed and is a harmonic function with respect to $z'\in \Omega \backslash \{z\}$ when $z\in \Omega$ is fixed. From this, we can conclude that:

Lemma 5.1

$G_{\Omega }(z,z')$ is a smooth function on $(\Omega \times \Omega )\backslash Diag_{\Omega }$, where $Diag_{\Omega }=\{(z,z)|z\in \Omega \}$.

Proof

For an arbitrary point $(z_0,z_0')\in (\Omega \times \Omega )\backslash Diag_{\Omega }$, choose local coordinate neighborhoods $V_{z_0}$ of $z_0$ and $V_{z_0'}$ of $z_0'$ such that $\overline{V}_{z_0}\cap \overline{V}_{z_0'}=\emptyset$. After identifying $V_{z_0}$ and $V_{z_0'}$ with their images under local coordinate maps, we may assume that $V_{z_0}$ and $V_{z_0'}$ are disjoint open subsets of $\mathbb {C}$.

Let $\xi (z)=\exp \left( \frac{1}{|z|^2-1}\right)$ for $|z|<1$ and $\xi (z)=0$ for $|z|\ge 1$, then $\xi (z)$ is a smooth function depending only on |z| and $supp\xi =\{z\in C:|z|\le 1\}$. Let $\mu (z):=\xi (z)/\int _{\mathbb {C}}\xi (w)dV(w)$, where dV denotes the Lebesgue measure. Denote that

$$\begin{aligned} \mu _{\epsilon }(z):=\frac{1}{\epsilon ^2}\mu \left( \frac{z}{\epsilon }\right) , \end{aligned}$$

where $0<\epsilon <1$, then $\mu _{\epsilon }(z)$ is a smooth function depending only on |z|, $supp \mu _{\epsilon }=\{z\in \mathbb {C}:|z|\le \epsilon \}$ and $\int _{\mathbb {C}}\mu _{\epsilon }(z)dV(z)=1$.

Let $z_1\in V_{z_0}$ be a fixed point, then $G_{\Omega }(z_1,z')$ is a continuous function with respect to $z'\in \overline{V}_{z_0'}$, hence $G_{\Omega }(z,z')$ is bounded on $\{z_1\}\times \overline{V}_{z_0'}$. For a fixed point $z_1'\in \overline{V}_{z_0'}$, $G_{\Omega }(z,z_1')$ is a negative harmonic function on a neighborhood of $\overline{V}_{z_0}$, $G_{\Omega }(z,z')$ is therefore bounded on $\overline{V}_{z_0}\times \overline{V}_{z_0'}$ by Lemma 2.9. Take $\epsilon <\frac{1}{2}\min \{dist(z_0,\partial {V_{z_0}}),dist(z_0',\partial {V_{z_0'}})\}$, then we have

$$\begin{aligned} G_{\Omega }(z,z')&=\int _{\mathbb {C}}G_{\Omega }(z-w,z')\mu _{\epsilon }(w)dV(w)\\&=\int _{\mathbb {C}}\int _{\mathbb {C}}G_{\Omega }(z-w,z'-w')\mu _{\epsilon }(w)\mu _{\epsilon }(w')dV(w)dV(w')\\&=\int _{\mathbb {C}}\int _{\mathbb {C}}G_{\Omega }(w,w')\mu _{\epsilon }(z-w)\mu _{\epsilon }(z'-w')dV(w)dV(w') \end{aligned}$$

for $(z,z')\in B(z_0,\epsilon )\times B(z_0',\epsilon )$, which implies that $G_{\Omega }(z,z')$ is smooth on a neighborhood of $(z_0,z_0')$. Since $(z_0,z_0')\in (\Omega \times \Omega )\backslash Diag_{\Omega }$ is arbitrarily chosen, $G_{\Omega }(z,z')$ is smooth on $(\Omega \times \Omega )\backslash Diag_{\Omega }$. $\square$

Let w be a local coordinate on $U\subset \Omega$. The following lemma shows the smoothness of $G_{\Omega }(z,z')-\log |w(z)-w(z')|$ on $U\times U$.

Lemma 5.2

$G_{\Omega }(z,z')-\log |w(z)-w(z')|$ can be extended to be a smooth function on $U\times U$.

Proof

Without loss of generality, we can assume that $w(U)\subset \mathbb {C}$ is unit disc $\Delta$.

For each $z_0'\in U$, $G_{\Omega }(z,z_0')-\log |w(z)-w(z_0')|$ is a harmonic function on $U\backslash \{z_0'\}$ and is bounded near $z_0'$, hence can be extended to be a harmonic function on U. Since $z_0'\in \Delta$ is arbitrarily chosen, $G_{\Omega }(z,z')-\log |w(z)-w(z')|$ can be extended to be a function on $U\times U$ such that $G_{\Omega }(z,z_0')-\log |w(z)-w(z_0')|$ is a harmonic function on U for any fixed $z_0'\in U$. Denote that

$$\begin{aligned} H(z,z'):=G_{\Omega }(z,z')-\log |w(z)-w(z')|. \end{aligned}$$

Notice it that $G_{\Omega }(z,z')$ and $\log |w(z)-w(z')|$ are symmetric with respect to its two variables, then we know that $G_{\Omega }(z_0,z)-\log |w(z_0)-w(z)|$ is also a harmonic function on U for any fixed $z_0\in U$.

Recall that for unit disc, the Green function is

$$\begin{aligned} G_{\Delta }(z,z')=\frac{1}{2}\log \frac{|z-z'|^2}{|z-z'|^2+(1-|z|^2)(1-|z'|^2)} \end{aligned}$$

(see [5]). Note that $G_{\Omega }(z,z')<0$, then by the property of Green function, we have

$$\begin{aligned} G_{\Omega }(z,z')\le G_{\Delta }(w(z),w(z')) \end{aligned}$$

on $U \times U$, which implies that

$$\begin{aligned} G_{\Omega }(z,z')-\log |w(z)-w(z')|+\frac{1}{2}\log \left( |w(z)-w(z')|^2+(1-|w(z)|^2)(1-|w(z')|^2)\right) \le 0 \end{aligned}$$

on $U \times U$. As $\frac{1}{2}\log \left( |w(z)-w(z')|^2+(1-|w(z)|^2)(1-|w(z')|^2)\right)$ is smooth on $U\times U$, we have $H(z,z')$ is bounded from above on $\overline{V}_{z_0}\times \overline{V}_{z_0}$, where $z_0$ is any point in U and $V_{z_0}\Subset U$ is a neighborhood of $z_0$. For each fixed $z_0'\in U$, $H(z,z_0')$ is a harmonic function on a neighborhood of $\overline{V}_{z_0}$, $H(z,z')$ is therefore bounded on $\overline{V}_{z_0}\times \overline{V}_{z_0}$ by Lemma 2.9. By the same convolution method in Lemma 5.1, we get that $H(z,z')$ is a smooth function on a neighborhood of $(z_0,z_0)$. Since smoothness is a local property, Lemma 5.2 has been proved. $\square$

Let $\gamma$ be a piecewise smooth closed curve in $\Omega$. Using Lemma 5.1, we can get the following lemma.

Lemma 5.3

$\int _{\gamma }\widetilde{d}G_{\Omega }(\cdot ,z')$ is a harmonic function with respect to $z'$ on $\Omega \backslash \gamma$, where $\widetilde{d}=\frac{\partial -{\bar{\partial }}}{i}$.

In the following, we discuss the harmonic function $\int _{\gamma }\widetilde{d}G_{\Omega }(\cdot ,z')$. The following lemma shows that we only need to consider the curves which intersect with themselves at finite many points.

Lemma 5.4

Suppose that $\gamma$ is a piecewise smooth closed curve in $\Omega$. Then there exists a piecewise smooth closed curve $\tilde{\gamma }$ which is homotopic to $\gamma$ in $\Omega$, and $\tilde{\gamma }$ intersects with itself at finite many points.

Proof

$\gamma :[0,1]\rightarrow \Omega \backslash \{z_1\}$ is a piecewise smooth map, hence [0, 1] can be divided into finite many sections $I_i=[x_i-1,x_i]$, $1\le i\le N$, such that each $\gamma |_{I_i}$ is contained in a simply connected coordinate neighborhood $V_i$ in $\Omega$.

Now construct a smooth curve $\tilde{\gamma }:[0,1]\rightarrow \Omega \backslash \{z_1\}$ as follows: $\tilde{\gamma }|_{I_1}:I_1\rightarrow V_1$ is a smooth mapping such that $\tilde{\gamma }(x_0)=\gamma (x_0)$, $\tilde{\gamma }(x_1)=\gamma (x_1)$ and $\tilde{\gamma }|_{I_1}$ does not intersect with itself unless $\gamma (x_0)=\gamma (x_1)$; Suppose that $\tilde{\gamma }|_{I_i}$ has been defined for $1\le i\le k-1$, $k\le N$, then define $\tilde{\gamma }|_{I_k}:I_k\rightarrow V_k$ to be a smooth mapping such that $\tilde{\gamma }(x_{k-1})=\gamma (x_{k-1})$, $\tilde{\gamma }(x_k)=\gamma (x_k)$ and $\tilde{\gamma }|_{I_k}$ intersects with $\tilde{\gamma }|_{I_i}$ at finite many points for $1\le i\le k$.

Since $V_i$ is simply connected, $\tilde{\gamma }|_{I_i}$ is homotopic to $\gamma |_{I_i}$ for any $1\le i\le N$. Thus, $\tilde{\gamma }$ is homotopic to $\gamma$ and intersects with itself at finite many points. $\square$

Lemma 5.5

Suppose that $\gamma$ is a piecewise smooth closed curve. Then there exists a harmonic function $H(z')$ on $\Delta$ such that $H(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z'))+2k\pi$ for $z'\in \Delta \backslash p^{-1}(\gamma )$, where k is an integer depending on $z'$ and $p:\Delta \rightarrow \Omega$ is the universal covering from unit disc $\Delta$ to $\Omega$.

Proof

Note that $d\widetilde{d}=2i\partial {\bar{\partial }}$ and $\int _{\gamma _{0}}\widetilde{d}G_{\Omega }(\cdot ,z_0)\in 2\pi \mathbb {Z}$, where $\gamma _0\subset U_{z_0}\backslash \{z_0\}$ is a closed piecewise smooth curve and $U_{z_0}$ is a coordinate disc centered on $z_0\in \Omega$. By Lemma 5.4, we only need to consider piecewise smooth closed curve $\gamma$ which intersects with itself at finite many points.

Assume that $\gamma$ intersects with itself at n points, and we prove this Lemma by induction on n.

When $n=1$, $\gamma$ is a simple closed curve. Firstly, suppose that $\Omega \backslash \gamma$ is not connected. Note that $\Omega$ is orientable, then we denote the two connected components of $\Omega \backslash \gamma$ by $\Omega _1$ and $\Omega _2$, where $\Omega _1$ is left to $\gamma$ and $\Omega _2$ is right to $\gamma$. Define

$$\begin{aligned} h(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z') \end{aligned}$$

for $z'\in \Omega _1$ and

$$\begin{aligned} h(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+2\pi \end{aligned}$$

for $z'\in \Omega _2$. $h(z')$ is a harmonic function on $\Omega \backslash \gamma$ by Lemma 5.3. For any point $x_0\in \gamma$, choose a local coordinate neighborhood $V_{x_0}$ of $x_0$ and a segment $\gamma _1$ of $\gamma$ such that $\gamma _1\subset V_{x_0}$. Assume that $\gamma =\beta _1\gamma _1\beta _2$. Construct a smooth curve $\gamma _2$ in $V_{x_0}\cap \Omega _2$ such that the starting point and end point of $\gamma _2$ coincide with $\gamma _1$. Denote the open set in $V_{x_0}\cap \Omega _2$ bounded by $\gamma _2\left( \gamma _1\right) ^{-1}$ by G, where $\left( \gamma _1\right) ^{-1}(t)=\gamma _1(1-t)$ on [0, 1]. Note that $\tilde{\gamma }:=\beta _1\gamma _2\beta _2$ is a piecewise smooth closed curve in $\Omega$, then $\int _{\tilde{\gamma }}\widetilde{d} G_{\Omega }(\cdot ,z')$ is a harmonic function on $\Omega _1\cup \gamma _1\cup G$, which is a connected component of $\Omega \backslash \tilde{\gamma }$. For $z'\in \Omega _1$, we have

$$\begin{aligned} \int _{\tilde{\gamma }}\widetilde{d} G_{\Omega }(\cdot ,z')&=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+\int _{\gamma _2\left( \gamma _1\right) ^{-1}}\widetilde{d} G_{\Omega }(\cdot ,z')\\&=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')\\&=h(z'). \end{aligned}$$

For $z'\in G\subset \Omega _2$, we have

$$\begin{aligned} \int _{\tilde{\gamma }}\widetilde{d} G_{\Omega }(\cdot ,z')&=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+\int _{\gamma _2\left( \gamma _1\right) ^{-1}}\widetilde{d} G_{\Omega }(\cdot ,z')\\&=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+2\pi \\&=h(z'). \end{aligned}$$

As a result, $h(z')$ can be extended to a harmonic function on a neighborhood of $x_0$. Since $x_0\in \gamma$ is arbitrarily chosen, $h(z')$ can be extended to be a harmonic function on $\Omega$. $H(z'):=h(p(z'))$ is therefore a harmonic function on $\Delta$ such that

$$\begin{aligned} H(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+2k\pi , \end{aligned}$$

where $k=0$ if $z'\in p^{-1}(\Omega _1)$ and $k=1$ if $z'\in p^{-1}(\Omega _2)$.

Now suppose that $\Omega \backslash \gamma$ is connected. $d\left( \int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')\right)$ is a smooth closed differential form on $\Omega \backslash \gamma$. For any point $x_0\in \gamma$, choose a connected coordinate neighborhood $\Omega '$ of $x_0$ which can be divided by $\gamma \cap \Omega '$ into two connected components denoted by $\Omega '_1$ and $\Omega '_2$, where $\Omega '_1$ is left to $\gamma \cap \Omega '$, $\Omega '_2$ is right to $\gamma \cap \Omega '$. Define

$$\begin{aligned} h(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z') \end{aligned}$$

for $z'\in \Omega '_1$ and

$$\begin{aligned} h(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+2\pi \end{aligned}$$

for $z'\in \Omega '_2$. Similarly, $h(z')$ can be extended to a harmonic function on a neighborhood $\Omega '$ of $x_0$. Notice it that $d\left( \int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')\right) =dh(z')$ on $\Omega '$, and $x_0\in \gamma$ is arbitrarily chosen. $d\left( \int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')\right)$ can be extended to be a smooth closed form on $\Omega$, and we denote it by $\omega$. Then $p^*\omega$ is a smooth closed form on $\Delta$.

Fix a connected component U of $\Delta \backslash p^{-1}(\gamma )$, then $\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z'))$ is a harmonic function on U, and its differential is $p^*\omega$. Since $\Delta$ is simply connected, $\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z'))$ can be extended to a harmonic function denoted by $H(z')$ on $\Delta$ through $p^*\omega$.

It is clear that $p(U)=\Omega \backslash \gamma$. Thus, for any point $z_1'\in \Delta \backslash p^{-1}(\gamma )$, there exists a point $z_2'\in U\cap p^{-1}(p(z_1'))$, such that $p(z_1')=p(z_2')$. Let $\tilde{\gamma }$ be a curve in $\Delta$ from $z_2'$ to $z_1'$. $p_*\tilde{\gamma }$ is a smooth closed curve in $\Omega$, and after a homotopy if necessary (similarly to the proof of Lemma 5.4), we can assume that $p_*\tilde{\gamma }$ intersects $\gamma$ at finite many points. Divide [0, 1] into $I_j=[t_{j-1},t_j]$ for $1\le j\le N$, such that $p_*\tilde{\gamma }|_{(t_{j-1},t_j)}$ intersects $\gamma$ at one point and $p_*\tilde{\gamma }(t_j)\notin \gamma$. By the construction of $\omega$ and $H(z')$, we have

$$\begin{aligned} H(\tilde{\gamma }(t_j))&=H(\tilde{\gamma }(t_{j-1}))+\int _{\tilde{\gamma }|_{I_j}}p^*\omega \\&=H(\tilde{\gamma }(t_{j-1}))+\int _{p_*\tilde{\gamma }|_{I_j}}\omega \\&=H(\tilde{\gamma }(t_{j-1}))+\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p_*\tilde{\gamma }(t_j))-\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p_*\tilde{\gamma }(t_{j-1}))+2k_j\pi , \end{aligned}$$

where $k_j=1$ or $-1$, depending on whether $p_*\tilde{\gamma }|_{I_{j}}$ crossing $p_*\tilde{\gamma }$ from right to left or not. Since $p_*\tilde{\gamma }(t_N)=p_*\tilde{\gamma }(t_0)$,

$$\begin{aligned} H(z_1')-H(z_2')&=\sum _{i=1}^N (H(\tilde{\gamma }(t_i))-H(\tilde{\gamma }(t_{i-1})))\\&=\sum _{i=1}^N \left( \int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p_*\tilde{\gamma }(t_j))-\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p_*\tilde{\gamma }(t_{j-1}))+2k_j\pi \right) \\&=\sum _{i=1}^N 2k_i\pi . \end{aligned}$$

Since $z_2'\in U$ and $p(z_2')=p(z_1')$, we have $H(z_2')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z_2'))=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z_1'))$. Therefore

$$\begin{aligned} H(z_1')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z_1'))+2k\pi , \end{aligned}$$

where $k=\sum _{j=1}^N k_j$ ia an integer.

Suppose that this Lemma has been proved when $n=k-1$, $k\ge 2$. Assume that $\gamma$ intersects with itself at k points. Denote $\gamma (t_1)$ to be the first point of $\gamma$ at which $\gamma$ intersects with itself, and $\gamma (t_1)=\gamma (t_0)$, $0\le t_0<t_1$.

Let $\tilde{\gamma }_1(t):[0,t_1-t_0]\rightarrow \Omega$ satisfy $\tilde{\gamma }_1(t)=\gamma (t-t_0)$; Let $\tilde{\gamma }_2(t):[0,1-t_1+t_0]$ satisfy $\tilde{\gamma }_2(t)=\gamma (t)$ on $[0,t_0]$ and $\tilde{\gamma }_2(t)=\tilde{\gamma }(t+t_1-t_0)$ on $[t_0,1-t_1+t_0]$. Note that $\tilde{\gamma }_1$ is a smooth simple closed curve and $\tilde{\gamma }_2$ is a smooth closed curve which intersects with itself at most $k-1$ points. By assumption, for $1\le j\le 2$, there exists a harmonic function $H_j$ on $\Delta$ such that for any point $z'\in \Delta \backslash p^{-1}(\tilde{\gamma }_j)$,

$$\begin{aligned} H_j(z')=\int _{\tilde{\gamma }_j}\widetilde{d} G_{\Omega }(\cdot ,p(z'))+2k_j'\pi , \end{aligned}$$

where $k_j'$ is an integer depending on $z'$. Define $H(z')=H_1(z')+H_2(z')$ on $\Delta \backslash p^{-1}(\gamma )$,

$$\begin{aligned} H_1(z')+H_2(z')&=\int _{\tilde{\gamma }_1}\widetilde{d} G_{\Omega }(\cdot ,p(z'))+2k_1'\pi +\int _{\tilde{\gamma }_2}\widetilde{d} G_{\Omega }(\cdot ,p(z'))+2k_2'\pi \\&=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,p(z'))+2(k_1'+k_2')\pi , \end{aligned}$$

where $k_1'$ and $k_2'$ are two integers depending on $z'$, which completes the induction.

Thus, Lemma 5.5 has been proved. $\square$

Remark 5.6

From the proof of Lemma 5.5, we can see that the following two statements are equivalent:

(1) each simple closed curve divides $\Omega$ into two disconnected sets;

(2) for every piecewise smooth closed curve $\gamma$, there exists a harmonic function $h(z')$ on $\Omega$ such that $h(z')=\int _{\gamma }\widetilde{d} G_{\Omega }(\cdot ,z')+2k\pi$ for $z'\in \Omega \backslash \gamma$, where k is an integer depending on $z'$.

It is clear that statement (1) holds for any open subsets of $\mathbb {C}$, and statement (1) does not hold when $\Omega$ is a torus less a small closed disc.