$C-$Robin Functions and Applications

TL;DR

This paper explores $C$-Robin functions within pluripotential theory, generalizing previous results to construct polynomial families that recover extremal functions for certain compact sets in complex space.

Contribution

It introduces and studies $C$-Robin functions in pluripotential theory and generalizes Bloom's results to construct polynomial families for extremal functions.

Findings

Construction of polynomial families for $C$-extremal functions

Generalization of Bloom's results to new convex bodies

Applications to nonpluripolar compact sets in complex space

Abstract

We continue the study in the setting of pluripotential theory arising from polynomials associated to a convex body $C$ in $({\bf R}^+)^d$. Here we discuss $C-$Robin functions and their applications. In the particular case where $C$ is a simplex in $({\bf R}^+)^2$ with vertices $(0,0),(b,0),(a,0)$, $a,b>0$, we generalize results of T. Bloom to construct families of polynomials which recover the $C-$extremal function $V_{C,K}$ of a nonpluripolar compact set $K\subset {\bf C}^d$.

Equations

$$H_C(z):=\underset{J\in C}{\sup }\log |z^J|:=\underset{(j_1,...,j_d)\in C}{\sup }\log [|z_1{|}^{j_1}\cdots |z_d{|}^{j_d}].$$

$$\Sigma \subset kCfor\; somek\in {\mathbb{Z}}^{+}$$

$$\Sigma :=\{(x_1,...,x_d)\in {\mathbb{R}}^d:0\le x_i\le 1,\sum _{j=1}^d{x}_i\le 1\}.$$

$$H_C(z)\ge \frac{1}{k}\underset{j=1,...,d}{\max }{\log }^{+}|z_j|=\frac{1}{k}{H}_{\Sigma }(z)$$

$$L_C=L_C({\mathbb{C}}^d):=\{u\in PSH({\mathbb{C}}^d):u(z)-H_C(z)=O(1),|z|\to \infty \},$$

$$L_{C,+}=L_{C,+}({\mathbb{C}}^d)=\{u\in L_C({\mathbb{C}}^d):u(z)\ge H_C(z)+C_u\}$$

$$Poly(nC):=\{p\in \mathbb{C}[z]:p(z)=\sum _{\alpha \in nC}{a}_{\alpha }{z}^{\alpha }\}.$$

$${\mathrm{deg}}_C(p)=\min \{n\in \mathbb{N}:p\in Poly(nC)\}.$$

$$V_{C,K}(z):=\sup \{u(z):u\in L_C,u\le 0\; onK\}.$$

$${\rho }_{\mathbf{u}}(z):=\underset{|\lambda |\to \infty }{\mathrm{limsup}}[u(\lambda z)-\log |\lambda |]$$

$$H_C(\lambda \circ (z_1,z_2))=H_C(z_1,z_2)+ab\log |\lambda |$$

$${\rho }_u(z_1,z_2):=\underset{|\lambda |\to \infty }{\mathrm{limsup}}[u(\lambda \circ (z_1,z_2))-ab\log |\lambda |]$$

$$C=co\{(0,...,0),(a_1,0,...,0),...,(0,...,0,a_d)\}$$

$$Poly(nC)=span\{e_1,...,e_{N_n}\}$$

$$VDM({\zeta }_1,...,{\zeta }_{N_n}):=\det [e_i({\zeta }_j){]}_{i,j=1,...,N_n}$$

$$=\det\left[\begin{array}[]{ccccc}e_{1}(\zeta_{1})&e_{1}(\zeta_{2})&\ldots&e_{1}(\zeta_{N_{n}})\\ \vdots&\vdots&\ddots&\vdots\\ e_{N_{n}}(\zeta_{1})&e_{N_{n}}(\zeta_{2})&\ldots&e_{N_{n}}(\zeta_{N_{n}})\end{array}\right]$$

$$V_n=V_n(K):=\underset{{\zeta }_1,...,{\zeta }_{N_n}\in K}{\max }|VDM({\zeta }_1,...,{\zeta }_{N_n})|.$$

$${\delta }_C(K):=\underset{n\to \infty }{\mathrm{limsup}}{V}_n^{1/l_n}$$

$$\mathcal{E}(u,v):={\int }_{{\mathbb{C}}^d}(u-v)\sum _{j=0}^d(d{d}^{c}u{)}^j\wedge (d{d}^{c}v{)}^{d-j}.$$

$$\mathcal{E}(u,v)+\mathcal{E}(v,w)+\mathcal{E}(w,u)=0.$$

$$\log {\delta }_C(K)=\frac{-1}{c}\mathcal{E}(V_{C,K}^{\ast },H_C)$$

$$0=\mathcal{E}(V_{C,E}^{\ast },H_C)+\mathcal{E}(H_C,V_{C,F}^{\ast })+\mathcal{E}(V_{C,F}^{\ast },V_{C,E}^{\ast })$$

$$=\mathcal{E}(V_{C,E}^{\ast },H_C)-\mathcal{E}(V_{C,F}^{\ast },H_C)+\mathcal{E}(V_{C,F}^{\ast },V_{C,E}^{\ast })$$

$$=\mathcal{E}(V_{C,F}^{\ast },V_{C,E}^{\ast }).$$

$$0=\mathcal{E}(V_{C,F}^{\ast },V_{C,E}^{\ast })={\int }_{{\mathbb{C}}^d}(V_{C,F}^{\ast }-V_{C,E}^{\ast })\sum _{j=0}^d(d{d}^c{V}_{C,F}^{\ast }{)}^j\wedge (d{d}^c{V}_{C,E}^{\ast }{)}^{d-j}$$

$$={\int }_{{\mathbb{C}}^d}(V_{C,F}^{\ast }-V_{C,E}^{\ast })(d{d}^c{V}_{C,F}^{\ast }{)}^d+{\int }_{{\mathbb{C}}^d}(V_{C,F}^{\ast }-V_{C,E}^{\ast })(d{d}^c{V}_{C,E}^{\ast }{)}^d$$

$$+{\int }_{{\mathbb{C}}^d}(V_{C,F}^{\ast }-V_{C,E}^{\ast })\sum _{j=1}^{d-1}(d{d}^c{V}_{C,F}^{\ast }{)}^j\wedge (d{d}^c{V}_{C,E}^{\ast }{)}^{d-j}.$$

$$V_{C,F}^{\ast }=V_{C,E}^{\ast }=0q.e.\; on\; supp(d{d}^c{V}_{C,E}^{\ast }{)}^d$$

$$V_{C,F}^{\ast }=0q.e.\; on\; supp(d{d}^c{V}_{C,F}^{\ast }{)}^d.$$

$$0={\int }_{{\mathbb{C}}^d}(-V_{C,E}^{\ast })(d{d}^c{V}_{C,F}^{\ast }{)}^d+{\int }_{{\mathbb{C}}^d}(V_{C,F}^{\ast }-V_{C,E}^{\ast })\sum _{j=1}^{d-1}(d{d}^c{V}_{C,F}^{\ast }{)}^j\wedge (d{d}^c{V}_{C,E}^{\ast }{)}^{d-j}$$

Full text

*Supported by Simons Foundation grant No. 354549

1. 1 Introduction
2. 2 Rumely formula and transfinite diameter
3. 3 Other preliminary results: General
4. 4 $C-$Robin function
5. 5 Preliminary results: Triangle case
6. 6 The integral formula
7. 7 $C-$transfinite diameter and directional Chebyshev constants
8. 8 Polynomials approximating $V_{C,K}$
9. 9 Further directions

1. Introduction

As in [1], we fix a convex body $C\subset({\mathbb{R}}^{+})^{d}$ and we define the logarithmic indicator function

[TABLE]

We assume throughout that

[TABLE]

where

[TABLE]

Then

[TABLE]

where $\log^{+}|z_{j}|=\max[0,\log|z_{j}|]$. We define

[TABLE]

and

[TABLE]

where $PSH({\mathbb{C}}^{d})$ denotes the class of plurisubharmonic functions on ${\mathbb{C}}^{d}$. These are generalizations of the classical Lelong classes $L:=L_{\Sigma},\ L^{+}:=L_{\Sigma,+}$ when $C=\Sigma$. Let ${\mathbb{C}}[z]$ denote the polynomials in $z$ and

[TABLE]

For a nonconstant polynomial $p$ we define

[TABLE]

If $p\in Poly(nC),\ n\geq 1$ we have $\frac{1}{n}\log|p|\in L_{C}$; also each $u\in L_{C,+}({\mathbb{C}}^{d})$ is locally bounded in ${\mathbb{C}}^{d}$. For $C=\Sigma$, we write $Poly(nC)=\mathcal{P}_{n}$.

The $C$-extremal function of a compact set $K\subset{\mathbb{C}}^{2}$ is defined as the uppersemicontinuous (usc) regularization $V_{C,K}^{*}(z):=\limsup_{\zeta\to z}V_{C,K}(\zeta)$ of

[TABLE]

If $K$ is regular ($V_{K}:=V_{\Sigma,K}$ is continuous), then $V_{C,K}=V_{C,K}^{*}$ is continuous (cf., [9]). In particular, for $K=T^{d}=\{(z_{1},...,z_{d})\in{\mathbb{C}}^{d}:|z_{j}|=1,\ j=1,...,d\}$, $V_{C,T^{d}}=V_{C,T^{d}}^{*}=H_{C}$ (cf., (2.7) in [1]). If $K$ is not pluripolar, i.e., for any $u$ psh with $u=-\infty$ on $K$ we have $u\equiv-\infty$, the Monge-Ampère measure $(dd^{c}V_{C,K}^{*})^{d}$ is a positive measure with support in $K$ and $V_{C,K}^{*}=0$ quasi-everywhere (q.e.) on supp$(dd^{c}V_{C,K}^{*})^{d}$ (i.e., everywhere except perhaps a pluripolar set).

Much of the recent development of this $C-$pluripotential theory can be found in [9], [1] and [2]. One noticeable item lacking from these works is a constructive approach to finding natural concrete families of polynomials associated to $K,C$ which recover $V_{C,K}$. In order to do this, following the approach of Tom Bloom in [4] and [5], we introduce a $C-$Robin function $\rho_{u}$ for a function $u\in L_{C}$. The “usual” Robin function ${\bf\rho_{u}}$ associated to $u\in L_{\Sigma}$ is defined as

[TABLE]

and this detects the asymptotic behavior of $u$. This definition is natural since the “growth function” $H_{\Sigma}(z)=\max_{j=1,...,d}\log^{+}|z_{j}|$ satisfies $H_{\Sigma}(\lambda z)=H_{\Sigma}(z)+\log|\lambda|$. Let $C$ be the triangle in ${\mathbb{R}}^{2}$ with vertices $(0,0),(b,0),(0,a)$ where $a,b$ are relatively prime positive integers. Then

1. (1)

$H_{C}(z_{1},z_{2})=\max[\log^{+}|z_{1}|^{b},\log^{+}|z_{2}|^{a}]$ (note $H_{C}=0$ on the closure of the unit polydisk $P^{2}:=\{(z_{1},z_{2}):|z_{1}|,|z_{2}|<1\}$); 2. (2)

defining $\lambda\circ(z_{1},z_{2}):=(\lambda^{a}z_{1},\lambda^{b}z_{2})$, we have

[TABLE]

for $(z_{1},z_{2})\in{\mathbb{C}}^{2}\setminus P^{2}$ and $|\lambda|\geq 1$.

Given $u\in L_{C}({\mathbb{C}}^{2})$, we define the $C-$Robin function of $u$ (Definition 4.2) as

[TABLE]

for $(z_{1},z_{2})\in{\mathbb{C}}^{2}$. This agrees with (1.5) when $a=b=1$; i.e., when $C=\Sigma\subset({\mathbb{R}}^{+})^{2}$. For general convex bodies $C$, it is unclear how to define an analogue to recover the asymptotic behavior of $u\in L_{C}$.

The next two sections give some general results in $C-$pluripotential theory which will be used further on but are of independent interest. Section 4 begins in earnest with the case where $C$ is a triangle in ${\mathbb{C}}^{2}$. The key results utilized in our analysis are the use of an integral formula of Bedford and Taylor [3], Theorem 6.1 in section 6, yielding the fundamental Corollary 6.4, and recent results on $C-$transfinite diameter in [12] and [13] of the second author in section 7. Our arguments in Sections 5 and 8 follow closely those of Bloom in [4] and [5]. The main theorem, Theorem 8.3, is stated and proved in section 8; then explicit examples of families of polynomials which recover $V_{C,K}$ are provided. We mention that the results given here for triangles $C$ in ${\mathbb{R}}^{2}$ with vertices $(0,0),(b,0),(0,a)$ where $a,b$ are relatively prime positive integers should generalize to the case of a simplex

[TABLE]

in $({\mathbb{R}}^{+})^{d},\ d>2$ where $a_{1},...,a_{d}$ are pairwise relatively prime (cf., Remark 4.5). Section 9 indicates generalizations to weighted situations.

2. Rumely formula and transfinite diameter

We recall the definition of $C-$transfinite diameter $\delta_{C}(K)$ of a compact set $K\subset{\mathbb{C}}^{d}$ where $C$ satisfies (1.2). Letting $N_{n}$ be the dimension of $Poly(nC)$ in (1.3), we have

[TABLE]

where $\{e_{j}(z):=z^{\alpha(j)}=z_{1}^{\alpha_{1}(j)}\cdots z_{d}^{\alpha_{d}(j)}\}_{j=1,...,N_{n}}$ are the standard basis monomials in $Poly(nC)$ in any order. For points $\zeta_{1},...,\zeta_{N_{n}}\in{\mathbb{C}}^{d}$, let

[TABLE]

[TABLE]

and for a compact subset $K\subset{\mathbb{C}}^{d}$ let

[TABLE]

Then

[TABLE]

is the $C-$transfinite diameter of $K$ where $l_{n}:=\sum_{j=1}^{N_{n}}{\rm deg}_{C}(e_{j})$. The existence of the limit is not obvious but in this setting it is proved in [1]. We return to this issue in section 7.

Next, for $u,v\in L_{C,+}$, we define the mutual energy

[TABLE]

Here $dd^{c}=i\partial\overline{\partial}$ and for locally bounded psh functions, e.g., for $u,v\in L_{C,+}$, the complex Monge-Ampère operators $(dd^{c}u)^{j}\wedge(dd^{c}v)^{d-j}$ are well-defined as positive measures. We have that $\mathcal{E}$ satisfies the cocycle property; i.e., for $u,v,w\in L_{C,+}$, (cf., [1], Proposition 3.3)

[TABLE]

Connecting these notions, we recall the following formula from [1].

Theorem 2.1**.**

Let $K\subset{\mathbb{C}}^{d}$ be compact and nonpluripolar. Then

[TABLE]

where $c$ is a positive constant depending only on $d$ and $C$.

We will use the global domination principle for general $L_{C}$ and $L_{C,+}$ classes associated to convex bodies satisfying (1.2) (cf., [11]):

Proposition 2.2**.**

For $C\subset({\mathbb{R}}^{+})^{d}$ satisfying (1.2), let $u\in L_{C}$ and $v\in L_{C,+}$ with $u\leq v$ a.e.-$(dd^{c}v)^{d}$. Then $u\leq v$ in ${\mathbb{C}}^{d}$.

We use these ingredients to prove the following.

Proposition 2.3**.**

Let $E\subset F$ be compact and nonpluripolar. If $\delta_{C}(E)=\delta_{C}(F)$ then $V_{C,E}^{*}=V_{C,F}^{*}$.

Proof.

By Theorem 2.1, the hypothesis implies that $\mathcal{E}(V_{C,E}^{*},H_{C})=\mathcal{E}(V_{C,F}^{*},H_{C})$. Using the cocycle property,

[TABLE]

[TABLE]

[TABLE]

From the definition (2.1),

[TABLE]

[TABLE]

[TABLE]

Now $E\subset F$ implies $V_{C,F}^{*}\leq V_{C,E}^{*}$; i.e., $V_{C,F}^{*}-V_{C,E}^{*}\leq 0\ \hbox{on}\ {\mathbb{C}}^{d}$. Also,

[TABLE]

and

[TABLE]

Thus we see that

[TABLE]

where each term on the right-hand-side is nonpositive. Hence each term vanishes. In particular,

[TABLE]

implies that $V_{C,E}^{*}=0$ q.e. on supp$(dd^{c}V_{C,F}^{*})^{d}$ (and hence a.e.-$(dd^{c}V_{C,F}^{*})^{d}$).

We finish the proof by using the domination principle (Proposition 2.2): we have $V_{C,E}^{*},V_{C,F}^{*}\in L_{C,+}({\mathbb{C}}^{d})$ with

[TABLE]

and hence $V_{C,E}^{*}\leq V_{C,F}^{*}$ on ${\mathbb{C}}^{d}$; i.e., $V_{C,E}^{*}=V_{C,F}^{*}$ on ${\mathbb{C}}^{d}$.

∎

Remark 2.4**.**

For $C=\Sigma$, Proposition 2.3 was proved for regular compact sets $E,F$ in [5] and in general (compact and nonpluripolar sets) in [6]. Both results utilized the “usual” Robin functions (1.5) of $V_{E}^{*},V_{F}^{*}$.

3. Other preliminary results: General

Let $K\subset{\mathbb{C}}^{d}$ be compact and nonpluripolar and let $\mu$ be a positive measure on $K$ such that one can form orthonormal polynomials $\{p_{\alpha}\}$ using Gram-Schmidt on the monomials $\{z^{\alpha}\}$. We use the notion of degree given in (1.4): $\deg_{C}(p)=\min\{n\in{\mathbb{N}}\colon p\in Poly(nC)\}$. We have the Siciak-Zaharjuta type polynomial formula

[TABLE]

(cf., [1], Proposition 2.3). It follows that $\{z\in{\mathbb{C}}^{d}:V_{C,K}(z)=0\}=\widehat{K}$, the polynomial hull of $K$:

[TABLE]

In this section, we follow the arguments of Zeriahi in [17].

Proposition 3.1**.**

In this setting,

[TABLE]

Proof.

Let $Q_{n}:=\sum_{\alpha\in nC}c_{\alpha}p_{\alpha}\in Poly(nC)$ with $||Q_{n}||_{K}\leq 1$. Then

[TABLE]

by Cauchy-Schwarz. Hence

[TABLE]

where recall $N_{n}=$dim$(Poly(nC))$.

Now fix $z_{0}\in{\mathbb{C}}^{d}\setminus\widehat{K}$ and let $\alpha_{n}\in nC$ be the multiindex with $\deg_{C}(p_{\alpha_{n}})$ largest such that

[TABLE]

We claim that taking any sequence $\{Q_{n}\}$ with $||Q_{n}||_{K}\leq 1$ for all $n$,

[TABLE]

For if not, then by the above argument, there exists $A<+\infty$ such that for any $n$ and any $Q_{n}\in Poly(nC)$ with $||Q_{n}||_{K}\leq 1$,

[TABLE]

where $M(z_{0})$ is independent of $n$. But then

[TABLE]

[TABLE]

which contradicts $z_{0}\in{\mathbb{C}}^{d}\setminus\widehat{K}$. We conclude that for any $z\in{\mathbb{C}}^{d}\setminus\widehat{K}$, for any $n$ and any $Q_{n}\in Poly(nC)$ with $||Q_{n}||_{K}\leq 1$,

[TABLE]

where we can assume $\deg_{C}(p_{\alpha_{n}})\uparrow+\infty$. Hence, for such $z$,

[TABLE]

[TABLE]

where we have used $\deg_{C}(p_{\alpha_{n}})\leq n$.

∎

Suppose $\mu$ is any Bernstein-Markov measure for $K$; i.e., for any $\epsilon>0$, there exists a constant $c_{\epsilon}$ so that

[TABLE]

From (1.2), $\Sigma\subset kC\subset m\Sigma$ for some $k,m$ and we can replace $(1+\epsilon)^{n}$ by $(1+\epsilon)^{\deg_{C}(p_{n})}$. In particular, for the orthonormal polynomials $\{p_{\alpha}\}$,

[TABLE]

Thus

[TABLE]

and we obtain equality in the previous result:

Corollary 3.2**.**

In this setting, if $\mu$ is any Bernstein-Markov measure for $K$,

[TABLE]

We remark that Bernstein-Markov measures exist in abundance; cf., [8]. Our goal in subsequent sections is to generalize the results in [4] and [5] of T. Bloom to give more constructive ways of recovering $V_{C,K}$ from special families of polynomials.

4. $C-$Robin function

We begin with the observation that a proof similar to that of Theorem 5.3.1 of [10] yields the following result.

Theorem 4.1**.**

Let $C,C^{\prime}\subset({\mathbb{R}}^{+})^{d}$ be convex bodies and let $F:{\mathbb{C}}^{d}\to{\mathbb{C}}^{d}$ be a proper polynomial mapping satisfying

[TABLE]

Then for $K\subset{\mathbb{C}}^{d}$ compact,

[TABLE]

Proof.

Since $H_{C}(z):=\sup_{J\in C}\log|z^{J}|$, the hypothesis can be written

[TABLE]

We first show that $\liminf_{|z|\to\infty}\frac{e^{H_{C}(F(z))}}{e^{H_{C^{\prime}}(z)}}>0$ implies

[TABLE]

Indeed, starting with $u\in L_{C^{\prime}}$ with $u\leq 0$ on $F^{-1}(K)$, take

[TABLE]

where the supremum is over all preimages of $z$. Then $v\in PSH({\mathbb{C}}^{d})$ and $v\leq 0$ on $K$. Note that $v(F(z))=u(z)$. Now $u\in L_{C^{\prime}}$ implies

[TABLE]

To show $v\in L_{C}$, since $F$ is proper it suffices to show

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

from the hypothesized condition in (4.1) so $v\in L_{C}$ and (4.2) follows.

Next we show that

[TABLE]

Letting $u\in L_{C}$ with $u\leq 0$ on $K$, we have $u(F(z))\in PSH({\mathbb{C}}^{d})$ and $u(F(z))\leq 0$ on $F^{-1}(K)$ and we are left to show $u(F(z))\in L_{C^{\prime}}$. Now

[TABLE]

[TABLE]

[TABLE]

from the hypothesized condition in (4.1) and $u\in L_{C}$.

∎

We can apply this in ${\mathbb{C}}^{d}$ with $C^{\prime}=c\Sigma$ where $c\in{\mathbb{Z}}^{+}$ and $C$ is an arbitrary convex body in $({\mathbb{R}}^{+})^{d}$. Given $K\subset{\mathbb{C}}^{d}$ compact, provided we can find $F$ satisfying the hypotheses of Theorem 4.1, from the relation

[TABLE]

we can form a scaling of the standard Robin function (1.5) for $V_{F^{-1}(K)}$, i.e., $\rho_{F^{-1}(K)}:=\rho_{\Sigma,F^{-1}(K)}$, and we have

[TABLE]

This gives a connection between the standard Robin function $\rho_{F^{-1}(K)}$ and something resembling a possible definition of a $C-$Robin function $\rho_{C,K}$ (the right-hand-side). Given $K\subset{\mathbb{C}}^{d}$, the set $F^{-1}(K)$ can be very complicated so that, apriori, this relation has little practical value.

For the rest of this section, and for most of the subsequent sections, we work in ${\mathbb{C}}^{2}$ with variables $z=(z_{1},z_{2})$ and we let $C$ be the triangle with vertices $(0,0),(b,0),(0,a)$ where $a,b$ are relatively prime positive integers. We recall from the introduction:

1. (1)

$H_{C}(z_{1},z_{2})=\max[\log^{+}|z_{1}|^{b},\log^{+}|z_{2}|^{a}]$ (note $H_{C}=0$ on the closure of the unit polydisk $P^{2}:=\{(z_{1},z_{2}):|z_{1}|,|z_{2}|<1\}$); 2. (2)

defining $\lambda\circ(z_{1},z_{2}):=(\lambda^{a}z_{1},\lambda^{b}z_{2})$, we have

[TABLE]

for $(z_{1},z_{2})\in{\mathbb{C}}^{2}\setminus P^{2}$ and $|\lambda|\geq 1$.

Definition 4.2**.**

Given $u\in L_{C}$, we define the $C-$Robin function of $u$:

[TABLE]

for $(z_{1},z_{2})\in{\mathbb{C}}^{2}$.

We claim that $\rho_{u}\in L_{C}$. To see this, we lift the circle action on ${\mathbb{C}}^{2}$,

[TABLE]

to ${\mathbb{C}}^{3}$ in the following manner:

[TABLE]

Given a function $u\in L_{C}({\mathbb{C}}^{2})$, we can associate a function $h$ on ${\mathbb{C}}^{3}$ which satisfies

1. (1)

$h(1,z_{1},z_{2})=u(z_{1},z_{2})$ for all $(z_{1},z_{2})\in{\mathbb{C}}^{2}$; 2. (2)

$h\in L_{\tilde{C}}({\mathbb{C}}^{3})$ where $\tilde{C}=co\{(0,0,0),(1,0,0),(0,b,0),(0,0,a)\}$; 3. (3)

$h$ is $ab-$log-homogeneous:

[TABLE]

Indeed, we simply set

[TABLE]

[TABLE]

Now since $h$ is psh on ${\mathbb{C}}^{3}$, we have

[TABLE]

Proposition 4.3**.**

For $u\in L_{C}$, we have $\rho_{u}\in L_{C}$. In particular, $\rho_{u}$ is plurisubharmonic.

Proof.

The psh of $\rho_{u}$ follows directly from (4.4) since $h$ is psh on ${\mathbb{C}}^{3}$. To show $\rho_{u}\in L_{C}$, note that

[TABLE]

From (4.3) $H_{C}$ satisfies the same relation for $(z_{1},z_{2})\in{\mathbb{C}}^{2}\setminus P^{2}$ and $|\lambda|\geq 1$ which gives the result. ∎

Remark 4.4**.**

Since $\rho_{u}(\lambda\circ(z_{1},z_{2}))=\rho_{u}(z_{1},z_{2})+ab\log|\lambda|$, in particular,

[TABLE]

Moreover, any point $(z_{1},z_{2})\in{\mathbb{C}}^{2}$ is of the form $(\lambda^{a}\zeta_{1},\lambda^{b}\zeta_{2})$ for some $(\zeta_{1},\zeta_{2})\in\partial P^{2}$ and some $\lambda\in{\mathbb{C}}$. Indeed, if $b\geq a$ then we get all points $(z_{1},z_{2})\in{\mathbb{C}}^{2}$ with $|z_{2}|^{a}\leq|z_{1}|^{b}$ as $(\lambda^{a}\zeta_{1},\lambda^{b}\zeta_{2})$ for some $(\zeta_{1},\zeta_{2})$ with $|\zeta_{1}|=1$ and $|\zeta_{2}|\leq 1$ and we get all points $(z_{1},z_{2})\in{\mathbb{C}}^{2}$ with $|z_{2}|^{a}\geq|z_{1}|^{b}$ as $(\lambda^{a}\zeta_{1},\lambda^{b}\zeta_{2})$ for some $(\zeta_{1},\zeta_{2})$ with $|\zeta_{1}|\leq 1$ and $|\zeta_{2}|=1$. Thus we recover the values of $\rho_{u}$ on ${\mathbb{C}}^{2}$ from its values on $\partial P^{2}$.

Remark 4.5**.**

In the general case where

[TABLE]

where $a_{1},...,a_{d}$ are pairwise relatively prime, we have

[TABLE]

and we define

[TABLE]

so that

[TABLE]

for $(z_{1},...,z_{d})\in{\mathbb{C}}^{d}\setminus P^{d}$ and $|\lambda|\geq 1$. Then given $u\in L_{C}$, we define the $C-$Robin function of $u$ as

[TABLE]

for $(z_{1},...,z_{d})\in{\mathbb{C}}^{d}$.

We recall the Siciak-Zaharjuta formula (3.1) for $K\subset{\mathbb{C}}^{d}$ compact:

[TABLE]

For simplicity in notation, we write $\rho_{C,K}:=\rho_{V_{C,K}^{*}}$. The following result will be used in section 8.

Theorem 4.6**.**

Let $K\subset{\mathbb{C}}^{2}$ be nonpluripolar and satisfy

[TABLE]

Then $K=\{\rho_{C,K}\leq 0\}$ and $V_{C,K}^{*}=\rho^{+}_{C,K}:=\max[\rho_{C,K},0]$.

Proof.

We first define a $C-$homogeneous extremal function $H_{C,K}$ associated to a general compact set $K$. To this end, for each $n\in{\mathbb{N}}$ we define the collection of $nC$-homogeneous polynomials by

[TABLE]

Note that for $h_{n}\in H_{n}(C)$,

[TABLE]

and thus $u:=\frac{1}{n}\log|h_{n}|$ satisfies

[TABLE]

Define

[TABLE]

Then $H_{C,K}$ satisfies the property in (4.6). Clearly

[TABLE]

and hence $K\subset\{H_{C,K}\leq 0\}$.

For a polynomial $p\in Poly(nC)$, we write

[TABLE]

where $\tilde{h}_{l}(z_{1},z_{2}):=\sum_{aj+bk=l}c_{jk}z_{1}^{j}z_{2}^{k}$ satisfies

[TABLE]

Then for each $l=0,1,...,nab$,

[TABLE]

To prove (4.9), note that

[TABLE]

Take $(z_{1},z_{2})\in K$ at which $|\tilde{h}_{l}(z_{1},z_{2})|=||\tilde{h}_{l}||_{K}$. Then by the Cauchy estimates for $\lambda\to F(\lambda):=p(\lambda\circ(z_{1},z_{2}))$ on the unit circle,

[TABLE]

proving (4.9).

We define

[TABLE]

If $a=b=1$, $\tilde{H}_{l}=H_{l}(C)=H_{l}(\Sigma)$ are the usual homogeneous polynomials of degree $l$ in ${\mathbb{C}}^{d}$. Moreover, if $\tilde{h}_{l}\in\tilde{H}_{l}$, then $\tilde{h}_{l}^{ab}\in H_{l}(C)$. Since $||\tilde{h}_{l}||_{K}\leq 1$ if and only if $||\tilde{h}_{l}^{ab}||_{K}\leq 1$, this shows

[TABLE]

We define the $C-$homogeneous polynomial hull $\widehat{K}_{C}$ of a compact set $K$ as

[TABLE]

It is clear $\widehat{K}\subset\widehat{K}_{C}$ for any compact set $K$. We show the reverse inclusion, and hence equality, for $K$ satisfying (4.5). To this end, let $a\in\widehat{K}_{C}$. For $p\in Poly(nC)$, write $p=\sum_{l=0}^{nab}\tilde{h}_{l}$ as in (4.8). Then

[TABLE]

Thus

[TABLE]

Apply this to $p^{m}\in Poly(nmC)$:

[TABLE]

Letting $m\to\infty$, we obtain $|p(a)|\leq||p||_{K}$ and hence $a\in\widehat{K}$.

We use this to show

[TABLE]

for sets satisfying (4.5). To see this, we observe from (4.10) that the right-hand-side of (4.11) is the $C-$homogeneous polynomial hull $\widehat{K}_{C}$ of $K$ while the left-hand-side is the polynomial hull $\widehat{K}$ of $K$. Thus (4.11) follows from the previous paragraph.

Now we claim that $V_{C,K}^{*}=H_{C,K}^{+}$. We observed that $H_{C,K}^{+}\leq V_{C,K}\leq V_{C,K}^{*}$; for the reverse inequality, we observe that $H_{C,K}^{+}$ is in $L_{C}$ and since $H_{C,K}$ satisfies (4.6), we have $H_{C,K}^{+}$ is maximal outside $\widehat{K}$. From (4.11) we can apply the global domination principle (Proposition 2.2) to conclude that $H_{C,K}^{+}\geq V_{C,K}^{*}$ and hence $H_{C,K}^{+}=V_{C,K}^{*}$.

Using $H_{C,K}^{+}=V_{C,K}^{*}$,

[TABLE]

[TABLE]

for $(z_{1},z_{2})\in{\mathbb{C}}^{2}\setminus K$ by the invariance of $H_{C,K}$ (i.e., it satisfies (4.6)). Thus, from Proposition 4.3 (and the invariance of $\rho_{C,K}$) we have

[TABLE]

This shows $K=\{\rho_{C,K}\leq 0\}$ and $V_{C,K}^{*}=\rho^{+}_{C,K}:=\max[\rho_{C,K},0]$. ∎

Remark 4.7**.**

It follows that for

[TABLE]

where $h_{n}:=\tilde{h}_{nab}\in H_{n}(C)$ and $r_{n}=p-h_{n}=\sum_{aj+bk<nab}c_{jk}z_{1}^{j}z_{2}^{k}$ , if $u:=\frac{1}{n}\log|p_{n}|$ then

[TABLE]

We write $\widehat{p}_{n}:=h_{n}=\tilde{h}_{nab}$; thus $\rho_{u}=\frac{1}{n}\log|\widehat{p}_{n}|$.

In the case $a=b=1$ where $C=\Sigma$, we know from Corollary 4.6 of [7] that $K$ regular implies $\rho_{K}:=\rho_{\Sigma,K}$ is continuous. We need to know that for our triangles $C$ where $a,b$ are relatively prime positive integers we also have $\rho_{C,K}$ is continuous. To this end, we begin with the observation that applying Theorem 4.1 in the special case where $d=2$ and $C$ is our triangle with vertices $(0,0),(b,0),(0,a)$, we can take

[TABLE]

and $c=ab$ to obtain

[TABLE]

[TABLE]

[TABLE]

We use this connection between $\rho_{C,K}$ and the standard Robin function $\rho_{F^{-1}(K)}$ to show that $\rho_{C,K}$ is continuous if $K$ is regular.

Proposition 4.8**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and regular. Then $\rho_{C,K}$ is uniformly continuous on $\partial P^{2}$.

Proof.

With $F(z_{1},z_{2})=(z_{1}^{a},z_{2}^{b})$ as above, from Theorem 5.3.6 of [10], we have $F^{-1}(K)$ is regular. Thus, from Corollary 4.6 of [7], $\rho_{F^{-1}(K)}$ is continuous. Hence $\rho_{C,K}(z_{1}^{a},z_{2}^{b})=\rho_{C,K}(F(z_{1},z_{2}))$ is continuous. To show $\zeta\to\rho_{C,K}(\zeta)$ is continuous at $\zeta=(\zeta_{1},\zeta_{2})\in\partial P^{2}$, we use the fundamental relationship that

[TABLE]

To this end, let $\zeta^{n}=(\zeta^{n}_{1},\zeta^{n}_{2})\in\partial P^{2}$ converge to $\zeta=(\zeta_{1},\zeta_{2})$. Then

[TABLE]

for any $a-$th root $(\zeta^{n}_{1})^{1/a}$ of $\zeta^{n}_{1}$ and any $b-$th root $(\zeta^{n}_{2})^{1/b}$ of $\zeta^{n}_{2}$. But

[TABLE]

By continuity of $\rho_{F^{-1}(K)}$,

[TABLE]

[TABLE]

for the appropriate choice of $(\zeta_{1})^{1/a}$ and $(\zeta_{2})^{1/b}$. But

[TABLE]

Note that this also yields that the value of $\rho_{F^{-1}(K)}((\zeta_{1})^{1/a},(\zeta_{2})^{1/b})$ is independent of the choice of the roots $(\zeta_{1})^{1/a}$ and $(\zeta_{2})^{1/b}$. This can also be seen from the definitions of $\rho_{F^{-1}(K)}$ and $F$.

∎

Remark 4.9**.**

The relationship

[TABLE]

is a special case of a more general result. Let $u\in L_{C}$. Then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $\tilde{u}\in abL$, this last line is equal to the “usual” Robin function of $\tilde{u}$ in the sense of (1.5). To be precise, it is equal to $ab{\bf\rho_{\tilde{u}/ab}}$ where ${\bf\rho_{\tilde{u}/ab}}$ is the standard Robin function (1.5) of $\tilde{u}/ab\in L$. This observation will be crucial in section 6.

We need an analogue of formula (18) in [17] in order to verify a calculation in the next section. We follow the arguments in [17]. Recall we may lift the circle action on ${\mathbb{C}}^{2}$ to ${\mathbb{C}}^{3}$ via

[TABLE]

This gave a correspondence between $L_{C}({\mathbb{C}}^{2})$ and $L_{\tilde{C}}({\mathbb{C}}^{3})$ where $\tilde{C}=co\{(0,0,0),(1,0,0),(0,b,0),(0,0,a)\}$. In analogy with our class

[TABLE]

in ${\mathbb{C}}^{2}$, we can consider

[TABLE]

in ${\mathbb{C}}^{3}$. For $h_{n}\in H_{n}(\tilde{C})$, we have

[TABLE]

belongs to $L_{\tilde{C}}({\mathbb{C}}^{3})$ and $u_{n}$ is $ab-$log-homogeneous. That $u_{n}\in L_{\tilde{C}}({\mathbb{C}}^{3})$ is clear; to show the $ab-$log-homogeneity, note that

[TABLE]

[TABLE]

[TABLE]

so that

[TABLE]

Moreover, for $h_{n}\in H_{n}(\tilde{C})$, the polynomial

[TABLE]

conversely, if $p_{n}(z_{1},z_{2})=\sum_{(j,k):aj+bk\leq nab}c_{jk}z_{1}^{j}z_{2}^{k}\in Poly(nC)$ then

[TABLE]

Next, given a compact set $E\subset{\mathbb{C}}^{3}$, we define the $ab-$log-homogeneous $\tilde{C}-$extremal function

[TABLE]

and its usc regularization $H_{\tilde{C},E}^{*}$. Given the one-to-one correspondence between $Poly(nC)$ in ${\mathbb{C}}^{2}$ and $H_{n}(\tilde{C})$ in ${\mathbb{C}}^{3}$, we see that for $K\subset{\mathbb{C}}^{2}$ compact,

[TABLE]

and hence a similar equality holds for the usc regularizations of both sides. Using this, we observe that for $\zeta=(\zeta_{1},\zeta_{2})\not=(0,0)$, we have

[TABLE]

[TABLE]

[TABLE]

Here we have used the fact that

[TABLE]

We state this as a proposition:

Proposition 4.10**.**

For $K\subset{\mathbb{C}}^{2}$ compact,

[TABLE]

Remark 4.11**.**

Using the relation (4.12) and following the reasoning in [15], Proposition 2.3, it follows that a compact set $K\subset{\mathbb{C}}^{2}$ is regular; i.e., $V_{C,K}$ is continuous in ${\mathbb{C}}^{2}$, if and only if $H_{\tilde{C},\{1\}\times K}$ is continuous in ${\mathbb{C}}^{3}$. Thus we get an alternate proof of Proposition 4.8.

5. Preliminary results: Triangle case

We continue to let $C$ be the triangle with vertices at $(0,0),(b,0)$, and $(0,a)$ where $a,b$ are relatively prime positive integers. For $K\subset{\mathbb{C}}^{2}$ compact and $\zeta:=(\zeta_{1},\zeta_{2})\in\partial P^{2}$, we define Chebyshev constants

[TABLE]

We note that $\kappa_{n+m}\leq\kappa_{n}\kappa_{m}$: if we take $t_{n},t_{m}$ achieving $\kappa_{n},\kappa_{m}$, then $t_{n}t_{m}\in Poly(n+m)C$ and $\widehat{t_{n}t_{m}}=\widehat{t}_{n}\widehat{t}_{m}$ (see Remark 4.7) so that

[TABLE]

Thus $\lim_{n\to\infty}\kappa_{n}^{1/n}$ exists and we set

[TABLE]

The following relation between $\kappa(K,\zeta)$ and $\rho_{C,K}(\zeta)$ is analogous to Proposition 4.2 of [14].

Proposition 5.1**.**

For $\zeta\in\partial P^{2}$,

[TABLE]

Proof.

We first note that

[TABLE]

[TABLE]

Thus for any $p_{n}\in Poly(nC)$ with $||p_{n}||_{K}\leq 1$, $\kappa_{n}(K,\zeta)\leq\frac{1}{|\widehat{p}_{n}(\zeta)|}$. For such $p_{n}$, $\frac{1}{n}\log|p_{n}(z)|\leq V_{C,K}(z)$ for all $z\in{\mathbb{C}}^{2}$ so that

[TABLE]

for all $\zeta\in\partial P^{2}$. Taking the infimum over all such $p_{n}$,

[TABLE]

for all $n$; taking the limit as $n\to\infty$ gives

[TABLE]

To prepare for the reverse inequality, we let $\{b_{j}\}$ be an orthonormal basis of $\bigcup_{n}Poly(nC)$ in $L^{2}(\mu)$ where $\mu$ is any Bernstein-Markov measure for $K$: thus for any $\epsilon>0$, there exists a constant $c_{\epsilon}$ so that

[TABLE]

In particular,

[TABLE]

and from Corollary 3.2,

[TABLE]

We next show that for $\zeta\in\partial P^{2}$,

[TABLE]

For one inequality, we use the fact that for a function $u$ subharmonic on ${\mathbb{C}}$ with $u\in L$, the function $r\to\max_{|t|=r}u(t)$ is a convex function of $\log r$. Hence

[TABLE]

Thus if $u\in abL$; i.e., $u(z)-ab\log|z|=0(1),\ |z|\to\infty$, we have

[TABLE]

Fix $\zeta\in\partial P^{2}$ and letting $d_{j}:=\deg_{C}(b_{j})$ apply this to the function

[TABLE]

We obtain (using also Remark 4.7), for any $r$,

[TABLE]

[TABLE]

Thus

[TABLE]

[TABLE]

where we used Hartogs lemma and (5.1). Thus, letting $r\to\infty$,

[TABLE]

In order to prove the reverse inequality in (5.2), we use Proposition 4.10. With the notation from the previous section, and following the proof of Théorème 2 in [17], let $h\in H_{n}(\tilde{C})$ with $||h||_{1\times K}\leq 1$. Then

[TABLE]

with $||p||_{K}\leq 1$. Writing $p=\sum_{j=1}^{N_{n}}c_{j}b_{j}$ where $N_{n}=$dim$(Poly(nC))$ as in the proof of Proposition 3.1, we have $|c_{j}|\leq 1$ and hence

[TABLE]

Then

[TABLE]

Fixing $(z_{1},z_{2})=(\zeta_{1},\zeta_{2})$ and letting $|\lambda|\to\infty$, we get

[TABLE]

where $N_{n-1}\leq j_{n}\leq N_{n}$. Using Proposition 4.10 we conclude that

[TABLE]

and (5.2) is proved.

We now use (5.2) to prove that $\kappa(K,\zeta)\leq e^{-\rho_{C,K}(\zeta)}$ for $\zeta\in\partial P^{2}$ which will finish the proof of the proposition. Fixing such a $\zeta$ and $\epsilon>0$, take a subsequence $\{b_{k_{j}}\}$ with $d_{j}:=\deg_{C}(b_{k_{j}})$ such that

[TABLE]

Letting

[TABLE]

we have $||p_{j}||_{K}\leq 1$ and

[TABLE]

Thus

[TABLE]

[TABLE]

Letting $j\to\infty$,

[TABLE]

which holds for all $\epsilon>0$. Letting $\epsilon\to 0$ completes the proof.

∎

Using this proposition, and the observation within its proof that

[TABLE]

we obtain a result which will be useful in proving Theorem 5.3.

Corollary 5.2**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and regular. Given $\epsilon>0$, there exists a positive integer $m$ and a finite set of polynomials $\{W_{1},...,W_{s}\}\subset Poly(mC)$ such that $||W_{j}||_{K}=1,\ j=1,...,s$ and

[TABLE]

Proof.

From Proposition 5.1, given $\epsilon>0$, for each $\zeta\in\partial P^{2}$ we can find a polynomial $p\in Poly(nC)$ for $n\geq n_{0}(\epsilon)$ with $||p||_{K}=1$ and

[TABLE]

By continuity of $\rho_{K,C}$, which follows from Proposition 4.8, such an inequality persists in a neighborhood of $\zeta$. We take a finite set $\{p_{1},...,p_{s}\}$ of such polynomials with $p_{i}\in Poly(n_{i}C)$ such that

[TABLE]

Raising the $p_{i}$’s to powers to obtain $W_{i}$’s of the same $C-$degree $m$, we still have $||W_{i}||_{K}=1$ and

[TABLE]

∎

Given $K\subset{\mathbb{C}}^{2}$ compact, and given $h_{n}\in H_{n}(C)$, we define

[TABLE]

and $||Tch_{K}h_{n}||_{K}=\inf\{||h_{n}+q_{n-1}||_{K}:q_{n-1}\in Poly(n-1)C\}$. The polynomial $Tch_{K}h_{n}$ need not be unique but each such polynomial yields the same value of $||Tch_{K}h_{n}||_{K}$. The next result is similar to Theorem 3.2 of [4].

Theorem 5.3**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact, regular and polynomially convex. If $\{Q_{n}\}$ is a sequence of polynomials with $Q_{n}\in H_{n}(C)$ satisfying

[TABLE]

then

[TABLE]

Proof.

We follow the proof in [15]. Given $\epsilon>0$, we start with polynomials $\{W_{1},...,W_{s}\}\subset Poly(mC)$ such that $||W_{j}||_{K}=1,\ j=1,...,s$ and

[TABLE]

(and hence on all of ${\mathbb{C}}^{2}$) from Corollary 5.2. From the hypotheses on $\{Q_{n}\}$ and the continuity of $\rho_{C,K}$ (Proposition 4.8), we apply Hartogs lemma to conclude

[TABLE]

Thus

[TABLE]

Note that $Q_{n}\in H_{n}(C)$ implies $Q_{n}(\lambda\circ\zeta)=\lambda^{nab}Q_{n}(\zeta)$ so that

[TABLE]

Similary $\widehat{W}_{j}\in H_{m}(C)$ implies

[TABLE]

so that (5.4) holds on all of ${\mathbb{C}}^{2}$.

We fix $R>1$ and define

[TABLE]

Since $\widehat{W}_{j}(\lambda\circ\zeta)=\lambda^{mab}\widehat{W}_{j}(\zeta)$, we have $e^{i\theta}\circ G=G$; since $\widehat{W}_{j}(0)=0$, we have $0\in G$. We claim $G$ is bounded. To see this, choose $r>0$ so that

[TABLE]

Then $V_{C,K}(z_{1},z_{2})\geq H_{C}(z_{1}/r,z_{2}/r)$ and hence

[TABLE]

Since

[TABLE]

$G$ is bounded.

Next, choose $\delta>0$ sufficiently large so that

[TABLE]

Define

[TABLE]

Given $\theta>1$, we can choose $p>0$ sufficiently large so that

[TABLE]

which is complete circled (in the ordinary sense) and strictly pseudoconvex, satisfies

[TABLE]

(note this is just a replacement of an $l^{\infty}-$norm with an $l^{p}-$norm).

We write $z:=(z_{1},z_{2})\in{\mathbb{C}}^{2}$ and $\widehat{W}(z):=(\widehat{W}_{1}(z),...,\widehat{W}_{s}(z))\in{\mathbb{C}}^{s}$ for simplicity in notation. Let

[TABLE]

Then $Y$ is a closed, complex submanifold of ${\mathbb{C}}^{2}\times{\mathbb{C}}^{s}$. Appealing to the bounded, holomorphic extension result stated as Theorem 3.1 in [15], there exists a positive constant $M$ such that for every $f\in H^{\infty}(Y\cap D)$ there exists $F\in H^{\infty}(D)$ with

[TABLE]

We will apply this to the polynomials $Q_{n}(z)$. First, we observe that if $\pi:{\mathbb{C}}^{2}\times{\mathbb{C}}^{s}\to{\mathbb{C}}^{2}$ is the projection $\pi(z,w)=z$, then

[TABLE]

To see the last inclusion – note we use $\theta\circ G$, not $\theta G$ – first note that

[TABLE]

and thus since $\widehat{W}_{j}(\frac{1}{\theta}\circ s)=\frac{1}{\theta^{mab}}\widehat{W}_{j}(s)$,

[TABLE]

On the other hand,

[TABLE]

[TABLE]

Applying the bounded holomorphic extension theorem to $f(z,w):=Q_{n}(z)$ for each $n$, we get $F_{n}(z,w)\in H^{\infty}(D)$ with

[TABLE]

for all $z\in\pi(Y\cap D)$ and

[TABLE]

Utilizing the set inclusion $\pi(Y\cap D)\subset\theta\circ G$, the definition of $\theta\circ G$ and (5.4) (which recall is valid on all of ${\mathbb{C}}^{2}$),

[TABLE]

Since $D$ is complete circled, we can expand $F_{n}$ into a series of homogeneous polynomials which converges locally uniformly on all of $D$. Rearranging into a multiple power series, we write

[TABLE]

Using $Q_{n}(z)=F_{n}(z,\delta\widehat{W}(z))$ for $z\in\pi(Y\cap D)$, we obtain for such $z$,

[TABLE]

where the prime denotes that the sum is taken over multiindices

[TABLE]

This is because $Q_{n}\in H_{n}(C)$ and $\widehat{W}_{j}\in H_{m}(C),\ j=1,...,s$. Precisely, each $\widehat{W}_{j}(z)$ is of the form $\sum_{a\alpha+b\beta=mab}c_{\alpha\beta}z_{1}^{\alpha}z_{2}^{\beta}$ so that if $J=(j_{1},...,j_{s})$, a typical monomial occurring in $Q_{n}(z)$ must be of the form

[TABLE]

where $a\alpha_{k}+b\beta_{k}=mab,\ k=1,...,s$; hence

[TABLE]

In order for (5.7) to (possibly) appear in $Q_{n}(z)$, we require (5.6). The positive integers $i$ in (5.6) are related to the lengths $|I|$ by $|I|=i_{1}+i_{2}\leq ai_{1}+bi_{2}=iab$; and if, say, $a\leq b$ we have a reverse estimate

[TABLE]

However, all we will need to use is the fact that the number of multiindices occurring in the sum for $Q_{n}(z)$ is at most $N_{n}=$ dim$(Poly(nC))$ and $\lim_{n\to\infty}N_{n}^{1/n}=1$.

Applying the Cauchy estimates on the polydisk $\Delta\subset D$, we obtain

[TABLE]

for $n\geq n_{0}(\epsilon)$.

We now define

[TABLE]

From (5.6) and the previously observed fact that

[TABLE]

we have $\widehat{p}_{n}(z)=Q_{n}(z)$. Using the estimates (5.5), (5.8), the facts that $||W_{j}||_{K}=1,\ j=1,...,s$ and

[TABLE]

we obtain

[TABLE]

[TABLE]

[TABLE]

where $C_{n}$ can be taken as $M$ times the cardinality of the set of multiindices in (5.6). Clearly $\lim_{n\to\infty}C_{n}^{1/n}=1$ so that

[TABLE]

Since $\epsilon>0,\ R>1$ and $\theta>1$ were arbitrary, the result follows.

∎

6. The integral formula

In the standard setting of the Robin function ${\bf\rho_{u}}$ associated to $u\in L({\mathbb{C}}^{2})$ (cf., 1.5), for $z=(z_{1},z_{2})\not=(0,0)$ we can define

[TABLE]

so that $\underline{\bf\rho_{u}}(tz)=\underline{\bf\rho_{u}}(z)$ for $t\in{\mathbb{C}}\setminus\{0\}$. Thus we can consider $\underline{\bf\rho_{u}}$ as a function on ${\mathbb{P}}^{1}$ where to $p=(p_{1},p_{2})\in\partial P^{2}$ we associate the point where the complex line $\lambda\to\lambda p$ hits $H_{\infty}$. The integral formula Theorem 5.5 of [3] in this setting is the following.

Theorem 6.1**.**

(Bedford-Taylor)* Let $u,v,w\in L^{+}({\mathbb{C}}^{2})$. Then*

[TABLE]

where $\Omega$ is the standard Kähler form on ${\mathbb{P}}^{1}$.

We use this to develop an integral formula for $u,v,w\in L_{C,+}$. Letting

[TABLE]

we recall that for $u\in L_{C}$, we have

[TABLE]

[TABLE]

where ${\bf\rho_{\tilde{u}/ab}}$ is the standard Robin function of $\tilde{u}/ab\in L$. It follows from the calculations in Remark 4.9 that if $u\in L_{C,+}$ then $\tilde{u}\in abL^{+}$. From (6.1), if $u,v,w\in L_{C,+}$,

[TABLE]

We apply Theorem 6.1 to the right-hand-side, multiplying by factors of $ab$ since $\tilde{u},\tilde{v},\tilde{w}\in abL^{+}$, to obtain the desired integral formula:

[TABLE]

Corollary 6.2**.**

Let $u,v\in L_{C,+}$ with $u\geq v$. Then

[TABLE]

[TABLE]

Proof.

From (6.3)

[TABLE]

[TABLE]

We observe that

[TABLE]

Using this and the hypothesis $u\geq v$ gives the result. ∎

We also obtain a generalization of Theorem 6.9 of of [3]:

Corollary 6.3**.**

Let $E,F$ be nonpluripolar compact subsets of ${\mathbb{C}}^{2}$ with $E\subset F$. We have $\rho_{C,E}=\rho_{C,F}$ if and only if $V_{C,E}^{*}=V_{C,F}^{*}$ and $\widehat{E}=\widehat{F}\setminus P$ where $P$ is pluripolar.

Proof.

The “if” direction is obvious. For “only if” we may assume $E=\widehat{E}$ and $F=\widehat{F}$ since $V_{C,K}^{*}=V_{C,\widehat{K}}^{*}$ for $K$ compact. It suffices to show $V_{C,E}^{*}\leq V_{C,F}^{*}$ as $E\subset F$ gives the reverse inequality. We have

[TABLE]

since $V_{C,F}^{*}=0$ q.e. on $F$ (and hence a.e-$(dd^{c}V_{C,E}^{*})^{2}$). Applying Corollary 6.2 with $u=V_{C,E}^{*}$ and $v=V_{C,F}^{*}$, the right-hand-side of the displayed inequality is nonpositive since $\rho_{C,E}=\rho_{C,F}$ implies ${\bf\rho_{\tilde{V}_{C,E}^{*}/ab}}={\bf\rho_{\tilde{V}_{C,F}^{*}/ab}}$ on ${\mathbb{C}}^{2}$ by (6.2) so that $\underline{\bf\rho_{\tilde{V}_{C,E}^{*}/ab}}=\underline{\bf\rho_{\tilde{V}_{C,F}^{*}/ab}}$ on ${\mathbb{P}}^{1}$. Hence $\int_{{\mathbb{C}}^{2}}V_{C,E}^{*}(dd^{c}V_{C,F}^{*})^{2}=0$. We conclude that $V_{C,E}^{*}=0$ a.e-$(dd^{c}V_{C,F}^{*})^{2}$. By Proposition 2.2, $V_{C,E}^{*}\leq V_{C,F}^{*}$. Then $\widehat{E}=\widehat{F}\setminus P$ follows since $\{z\in{\mathbb{C}}^{2}:V_{C,E}^{*}=0\}$ differs from $E=\widehat{E}$ by a pluripolar set.

∎

Again using Corollary 6.2 and Proposition 2.2, we get an analogue of Lemma 2.1 in [4], which is the key result for all that follows.

Corollary 6.4**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and nonpluripolar and let $v\in L_{C}$ with $v\leq 0$ on $K$. Suppose that $\rho_{v}=\rho_{C,K}$ on $\partial P^{2}$. Then $v=V_{C,K}^{*}$ on ${\mathbb{C}}^{2}\setminus\widehat{K}$.

Proof.

Fix a constant $c$ so that $H_{C}(z)<c$ on $K$ and let

[TABLE]

Then $w\in L_{C,+}$ with $w=0$ on $\widehat{K}$ and since $H_{C}-c\leq V_{C,K}$ we have $\rho_{w}=\rho_{C,K}$ on $\partial P^{2}$. Then $\rho_{w}=\rho_{C,K}$ on ${\mathbb{C}}^{2}$ (see Remark 4.4) and by (6.2), ${\bf\rho_{\tilde{w}/ab}}={\bf\rho_{\tilde{V}_{C,K}/ab}}$ on ${\mathbb{C}}^{2}$. Thus $\underline{\bf\rho_{\tilde{w}/ab}}=\underline{\bf\rho_{\tilde{V}_{C,K}/ab}}$ on ${\mathbb{P}}^{1}$. Since $w\leq V_{C,K}^{*}$, by Corollary 6.2,

[TABLE]

the last equality due to $w=0$ on supp$(dd^{c}V_{C,K}^{*})^{2}$. Thus $V_{C,K}^{*}=0$ a.e.-$(dd^{c}w)^{2}$ and hence $V_{C,K}^{*}\leq w$ a.e.-$(dd^{c}w)^{2}$. By Proposition 2.2, $V_{C,K}^{*}\leq w$ on all of ${\mathbb{C}}^{2}$. Since $V_{C,K}^{*}\geq H_{C}-c$, $v=V_{C,K}^{*}$ on ${\mathbb{C}}^{2}\setminus\widehat{K}$.

∎

As in Theorem 2.1 in [4], we get a sufficient condition for a sequence of polynomials to recover the $C-$extremal function of $K$ outside of $\widehat{K}$. This will be used in section 8.

Theorem 6.5**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and nonpluripolar. Let $\{p_{j}\}$ be a sequence of polynomials, $p_{j}\in Poly(d_{j}C)$, with $\deg_{C}(p_{j})=d_{j}$ such that

[TABLE]

[TABLE]

Then

[TABLE]

Remark 6.6**.**

Given an orthonormal basis $\{b_{j}\}$ of $\bigcup_{n}Poly(nC)$ in $L^{2}(\mu)$ where $\mu$ is a Bernstein-Markov measure for $K$, using

[TABLE]

from Corollary 3.2, in the proof of Proposition 5.1 we showed

[TABLE]

Theorem 6.5 is a type of reverse implication.

Proof.

The function

[TABLE]

is psh in ${\mathbb{C}}^{2}$. Given $\epsilon>0$,

[TABLE]

Thus

[TABLE]

We conclude that $v\in L_{C}$ and $v\leq V_{C,K}$. Hence $\rho_{v}\leq\rho_{C,K}$.

From Corollary 6.4, to show $v=V_{C,K}^{*}$ outside $\widehat{K}$ it suffices to show $\rho_{v}\geq\rho_{C,K}$ on $\partial P^{2}$. We use the argument from Proposition 5.1. Recall from (5.3) for $u\in abL({\mathbb{C}})$ we have

[TABLE]

Fix $\zeta\in\partial P^{2}$ and apply the above to the function

[TABLE]

We see that for any $r$

[TABLE]

[TABLE]

Thus

[TABLE]

[TABLE]

where we used Hartogs lemma. Thus, letting $r\to\infty$,

[TABLE]

Since $\rho_{v}$ is usc, \bigl{(}\limsup_{j\to\infty}\frac{1}{d_{j}}\log|\widehat{p}_{j}(\zeta)|\bigr{)}^{*}\leq\rho_{v}(\zeta) and using the hypothesis (6.4) finishes the proof.

∎

7. $C-$transfinite diameter and directional Chebyshev constants

From [13], we have a Zaharjuta-type proof of the existence of the limit

[TABLE]

(see section 2) in the definition of $C-$transfinite diameter $\delta_{C}(K)$ of a compact set $K\subset{\mathbb{C}}^{d}$ where $C$ satisfies (1.2). In the classical ($C=\Sigma$) case, Zaharjuta [16] verified the existence of the limit in (7.1) by introducing directional Chebyshev constants $\tau(K,\theta)$ and proving

[TABLE]

where $\sigma:=\{(x_{1},...,x_{d})\in{\mathbb{R}}^{d}:0\leq x_{i}\leq 1,\ \sum_{j=1}^{d}x_{i}=1\}$ is the extreme “face” of $\Sigma$; $\sigma^{0}=\{(x_{1},...,x_{d})\in{\mathbb{R}}^{d}:0<x_{i}<1,\ \sum_{j=1}^{d}x_{i}=1\}$; and $|\sigma|$ is the $(d-1)-$dimensional measure of $\sigma$.

In [13], a slight difference with the classical setting is that we have

[TABLE]

where the directional Chebyshev constants $\tau(K,\theta)$ and the integration in the formula are over the entire $d-$dimensional convex body $C$. Moreover in the definition of $\tau(K,\theta)$ the standard grevlex ordering $\prec$ on $({\mathbb{Z}}^{+})^{d}$ (i.e., on the monomials in ${\mathbb{C}}^{d}$) was used. This was required to obtain the submultiplicativity of the “monic” polynomial classes

[TABLE]

and corresponding Chebyshev constants

[TABLE]

However, in our triangle setting, following [13] we can also define an ordering $\prec_{C}$ on $({\mathbb{Z}}^{+})^{2}$ by $\alpha=(\alpha_{1},\alpha_{2})\prec_{C}\beta=(\beta_{1},\beta_{2})$ if

1. (1)

$\deg_{C}(z^{\alpha})<\deg_{C}(z^{\beta})$ or 2. (2)

when $\deg_{C}(z^{\alpha})=\deg_{C}(z^{\beta})$ we have $\alpha_{2}<\beta_{2}$.

Then

1. (1)

one has submultiplicativity of the corresponding “monic” polynomial classes defined as in (7.3) using $\prec_{C}$ (which we denote $M^{\prec_{C}}_{k}(\alpha)$) and one gets the formula (7.2) with $\theta\to\tau(K,\theta)$ continuous; and 2. (2)

if $\phi=(\phi_{1},\phi_{2})$ is on the open hypotenuse $\mathcal{C}$ of $C$; i.e., $a\phi_{1}+b\phi_{2}=ab$ with $\phi_{1}\phi_{2}>0$, and if $\phi=r\theta$ where $\theta$ lies on the interior of $C$ and $r>1$, then $r\log\tau(K,\theta)=\log\tau(K,\phi)$ (see [12], Lemma 5.4). (Note $\mathcal{C}=\sigma^{0}$ if $C=\Sigma\subset({\mathbb{R}}^{+})^{2}$).

As a consequence, in our triangle case $C=co\{(0,0),(b,0),(0,a)\}$,

[TABLE]

where the directional Chebyshev constants $\tau(K,\theta)$ in (7.4) and the integration in the formula are over $\mathcal{C}$ and $\theta\to\tau(K,\theta)$ is continuous on $\mathcal{C}$. In what follows, we fix our triangle $C$ and use this $\prec_{C}$ ordering to define these directional Chebyshev constants

[TABLE]

for $\theta\in\mathcal{C}$ (in which case the limit exists) where

[TABLE]

Using (7.4) we have a result similar to Proposition 3.1 in [5]:

Proposition 7.1**.**

For $E\subset F$ compact subsets of ${\mathbb{C}}^{2}$,

1. (1)

for all $\theta\in\mathcal{C}$, $\tau(E,\theta)\leq\tau(F,\theta)$ and 2. (2)

$\delta_{C}(E)=\delta_{C}(F)$* if and only if $\tau(E,\theta)=\tau(F,\theta)$ for all $\theta\in\mathcal{C}$.*

8. Polynomials approximating $V_{C,K}$

Following [5], given a nonpluripolar compact set $K\subset{\mathbb{C}}^{2}$ and $\theta\in\mathcal{C}$, a sequence of polynomials $\{Q_{n}\}$ is $\theta-$asymptotically Chebyshev (we write $\theta aT$) for $K$ if

1. (1)

for each $n$ there exists $k_{n}\in{\mathbb{Z}}^{+}$ and $\alpha_{n}$ with $Q_{n}\in M^{\prec_{C}}_{k_{n}}(\alpha_{n})$; 2. (2)

$\lim_{n\to\infty}k_{n}=+\infty$ and $\lim_{n\to\infty}\frac{\alpha_{n}}{k_{n}}=\theta$; and 3. (3)

$\lim_{n\to\infty}||Q_{n}||_{K}^{1/k_{n}}=\tau(K,\theta).$

Proposition 8.1**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and nonpluripolar and satisfy $e^{i\theta}\circ K=K$. Let $\{Q_{n}\}$ be $\theta aT$ for $K$.Then $\{\widehat{Q}_{n}\}$ is $\theta aT$ for $K$.

Proof.

This follows from (4.9) giving $||\widehat{Q}_{n}||_{K}\leq||Q_{n}||_{K}$ for such $K$. ∎

Given $K\subset{\mathbb{C}}^{2}$ compact and nonpluripolar, define

[TABLE]

Note that if $e^{i\theta}\circ K=K$ then Theorem 4.6 shows that $K=K_{\rho}$. Moreover, from Remark 4.4,

[TABLE]

so that $e^{i\theta}\circ K_{\rho}=K_{\rho}$ and $V_{C,K_{\rho}}^{*}=\rho^{+}_{C,K}:=\max[0,\rho_{C,K}]$.

Theorem 8.2**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and regular and let $\{Q_{n}\}$ be $\theta aT$ for $K$ with $Q_{n}\in M^{\prec_{C}}_{k_{n}}(\alpha_{n})$.Then $\{\widehat{Q}_{n}\}$ is $\theta aT$ for $K_{\rho}$. Conversely, if $\{H_{n}\}$ is $\theta aT$ for $K_{\rho}$ with $H_{n}\in H_{k_{n}}(C)\cap M^{\prec_{C}}_{k_{n}}(\alpha_{n})$, then the sequence $\{Tch_{K}H_{n}\}$ is $\theta aT$ for $K$. Moreover,

[TABLE]

Proof.

Given $\{Q_{n}\}$ which are $\theta aT$ for $K$ with $Q_{n}\in M^{\prec_{C}}_{k_{n}}(\alpha_{n})$, we have

[TABLE]

Thus

[TABLE]

Hence for $z\in K_{\rho}$,

[TABLE]

Since

[TABLE]

[TABLE]

On the other hand, considering $\{H_{n}\}$ which are $\theta aT$ for $K_{\rho}$ (we can assume $H_{n}\in H_{k_{n}}(C)$ from Proposition 8.1) we have

[TABLE]

Thus

[TABLE]

[TABLE]

By rescaling $K$; i.e., replacing $K$ by $rK$ for appropriate $r\geq 1$ if need be, we can assume that

[TABLE]

In particular, $\rho_{C,K}\geq 0$ on $\partial P^{2}$. From Theorem 5.3 we conclude that

[TABLE]

We have $Tch_{K}H_{n}\in M^{\prec_{C}}_{k_{n}}(\alpha_{n})$ since $H_{n}\in H_{k_{n}}(C)\cap M^{\prec_{C}}_{k_{n}}(\alpha_{n})$ and hence

[TABLE]

Together with (8.1) we conclude that $\tau(K_{\rho},\theta)=\tau(K,\theta)$ and the inequalities in (8.1) and (8.2) are equalities.

∎

Finally, we utilize Theorems 8.2 and 6.5 together with Propositions 2.3 and 7.1 to prove our main result.

Theorem 8.3**.**

Let $K\subset{\mathbb{C}}^{2}$ be compact and regular. Let $\{p_{n}\}$ be a countable family of polynomials with $p_{n}\in Poly(k_{n}C)$ such that for every $\theta\in\mathcal{C}$, there is a subsequence which is $\theta aT$ for $K$. Then

[TABLE]

Proof.

Let

[TABLE]

Clearly $v\leq V_{C,K}$ on all of ${\mathbb{C}}^{2}$. Let

[TABLE]

To finish the proof, it suffices, by Theorem 6.5, to show that

[TABLE]

Clearly $w\leq\rho_{C,K}$ in ${\mathbb{C}}^{2}$ since $v\leq V_{C,K}$. To show the reverse inequality, we proceed as follows. Let

[TABLE]

Then $Z$ is open since $w$ is usc. We claim that $int(K_{\rho})\subset Z$. For if $z\in int(K_{\rho})$, we have $\rho_{C,K}(z)=-a<0$. Thus $w(z)\leq-a<0$ and $z\in Z$. Moreover, both sets $K_{\rho}$ and $Z$ satisfy the invariance property

[TABLE]

(for $Z$ this follows since $\widehat{p}_{n}\in H_{k_{n}}(C)$). Thus to show the equality $w(z)=\rho_{C,K}(z)$ it suffices to verify the equality

[TABLE]

Suppose this is false. Then we take a point $z_{0}\in\partial K_{\rho}\cap Z$ and a closed ball $B$ centered at $z_{0}$ contained in $Z$. Since $B$ is regular and $K$ is assumed regular, by Proposition 4.8 together with Lemma 4.1 of [5], $B\cup K_{\rho}$ is regular.

Given $\theta\in\mathcal{C}$, by assumption there exists a subsequence ${\mathbb{N}}_{\theta}\subset{\mathbb{N}}$ such that $\{p_{n}\}_{n\in{\mathbb{N}}_{\theta}}$ is $\theta aT$ for $K$. From Theorem 8.2, $\{\widehat{p}_{n}\}_{n\in{\mathbb{N}}_{\theta}}$ is $\theta aT$ for $K_{\rho}$. Since $w\leq 0$ on $B\cup K_{\rho}$, for $z\in B\cup K_{\rho}$ we have

[TABLE]

Using Hartogs lemma, we conclude that

[TABLE]

Hence

[TABLE]

for all $\theta\in\mathcal{C}$. Since $\tau(B\cup K_{\rho},\theta)\geq\tau(K_{\rho},\theta)$ we see that

[TABLE]

for all $\theta\in\mathcal{C}$. From Propositions 2.3 and 7.1 (and regularity of the sets $K_{\rho},\ B\cup K_{\rho}$),

[TABLE]

But $V_{C,K_{\rho}}=\rho_{C,K}^{+}=\max[0,\rho_{C,K}]$ thus if $B\setminus K_{\rho}\not=\emptyset$, since $\rho_{C,K}>0$ on ${\mathbb{C}}^{2}\setminus K_{\rho}$ and $V_{C,B\cup K_{\rho}}=0$ on $B\cup K_{\rho}$, this is a contradiction.

∎

As examples of sequences of polynomials satisfying the hypotheses of Theorem 8.3, as in [5] we have

1. (1)

the family $\{t_{k,\alpha}\in M^{\prec_{C}}_{k}(\alpha)\}_{k,\alpha}$ of Chebyshev polynomials (minimial supremum norm) for $K$ in these classes; 2. (2)

for a Bernstein-Markov measure $\mu$ on $K$, the corresponding polynomials $\{q_{k,\alpha}\in M^{\prec_{C}}_{k}(\alpha)\}_{k,\alpha}$ of minimal $L^{2}(\mu)$ norm (see Corollary 3.2); 3. (3)

any sequence $p_{\alpha(s)}=z^{\alpha(s)}-L_{\alpha(s-1)}(z^{\alpha(s)})$ where $\{z^{\alpha(s)}\}$ is an enumeration of monomials with the $\prec_{C}$ order and $L_{\alpha(s-1)}(z^{\alpha(s)})$ is the Lagrange interpolating polynomial for the monomial $z^{\alpha(s)}$ at points $\{z_{s-1,j}\}_{j=1,...s-1}$ in the $(s-1)-$st row in a triangular array $\{z_{jk}\}_{j=1,2,...;\ k=1,...j}\subset K$ where the Lebesgue constants $\Lambda_{\alpha(s)}$ associated to the array grow subexponentially. Here,

[TABLE]

where l_{sj}\in Poly\bigl{(}\deg_{C}(z^{\alpha(s)})C\bigr{)} satisfies $l_{sj}(z_{sk})=\delta_{jk},\ j,k=1,...,s$ and we require

[TABLE]

We refer to Corollary 4.4 of [5] for details.

Remark 8.4**.**

Example (3) includes the case of a sequence of $C-$Fekete polynomials for $K$ (cf., p 1562 of [5]). The case of $C-$Leja polynomials for $K$, defined using $C-$Leja points as in [13], also satisfy the hypotheses of Theorem 8.3. This can be seen by following the proof of Corollary 4.5 in [5]. The proof that $C-$Leja polynomials satisfy the analogue of (4.28) in [5] is given in Theorem 1.1 of [13].

9. Further directions

We reiterate that the arguments given in the note for triangles $C$ in ${\mathbb{R}}^{2}$ with vertices $(0,0),(b,0),(0,a)$ where $a,b$ are relatively prime positive integers should generalize to the case of a simplex

[TABLE]

in $({\mathbb{R}}^{+})^{d}$ with $a_{1},...,a_{d}$ pairwise relatively prime using the definition of the $C-$Robin function in Remark 4.5. Indeed, following the arguments on pp. 72-82 of [6] one should also be able to prove weighted versions of the $C-$Robin results for such simplices $C$ in ${\mathbb{R}}^{d}$. We indicate the transition from the $C-$weighted situation for $d=2$ to a $\tilde{C}-$homogeneous unweighted situation for $d=3$. As in section 4, we lift the circle action on ${\mathbb{C}}^{2}$,

[TABLE]

to ${\mathbb{C}}^{3}$ via

[TABLE]

Given a compact set $K\subset{\mathbb{C}}^{2}$ and an admissible weight function $w\geq 0$ on $K$, i.e., $w$ is usc and $\{z\in K:w(z)>0\}$ is not pluripolar, we associate the set

[TABLE]

It follows readily that

[TABLE]

Setting $\tilde{C}=co\{(0,0,0),(1,0,0),(0,b,0),(0,0,a)\}$, we can relate a weighted $C-$Robin function $\rho^{w}_{C,K}$ to the $\tilde{C}-$Robin function $\rho_{\tilde{C},\tilde{K}_{w}}$. Using these weighted ideas, the converse to Proposition 2.3 should follow as in [6].