Polynomials with exponents in compact convex sets and associated weighted extremal functions -
Approximations and regularity

A useful tool in pluripotential theory is regularization by convolution. If $u\in\mathcal{PSH}(\Omega)$, for some open $\Omega\subset\mathbb{C}^{n}$, then $u*\chi_{\delta}\in\mathcal{PSH}\cap\mathcal{C}^{\infty}(\Omega_{\delta})$, for $\delta>0$, where we define

Here, $\Omega_{\delta}=\{z\in\Omega\,;\,B(z,\delta)\subset\Omega\}$, where $B(z,\delta)$ is the open euclidean ball with center $z$ and radius $\delta$, $\chi_{\delta}$ is a standard smoothing kernel in $\mathbb{C}^{n}$ with support in the closed ball $\overline{B}(0,\delta)$, and $\mathcal{C}^{\infty}(\Omega_{\delta})$ is the family of smooth function on $\Omega_{\delta}$. We also have that $u*\chi_{\delta}\searrow u$, as $\delta\searrow 0$. In the specific setting of $\Omega=\mathbb{C}^{n}$, we have $\Omega_{\delta}=\mathbb{C}^{n}$. So, every function in $\mathcal{PSH}(\mathbb{C}^{n})$ can approximated by a decreasing sequence in $\mathcal{PSH}\cap\mathcal{C}^{\infty}(\mathbb{C}^{n})$. For details on smoothing by convolution, see Klimek [9, Thm. 2.9.2].

In some cases, it is necessary to choose a method for regularizing plurisubharmonic functions that preserves a certain subclass of $\mathcal{PSH}(\mathbb{C}^{n})$. One such case is the Lelong class $\mathcal{L}(\mathbb{C}^{n})$. We say that $u\in\mathcal{PSH}(\mathbb{C}^{n})$ belongs to the Lelong class, denoted by $\mathcal{L}(\mathbb{C}^{n})$, if $u(z)\leq\log^{+}\|z\|_{\infty}+c_{u}$ for some constant $c_{u}$, where $\|\cdot\|_{\infty}$ is the supremum norm in $\mathbb{C}^{n}$ and $\log^{+}x=\max\{0,\log x\}$. A powerful tool in the study of pluripotential theory is the Siciak-Zakharyuta function of $E\subset\mathbb{C}^{n}$, defined by

One can show that smoothing with a standard smoothing kernel preserves the Lelong class. Namely, if $u\in\mathcal{L}(\mathbb{C}^{n})$ then $u*\chi_{\delta}\in\mathcal{L}(\mathbb{C}^{n})$. This is used to show that $V_{K}$ is lower semicontinuous, for compact $K\subset\mathbb{C}^{n}$. See Klimek [9, Section 5.1].

In the recent study of pluripotential theory related to convex sets, the Lelong classes are generalized by describing the growth more freely. We fix a compact convex set $0\in S\subset\mathbb{R}^{n}_{+}$ and recall that the supporting function of $S$, given by $\varphi_{S}(\xi)=\sup_{x\in S}\langle x,\xi\rangle$, is positively homogeneous and convex, which is equivalent to $\varphi_{S}(t\xi)=t\varphi_{S}(\xi)$ and $\varphi_{S}(\xi+\eta)\leq\varphi_{S}(\xi)+\varphi_{S}(\eta)$, for $t\in\mathbb{R}_{+}$ and $\xi,\eta\in\mathbb{R}^{n}$. We define the logarithmic supporting function of $S$ by

where $\mathbb{C}^{*n}=(\mathbb{C}^{*})^{n}$ and $\mathbb{C}^{*}=\mathbb{C}\setminus\{0\}$. This function is continuous and plurisubharmonic on $\mathbb{C}^{n}$ and its behavior on the coordinate hyperplanes $\mathbb{C}^{n}\setminus\mathbb{C}^{*n}$ can be described using logarithmic supporting functions of sets of lower dimensions. See [12, Props. 3.3 and 3.4]. We define the Lelong class given by $S$ as the set of all $u\in\mathcal{PSH}(\mathbb{C}^{n})$ satisfying $u\leq H_{S}+c_{u}$, for some constant $c_{u}$, and denote it by $\mathcal{L}^{S}(\mathbb{C}^{n})$. Similarly, we define $\mathcal{L}^{S}_{+}(\mathbb{C}^{n})$ as the set of all functions $u\in\mathcal{L}^{S}(\mathbb{C}^{n})$ satisfying $H_{S}+c_{u}\leq u$, for some constant $c_{u}$. This leads to our definition of the Siciak-Zakharyuta function of $E\subset\mathbb{C}^{n}$ and $S$ with weight $q\colon E\rightarrow\mathbb{R}\cup\{+\infty\}$,

We write $V^{S}_{K}=V^{S}_{K,0}$, $V_{K,q}=V^{\Sigma}_{K,q}$, and $V_{K}=V^{\Sigma}_{K,0}$, where $\Sigma$ is the standard simplex in $\mathbb{R}^{n}$ given by $\Sigma=\operatorname{ch}\{0,e_{1},\dots,e_{n}\}$, where $\operatorname{ch}A$ denotes the closed convex hull of the set $A$.

A weight $q\colon E\rightarrow\mathbb{R}\cup\{+\infty\}$ is called admissible if $q$ is lower semicontinuous, the set $\{z\in E\,;\,q(z)<+\infty\}$ is non-pluripolar, and $\lim_{E\ni z,|z|\rightarrow\infty}(H_{S}(z)-q(z))=-\infty$, if $E$ is unbounded.

Fundamental results of the Siciak-Zakharyuta function are developed in [12]. There, it is proven that $V^{S}_{K,q}$ is lower semicontinuous on $\mathbb{C}^{*n}$, but it is not proven that it is lower semicontinuous on all of $\mathbb{C}^{n}$. The obstruction to proving lower semicontinuity on $\mathbb{C}^{n}$ is a suitable method of approximating functions in $\mathcal{L}^{S}(\mathbb{C}^{n})$ by functions in $\mathcal{L}^{S}\cap\mathcal{C}(\mathbb{C}^{n})$, where $\mathcal{C}(\mathbb{C}^{n})$ denotes all continuous functions on $\mathbb{C}^{n}$, since smoothing by convolution with a standard smoothing kernel does not always preserve $\mathcal{L}^{S}(\mathbb{C}^{n})$. In fact, $\mathcal{L}^{S}(\mathbb{C}^{n})$ is preserved by the standard convolution operator if and only if $S$ is a lower set, that is when $S$ is such that for all $x\in S$ we have that $[0,x_{1}]\times\dots\times[0,x_{n}]\subset S$. See [12, Thm. 5.8].

In Section 2, we demonstrate the connection between the lower semicontinuity of $V^{S}_{K,q}$ and regularization in $\mathcal{L}^{S}(\mathbb{C}^{n})$. We also show that if $S$ contains a neighborhood of $0$ in $\mathbb{R}^{n}_{+}$ then the standard convolution operator, along with gluing, suffices to show that $V^{S}_{K,q}$ is lower semicontinuous. Namely, we prove the following.

This is significantly stronger than the case when $S$ is a lower set. For example, if $S_{1},S_{2},\dots$ are lower sets, then so is $S=\cap_{j=1}^{\infty}S_{j}$. However, for an arbitrary convex compact $S\subset\mathbb{R}^{n}_{+}$, with $0\in S$, we can define $S_{j}=\operatorname{ch}\big{(}(1/j)\Sigma\cup S\big{)}$. Then $S=\cap_{j=1}^{\infty}S_{j}$. This idea, along with [12, Prop. 4.8(i)], can sometimes be used to strengthen results. This is done in [19, the proof of Thm. 1.1].

An important consequence of Theorem 1.1 is sharpening the Siciak-Zakharyuta theorem proved in [11, Thm. 1.1].

In Sections 3-5 we study regularization operators other than standard convolution. In Section 3 we consider a generalization of an operator of Siciak [16, Prop. 1.3].

With $\mu=|\cdot|$, Siciak proved that $R^{a}_{\mu,\delta}u\in\mathcal{L}(\mathbb{C}^{n})\cap\mathcal{C}(\mathbb{C}^{n})$ and that $R^{a}_{\mu,\delta}u\searrow u$, as $\delta\searrow 0$. In [1] this process is referred to as Ferrier approximation, since Siciak developed it using a result from Ferrier [2, Lemma 2, p. 48]. Here it is attributed to Siciak, since Ferrier was not concerned with approximations. It is claimed in [1, Prop. 3.1] that if $\mu=|\cdot|$, $u\in\mathcal{L}^{S}(\mathbb{C}^{n})$, and $S$ is such that it contains a neighborhood of $0$ in $\mathbb{R}^{n}_{+}$, then $R^{a}_{\mu,\delta}u\in\mathcal{L}^{S}(\mathbb{C}^{n})\cap\mathcal{C}(\mathbb{C}^{n})$. The following example shows that this claim is false.

The infimum in the definition of the operator $R^{a}_{\mu,\delta}$ is an infimal convolution of the functions $e^{-u}$ and $\delta^{-1}|\cdot|$. Details on infimal convolutions and its application in convexity theory can be found in Rockafellar [15]. Applications in complex analysis can be found in Kiselman [8] and Halvarsson [3, 4, 5].

In Section 4 we consider the following supremal convolution.

The meaning of $Zw$ is justified in Section 4. We note that $\sigma_{S}$ may be $0$. This only happens when $S=\{0\}$, which is a case of no interest, since then $\mathcal{L}^{S}(\mathbb{C}^{n})$ would only contain the constant functions.

In Section 5 we consider two related operators. The first one was developed in [12, Section 5].

The assumption that $u$ is locally bounded below ensure that $R^{c}_{\delta}u>-\infty$. Taking $u(z)=\log|z_{1}|$, we have $R^{c}_{\delta}u(z)=-\infty$, for $z\in\{0\}\times\mathbb{C}^{n-1}$. So, $R^{c}_{\delta}u$ fails to be continuous.

A possible approach to showing continuity on all of $\mathbb{C}^{n}$ is to regularize $e^{u}$ instead of $u$. The second operator in Section 5 is given by $\log R^{c}_{\delta}e^{u}$. It must be shown that this function is plurisubharmonic.

In a recent paper of Perera [13], a classical result on the regularity of $V_{K}$ is considered, and attempts are made to generalize it to the setting of $V^{S}_{K,q}$. Perera [13] claims to give sufficient condition for $V^{S}_{K,q}$ to be Hölder continuous on $\mathbb{C}^{n}$, with the only assumption that $S$ is a convex body. In Section, 6 we show that this can not hold without additional restrictions on $S$. The main result, Corollary 6.2, states that if $S$ is not a lower set then $\mathcal{L}^{S}_{+}(\mathbb{C}^{n})$ contains no uniformly continuous functions. Since $V^{S*}_{K,q}\in\mathcal{L}^{S}_{+}(\mathbb{C}^{n})$, by [12, Prop. 4.5], we have that $V^{S}_{K,q}$ can not be Hölder continuous if $S$ is not a lower set.

The main motivation for studying the regularization operators in Sections 3-5 is proving that $V^{S}_{K,q}$ is lower semicontinuous on $\mathbb{C}^{n}$, not just $\mathbb{C}^{*n}$, for compact $K\subset\mathbb{C}^{n}$ and admissible $q$. To prove lower semicontinuity using regularizations, we use the following variant of Dini’s theorem.

To see how regularization is used to show that $V^{S}_{K,q}$ is lower semicontinuous, we fix an open set $\Omega\subset\mathbb{C}^{n}$ and assume that for all $u\in\mathcal{L}^{S}(\mathbb{C}^{n})$ there exists a decreasing sequence $(u_{j})_{j\in\mathbb{N}}$ in $\mathcal{L}^{S}(\mathbb{C}^{n})$ such that $u_{j}|_{\Omega}$ is continuous and $u_{j}\searrow u$. If we take $\varepsilon>0$ and assume that $u\leq q$ on $K$ then, by Dini’s theorem, we can find $j\in\mathbb{N}$, such that $u_{j}-\varepsilon\leq q$ on $K$. Hence, $V^{S}_{K,q}$ is given as the supremum over a family of functions continuous on $\Omega$, and is therefore lower semicontinuous on $\Omega$.

The regularizations considered in this paper work for $\Omega=\mathbb{C}^{*n}$. Whether these, or any other, regularizations work on all of $\mathbb{C}^{n}$ remains an open question.

We now turn to the main result in this section, Theorem 1.1. To prove it we will regularize using standard convolution, as discussed at the start on Section 1. This regularization will generally not have that right growth, but we will fix this using gluing.

Recall that Siciak’s infimal convolution was given by

with $\mu=|\cdot|$. In Ferrier [2], it is proven that $R^{a}_{\mu,\delta}u$ is plurisubharmonic and continuous when $u$ is plurisubharmonic, $\delta=1$, and $\mu=|\cdot|$. Generalizing for $\delta>0$ is not significant. In Siciak [16], it is shown that if

for some constant $c_{u}$, then $R^{a}_{\mu,\delta}u(z)\leq\log(|z|+1)+c_{u}$, for $\delta<e^{c_{u}}$, and consequently that $R^{a}_{\mu,\delta}u\in\mathcal{L}(\mathbb{C}^{n})$. He also shows that $R^{a}_{\mu,\delta}u\searrow u$, as $\delta\searrow 0$. The fact that $R^{a}_{\mu,\delta}u$ is plurisubharmonic is not obvious, but follows from geometric arguments. We will recall some useful results on pseudoconvex domains before going through the proof.

Let $u\in\mathcal{PSH}(\mathbb{C}^{n})$. Recall that $\Omega=\{z\in\mathbb{C}^{n}\,;\,u(z)<c\}$ is pseudoconvex. To see this, assume $u$ is continuous and take compact $K\subset\Omega$. Note that, $u\leq c-\delta$ on $K$, for some $\delta>0$, so $u\leq c-\delta$ on $\widehat{K}_{\mathcal{PSH}(\mathbb{C}^{n})}$, which is compact. So, we have that $\widehat{K}_{\mathcal{PSH}(\Omega)}\subset\widehat{K}_{\mathcal{PSH}(\mathbb{C}^{n})}\subset\Omega$, which implies that $\widehat{K}_{\mathcal{PSH}(\Omega)}$ is relatively compact in $\Omega$. Hence, $\Omega$ is pseudoconvex. In the case that $u$ is not continuous, we consider the pseudoconvex sets $\Omega_{\delta}=\{z\in\mathbb{C}^{n}\,;\,u*\chi_{\delta}(z)<c\}$, for $\delta>0$. Since $\Omega_{\delta}\nearrow\Omega$, as $\delta\searrow 0$, we have, by the Behnke-Stein theorem, that $\Omega$ is pseudoconvex.

We call a continuous function $\mu\colon\mathbb{C}^{n}\rightarrow\mathbb{R}_{+}$ a distance function if (i) $\mu(z)=0$ if and only if $z=0$ and (ii) $\mu(tz)=|t|\mu(z)$ for $z\in\mathbb{C}^{n}$ and $t\in\mathbb{C}$. Note that, by (ii), we have constants $r_{\mu}=\inf_{|z|=1}\mu(z)$ and $s_{\mu}=\sup_{|z|=1}\mu(z)$ such that $r_{\mu}|z|\leq\mu(z)\leq s_{\mu}|z|$, for $z\in\mathbb{C}^{n}$, and by (i) we have that $r_{\mu}>0$. For an open $\Omega\subset\mathbb{C}^{n}$, we define the $\mu$-distance to the boundary by

and note that it is a continuous function. Pseudoconvex domains are classically characterized by distances, namely an open $\Omega\subset\mathbb{C}^{n}$ is pseudoconvex if and only if $-\log\mu_{\Omega}$ is plurisubharmonic for every (or some) distance function $\mu$. See Klimek [9, Thm. 2.10.4].

Let us recall some topological properties of $\mathcal{PSH}(\Omega)$, with the goal of determining when $\sup\mathcal{F}$ is upper semicontinuous, for some family $\mathcal{F}\subset\mathcal{PSH}(\Omega)$ that is locally upper bounded. For more details, see Hörmander [6, Thms. 4.1.8-9] and [7, Thm. 3.2.11-13]. Recall that $\mathcal{PSH}(\Omega)$ is the family of plurisubharmonic functions on $\Omega$ that are not identically $-\infty$ on any connected component of $\Omega$. So,

The topology $\mathcal{PSH}(\Omega)$ inherits from the weak topology on $\mathcal{D}^{\prime}(\Omega)$ coincides with topology it inherits from $L_{\text{loc}}^{1}(\Omega)$, as a Fréchet space topology with semi-norms $f\mapsto\int_{K}|f|\ d\lambda$, for compact $K$. With this topology, $\mathcal{PSH}(\Omega)$ is a closed subspace of $L_{\text{loc}}^{1}(\Omega)$, so it is a complete metrizable space.

Furthermore, $\mathcal{PSH}(\Omega)$ has a Montel property, which says that every sequence $(u_{j})_{j\in\mathbb{N}}$ in $\mathcal{PSH}(\Omega)$, that is locally bounded above and does not converge to $-\infty$ uniformly on every compact subset of $\Omega$, has a subsequence that converges in $\mathcal{PSH}(\Omega)$.

Sometimes, $\sup\mathcal{F}=\sup\mathcal{F}_{0}$ for some $\mathcal{F}_{0}\subset\mathcal{F}$ containing a minimal element. The Montel property then states that $\mathcal{F}_{0}$ is relatively compact. If, in addition, $\mathcal{F}_{0}$ is closed then $\sup\mathcal{F}$ is upper semicontinuous. This follows from Sigurdsson [18, Prop. 2.1], which we include for the convenience of the reader.