# The Geometry of *m*-Hyperconvex Domains

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## Abstract

We study the geometry of *m*-regular domains within the Caffarelli–Nirenberg–Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every *m*-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly *m*-subharmonic, and has bounded *m*-Hessian measure.

## Keywords

Barrier function Caffarelli–Nirenberg–Spruck model Exhaustion function*m*-subharmonic function Jensen measure

## Mathematics Subject Classification

Primary 31C45 32F17 32U05 Secondary 31B25 32U10 32T35 46J10 46A20## 1 Introduction

The geometry of the underlying space is usually essential when studying a given problem in analysis. The starting point of this paper is the model presented by Caffarelli et al. [16] in 1985 that makes it possible to investigate the transition between potential and pluripotential theories. Their construction relies on Gårding’s research on hyperbolic polynomials [27]. The authors of [16] also provided a very nice application to special Lagrangian geometry, which was in itself introduced as an example within calibrated geometry [30]. With the publications of [9], and [45], many analysts and geometers became interested in the Caffarelli–Nirenberg–Spruck model. To mention some references [23, 37, 47, 49, 51, 64, 71]. A usual assumption in these studies is that the underlying domain should admit a continuous exhaustion function that is *m*-subharmonic in the sense of Caffarelli et al. (see Sect. 2 for the definition of *m*-subharmonic functions). In this paper, we shall study the geometric properties of these domains. Let us now give a thorough background on the motivation behind this paper. It all starts with the following theorem:

### Theorem A

Assume that \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\), \(N\ge 2\). Then the following assertions are equivalent.

- (1)\(\partial \Omega \) is
*regular*at every boundary point \(y_0\in \partial \Omega \), in the sense thatfor each continuous function \(f:\partial \Omega \rightarrow {\mathbb {R}}\). Here$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow y_0\\ x\in \Omega \end{array}} {\text {PWB}}_f(x) =f(y_0), \end{aligned}$$and \({\mathcal {SH}}(\Omega )\) is the space of subharmonic functions defined on \(\Omega \);$$\begin{aligned} {\text {PWB}}_f(x)=\sup \Bigg \{v(x): v\in {\mathcal {SH}}(\Omega ),\; \varlimsup _{\begin{array}{c} \zeta \rightarrow \xi \\ \zeta \in \Omega \end{array}}v(\zeta )\le f(\xi ), \;\; \forall \xi \in \partial \Omega \Bigg \}, \end{aligned}$$ - (2)\(\partial \Omega \) has a
*strong barrier*at every point \(y_0\in \partial \Omega \) that is subharmonic, i.e., there exists a subharmonic function \(u:\Omega \rightarrow {\mathbb {R}}\) such thatand$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow y_0\\ x\in \Omega \end{array}} u(x)=0, \end{aligned}$$$$\begin{aligned} \limsup _{\begin{array}{c} x\rightarrow y\\ x\in \Omega \end{array}} u(x)<0 \qquad \text { for all } y\in {\bar{\Omega }}\backslash \{y_0\}. \end{aligned}$$ - (3)\(\partial \Omega \) has a
*weak barrier*at every point \(y_0\in \partial \Omega \) that is subharmonic, i.e., there exists a subharmonic function \(u:\Omega \rightarrow {\mathbb {R}}\) such that \(u<0\) on \(\Omega \) and$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow y_0\\ x\in \Omega \end{array}} u(x)=0. \end{aligned}$$ - (4)
\(\Omega \) admits an

*exhaustion function*that is negative and subharmonic, i.e., there exists a non-constant function \(\psi :\Omega \rightarrow {\mathbb {R}}\) such that for any \(c\in {\mathbb {R}}\) the set \(\{x\in \Omega :\psi (x)<c\}\) is relatively compact in \(\Omega \). Furthermore, the exhaustion function should be negative and subharmonic. - (5)
\(\partial \Omega \) is equal to the

*Jensen boundary*w.r.t. the Jensen measures generated by the cone of functions that are continuous on \({{\bar{\Omega }}}\), and subharmonic on \(\Omega \) (see Sect. 2 for definitions).

Inspired by the beauty of the equivalences in Theorem A, analysts started to investigate these notions within the model introduced by Lelong [44] and Oka [50] in 1942, where subharmonic functions are changed to plurisubharmonic functions. The unit polydisc in \({\mathbb {C}}^n\), \(n\ge 2\), shows that the notions of weak and strong barrier for plurisubharmonic functions are not equivalent. Instead, we have Theorems B and C below, where we assume that \(n\ge 2\). If \(n=1\), then the two theorems become Theorem A since subharmonic functions are then the same as plurisubharmonic functions.

### Theorem B

Assume that \(\Omega \) is a bounded domain in \({\mathbb {C}}^n\), \(n\ge 2\). Then the following assertions are equivalent.

- (1)\(\partial \Omega \) is
*B-regular*at every boundary point \(z_0\in \partial \Omega \), in the sense thatfor each continuous function \(f:\partial \Omega \rightarrow {\mathbb {R}}\). Here$$\begin{aligned} \lim _{\begin{array}{c} z\rightarrow z_0\\ z\in \Omega \end{array}} {\text {PB}}_f(z) =f(z_0), \end{aligned}$$Here \(\mathcal {PSH}(\Omega )\) is the space of plurisubharmonic functions defined on \(\Omega \);$$\begin{aligned} {\text {PB}}_f(z)=\sup \Bigg \{v(z): v\in \mathcal {PSH}(\Omega ),\; \varlimsup _{\begin{array}{c} \zeta \rightarrow \xi \\ \zeta \in \Omega \end{array}}v(\zeta )\le f(\xi ), \;\; \forall \xi \in \partial \Omega \Bigg \}. \end{aligned}$$ - (2)
\(\partial \Omega \) has a strong barrier at every point that is plurisubharmonic;

- (3)
\(\Omega \) admits an exhaustion function \(\varphi \) that is negative, smooth, plurisubharmonic, and such that \(\left( \varphi (z)-|z|^2\right) \) is plurisubharmonic.

- (4)
\(\partial \Omega \) is equal to the Jensen boundary w.r.t. the Jensen measures generated by the cone of functions that are continuous on \({\bar{\Omega }}\), and plurisubharmonic on \(\Omega \).

*P*), and the \({\bar{\partial }}\)-Neumann problem. The last condition in assertion (3) means that we have that

*uniformly*strictly plurisubharmonic.

### Theorem C

Assume that \(\Omega \) is a bounded domain in \({\mathbb {C}}^n\), \(n\ge 2\). Then the following assertions are equivalent.

- (1)
\(\Omega \) is

*hyperconvex*in the sense that it admits an exhaustion function that is negative and plurisubharmonic; - (2)
\(\partial \Omega \) has a weak barrier at every point that is plurisubharmonic;

- (3)
\(\Omega \) admits an exhaustion function that is negative, smooth and strictly plurisubharmonic;

- (4)
For every \(z\in \partial \Omega \), and every Jensen measure \(\mu \), which is generated by the cone of functions that are continuous on \({{\bar{\Omega }}}\), and plurisubharmonic on \(\Omega \), we have that \(\mu \) is carried by \(\partial \Omega \).

As we mentioned at the beginning of this exposé, the purpose of this paper is to study the geometry of the corresponding notions *B*-regular and hyperconvex domains within the Caffarelli–Nirenberg–Spruck model. More precisely, in Theorem 4.3, we prove what degenerates into Theorem B when \(m=n\), and in Theorem 4.1, we prove what is Theorem C in the case \(m=n\), except for the corresponding implication \((1)\Rightarrow (3)\). This we prove in Sect. 5 due to the different techniques used, and the length of that proof. In the case when \(m=1\), our Theorems 4.3 and 4.1 (together with Theorem 5.4) merge into Theorem A above with \(N=2n\).

This article is organized as follows. In Sect. 2, we shall state the necessary definitions and some preliminaries needed for this paper, and then in Sect. 3, we shall prove some basic facts about *m*-hyperconvex domains (Theorem 3.5). From Sect. 3, and Theorem 3.5 we would like to emphasize property (3). Up until now authors have defined *m*-hyperconvex domains to be bounded domains that admit an exhaustion function that is negative, continuous, and *m*-subharmonic. We prove that the assumption of continuity is superfluous. This result is also the starting point of the proof of Theorem 5.4. In Sect. 4, we prove Theorems 4.3 and 4.1, as mentioned above, which correspond to Theorems B and C, respectively. We end this paper by showing that every *m*-hyperconvex domain admits a smooth and strictly *m*-subharmonic exhaustion function (Theorem 5.4; see implication \((1)\Rightarrow (3)\) in Theorem C).

We end this introduction by highlighting an opportunity for future studies related to this paper. As convex analysis and pluripotential theory lives in symbiosis, Trudinger and Wang [60] draw their inspiration from the work of Caffarelli et al., and in 1999 they presented a model that makes it possible to study the transition between convex analysis and potential theory. For further information see e.g., [59, 60, 61, 65]. As [63] indicates, further studies of the geometric properties of what could be named *k-convex domains* are of interest. We leave these questions to others.

We want to thank Urban Cegrell, Per-Håkan Lundow, and Håkan Persson for inspiring discussions related to this paper. We are also grateful for the comments and suggestions given by the anonymous referee that helped with the presentation of the final version of this paper.

## 2 Preliminaries

In this section, we shall present the necessary definitions and fundamental facts needed for the rest of this paper. For further information related to potential theory see e.g., [4, 24, 41], and for more information about pluripotential theory see e.g., [22, 40]. We also want to mention the highly acclaimed book written by Hörmander called “*Notions of convexity*” [36]. Abdullaev and Sadullaev [3] have written an article that can be used as an introduction to the Caffarelli–Nirenberg–Spruck model. We recommend also Lu’s doctoral thesis [46]. We would like to point out that *m*-subharmonic functions in the sense of Caffarelli et al. is not equivalent of being subharmonic on *m*-dimensional hyperplanes in \({\mathbb {C}}^n\) studies by others (see e.g., [1, 2]). For other models in connection to plurisubharmonicity see e.g., [31, 32, 33].

### Definition 2.1

*u*be a subharmonic function defined in \(\Omega \). Then we say that

*u*is

*m-subharmonic*, \(1\le m\le n\), if the following inequality holds

*m*-subharmonic functions defined on \(\Omega \). We say that a function

*u*is

*strictly m-subharmonic*if it is

*m*-subharmonic on \(\Omega \), and for every \(p \in \Omega \) there exists a constant \(c_p >0\) such that \(u(z)-c_p |z| ^2\) is

*m*-subharmonic in a neighborhood of

*p*.

### Remark

In Theorem 2.2, we give a list of well-known properties that *m*-subharmonic functions enjoy.

### Theorem 2.2

- (1)
If \(u,v\in {\mathcal {SH}}_m(\Omega )\), then \(su+tv\in {\mathcal {SH}}_m(\Omega )\), for constants \(s,t\ge 0\);

- (2)
If \(u,v\in {\mathcal {SH}}_m(\Omega )\), then \(\max \{u,v\}\in {\mathcal {SH}}_m(\Omega )\);

- (3)If \(\{u_{\alpha }\}\) is a locally uniformly bounded family of functions from \( {\mathcal {SH}}_m(\Omega )\), then the upper semicontinuous regularizationdefines a$$\begin{aligned} \left( \sup _{\alpha }u_{\alpha }\right) ^* \end{aligned}$$
*m*-subharmonic function; - (4)
If \(\{u_j\}\) is a sequence of functions in \({\mathcal {SH}}_m(\Omega )\) such that \(u_j \searrow u\) and there is a point \(z \in \Omega \) such that \(u(z) > - \infty \), then \(u \in {\mathcal {SH}}_m(\Omega )\);

- (5)
If \(u\in {\mathcal {SH}}_m(\Omega )\) and \(\gamma :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a convex and nondecreasing function, then \(\gamma \circ u\in {\mathcal {SH}}_m(\Omega )\);

- (6)If \(u\in {\mathcal {SH}}_m(\Omega )\), then the standard regularization given by the convolution \(u\star \rho _{\varepsilon }\) is
*m*-subharmonic in \(\{z\in \Omega :{\text {dist}}(z,\partial \Omega )>\varepsilon \}\). Here we have that\(\rho :{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is a smooth function such that \(\rho (z)=\rho (|z|)\) and$$\begin{aligned} \rho _{\varepsilon }=\varepsilon ^{-2n}\rho \left( \frac{z}{\varepsilon }\right) , \end{aligned}$$where$$\begin{aligned} \rho (t)= {\left\{ \begin{array}{ll} \frac{C}{(1-t)^2}\exp \left( {\frac{1}{t-1}}\right) &{} \text { when } t\in [0,1]\\ 0 &{} \text { when } t\in (1,\infty ), \end{array}\right. } \end{aligned}$$*C*is a constant such that \(\int _{{\mathbb {C}}^n}\rho (|z|^2)\beta ^n=1\); - (7)If \(\omega \Subset \Omega \), \(u\in {\mathcal {SH}}_m(\Omega )\), \(v\in {\mathcal {SH}}_m(\omega )\), and \(\varlimsup _{z\rightarrow w}v(z)\le u(w)\) for all \(w\in \partial \omega \), then the function defined byis$$\begin{aligned} \varphi = {\left\{ \begin{array}{ll} u, \, \text{ on } \, \Omega \setminus \omega ,\\ \max \{u,v\}, \; \text{ on } \, \omega , \end{array}\right. } \end{aligned}$$
*m*-subharmonic on \(\Omega \);

We shall need several different envelope constructions. We have gathered their definitions and notations in Definition 2.3.

### Definition 2.3

- (a) For \(f\in {\mathcal {C}}({{\bar{\Omega }}})\) we defineand similarly$$\begin{aligned} \mathbf{S }_f(z)=\sup \left\{ v(z): v\in \mathcal {SH}_m(\Omega ), v\le f\right\} , \end{aligned}$$$$\begin{aligned} \mathbf{S }^c_f(z)=\sup \left\{ v(z): v\in \mathcal {SH}_m(\Omega )\cap {\mathcal {C}}({{\bar{\Omega }}}), v\le f\right\} . \end{aligned}$$
- (b) If instead \(f\in {\mathcal {C}}(\partial \Omega )\), then we letand$$\begin{aligned} \mathbf{S }_f(z)=\sup \left\{ v(z): v\in \mathcal {SH}_m(\Omega ), v^*\le f\, \text{ on } \, \partial \Omega \right\} , \end{aligned}$$$$\begin{aligned} \mathbf{S }^c_f(z)=\sup \left\{ v(z): v\in \mathcal {SH}_m(\Omega )\cap {\mathcal {C}}({{\bar{\Omega }}}), v\le f\, \text{ on } \, \partial \Omega \right\} . \end{aligned}$$

### Remark

Assume that \(m=1\). If \(\Omega \subset {\mathbb {C}}^n\)\((\cong {\mathbb {R}}^{2n})\) is a regular domain in the sense of Theorem A, and if \(f\in {\mathcal {C}}(\partial \Omega )\), then \({\text {PWB}}_f\) (defined also in Theorem A) is the unique harmonic function on \(\Omega \), continuous on \({{\bar{\Omega }}}\), such that \({\text {PWB}}_f=f\) on \(\partial \Omega \). Therefore, we have that \(\mathbf{S }_f(z)=\mathbf{S }_{{\text {PWB}}_f}(z)\), and \(\mathbf{S }^c_f(z)=\mathbf{S }^c_{{\text {PWB}}_f}(z)\).

In Definition 2.4, we state the definition of relative extremal functions in our setting.

### Definition 2.4

### Remark

*E*, \(v\le 0\), then \(v\le h_E\) on \({\bar{\Omega }}\setminus E\) and therefore \(v\le \mathbf{H }_E\), which means that \(\mathbf{S }_E\le \mathbf{S }_{\mathbf{H }_E}\). On the other hand since \(\mathbf{S }_{\mathbf{H }_E}^*\in \mathcal {SH}_m(\Omega )\) and \(\mathbf{S }_{\mathbf{H }_E}^*\le -1\) on

*E*which implies that

Błocki’s generalization of Walsh’s celebrated Theorem [62], and an immediate consequence will be needed as well.

### Theorem 2.5

Let \(\Omega \) be a bounded domain in \({\mathbb {C}}^n\), and let \(f\in {\mathcal {C}}({{\bar{\Omega }}})\). If for all \(w\in \partial \Omega \) we have that \(\lim _{z\rightarrow w}\mathbf{S }_f(z)=f(w)\), then \(\mathbf{S }_f\in {\mathcal {SH}}_m(\Omega )\cap {\mathcal {C}}({{\bar{\Omega }}})\).

### Proof

See Proposition 3.2 in [9]. \(\square \)

A direct consequence of Theorem 2.5 is the following.

### Corollary 2.6

Let \(\Omega \) be a bounded domain in \({\mathbb {C}}^n\), and let \(f\in {\mathcal {C}}({\bar{\Omega }})\). If for all \(w\in \partial \Omega \) we have that \(\lim _{z\rightarrow w}\mathbf{S }^c_f(z)=f(w)\), then \(\mathbf{S }^c_f=\mathbf{S }_f\in {\mathcal {SH}}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\).

### Proof

In Sect. 4, we shall make use of techniques from Choquet theory, in particular Jensen measures w.r.t. the cone \({\mathcal {SH}}_m(\Omega ) \cap {\mathcal {C}}({\bar{\Omega }})\) of continuous functions. This is possible since \({\mathcal {SH}}_m(\Omega ) \cap {\mathcal {C}}({\bar{\Omega }})\) contains the constant functions and separates points in \({\mathcal {C}}({\bar{\Omega }})\). Our inspiration can be traced back to the works mentioned in the introduction, but maybe more to [17] and [35].

### Definition 2.7

*Jensen measure with barycenter*\(z_0\in \bar{\Omega }\)

*w.r.t.*\({\mathcal {SH}}_m(\Omega ) \cap {\mathcal {C}}({\bar{\Omega }})\) if

*Jensen boundary*w.r.t. \({\mathcal {J}}_{z_0}^m\) is defined as

### Remark

The Jensen boundary is another name for the Choquet boundary w.r.t. a given class of Jensen measures. For further information see e.g., [12, 48].

### Remark

There are many different spaces of Jensen measures introduced throughout the literature. Caution is advised.

The most important tool in working with Jensen measures is the Edwards’ duality theorem that origins from [25]. We only need a special case formulated in Theorem 2.8. For a proof, and a discussion, of Edwards’ theorem see [69] (see also [20, 21, 55]).

### Theorem 2.8

*g*be a real-valued lower semicontinuous function defined on \({\bar{\Omega }}\). Then for every \(z\in \bar{\Omega }\) we have that

We end this section with a convergence result.

### Theorem 2.9

Assume that \(\Omega \) is a domain in \({\mathbb {C}}^n\), and let \(\{z_n\} \subset {\bar{\Omega }}\) be a sequence of points converging to \(z\in {\bar{\Omega }}\). Furthermore, for each *n*, let \(\mu _n \in {\mathcal {J}}_{z_n}^m\). Then there exists a subsequence \(\{\mu _{n_j}\}\), and a measure \(\mu \in {\mathcal {J}}_z^m\) such that \(\{\mu _{n_j}\}\) converges in the weak-\(^*\) topology to \(\mu \).

### Proof

## 3 Basic Properties of *m*-Hyperconvex Domains

The aim of this section is to introduce *m*-hyperconvex domains (Definition 3.1) within the Caffarelli–Nirenberg–Spruck model, and prove Theorem 3.5 . If \(m=1\), then the notion will be the same as regular domains (see assertion (4) in Theorem A in the introduction), and if \(m=n\) then it is the same as hyperconvex domains (see (1) in Theorem C).

### Definition 3.1

Let \(\Omega \) be a bounded domain in \({\mathbb {C}}^n\). We say that \(\Omega \) is *m-hyperconvex* if it admits an exhaustion function that is negative and *m*-subharmonic.

Traditionally, in pluripotential theory, the exhaustion functions are assumed to be bounded. That assumption is obviously superfluous in Definition 3.1. Even though it should be mentioned once again that up until now authors have defined *m*-hyperconvex domains to be bounded domains that admit an exhaustion function that is negative, *continuous*, and *m*-subharmonic. We prove below in Theorem 3.5 that the assumption of continuity is not necessary. Before continuing with Theorem 3.5 let us demonstrate the concept of *m*-hyperconvexity in the following two examples. Example 3.2 demonstrates that Hartogs’ triangle is 1-hyperconvex, but not 2-hyperconvex.

### Example 3.2

In Example 3.3, we construct a domain in \({\mathbb {C}}^3\) that is 2-hyperconvex, but not 3-hyperconvex.

### Example 3.3

*m*-subharmonic function if, and only if, \(m\le k\). Let us now consider the following domain:

*k*-hyperconvex, since

*k*-subharmonic exhaustion function. In particular, we get that for \(n=3\), and \(k=2\), the domain \(\Omega _2\subset {\mathbb {C}}^3\) is 2-hyperconvex but not 3-hyperconvex. \(\square \)

We shall need the following elementary lemma. For completeness we include a proof.

### Lemma 3.4

### Proof

The aim of this section is to prove the following theorem, especially property (3).

### Theorem 3.5

*m*-hyperconvex domains in \({\mathbb {C}}^n\), \(n\ge 2\), \(1\le m\le n\). Then we have the following.

- (1)
If \(\Omega _1 \cap \Omega _2\) is connected, then the domain \(\Omega _1 \cap \Omega _2\) is

*m*-hyperconvex in \({\mathbb {C}}^n\). - (2)
The domain \(\Omega _1 \times \Omega _2\) is

*m*-hyperconvex in \({\mathbb {C}}^{2n}\). - (3)
The domain \(\Omega \) admits a negative exhaustion function that is strictly

*m*-subharmonic on \(\Omega \), and continuous on \({\bar{\Omega }}\). - (4)
If \(\Omega \) is a priori only a bounded domain in \({\mathbb {C}}^n\) such that for every \(z\in \partial \Omega \) there exists a neighborhood \(U_z\) such that \(\Omega \cap U_z\) is

*m*-hyperconvex, then \(\Omega \) is*m*-hyperconvex.

### Proof

*Part (1)* For each \(j=1,2\), assume that \(\psi _j \in {\mathcal {SH}}_m(\Omega _j)\) is a negative exhaustion function for the *m*-hyperconvex domain \(\Omega _j\), \(j=1,2\). Then \(\max \{\psi _1,\psi _2\} \in {\mathcal {SH}}_m(\Omega _1\cap \Omega _2)\) is a negative exhaustion function for \(\Omega _1 \cap \Omega _2\). Thus, \(\Omega _1 \cap \Omega _2\) is *m*-hyperconvex in \({\mathbb {C}}^n\).

*Part (2)*This part is concluded by defining a negative exhaustion function by

*Part (3)*The proof of this part is inspired by [19]. First, we shall prove that there exists a negative and continuous exhaustion function. We know that \(\Omega \) always admits a bounded, negative, exhaustion function \(\varphi \in {\mathcal {SH}}_m(\Omega )\). Fix \(w\in \Omega \) and \(r>0\) such that \(B(w,r)\Subset \Omega \), and note that there exists a constant \(M>0\) such that

*strictly*

*m*-subharmonic exhaustion function for \(\Omega \). From the first part of this theorem, we know that there is a negative and continuous exhaustion function \(u \in {\mathcal {SH}}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\) for \(\Omega \). Choose \(M>0\) such that \(|z|^2-M \le -1\) on \(\Omega \), and define

*m*-subharmonic functions on defined \(\Omega \). We can conclude that \(\psi \in {\mathcal {SH}}_m(\Omega )\), since \(\psi (z)> -\infty \) for \(z \in \Omega \). The continuity of \(\psi \) is obtained by the Weierstrass

*M*-test. To see that \(\psi \) is strictly

*m*-subharmonic, note that if \(\omega \Subset \Omega \), then there exists an index \(j_\omega \) such that on \(\omega \) we have that

*m*-subharmonic, we have that \(\psi \) is strictly

*m*-subharmonic on \(\Omega \). Finally, \(\psi \) is an exhaustion function for \(\Omega \), since \(\psi _j|_{\partial \Omega }=0\) for all

*j*.

*Part (4)*The idea of the proof of this part is from [7]. By the assumption there are neighborhoods \(U_{z_1},\dots ,U_{z_N}\) such that \(\partial \Omega \subset \bigcup _{j=1}^NU_{z_j}\), and each \(U_{z_j}\cap \Omega \) is

*m*-hyperconvex. Let \(u_j:\Omega \rightarrow [-1,0]\) be a negative and continuous

*m*-subharmonic exhaustion function for \(U_{z_j}\cap \Omega \). Let \(V_j\Subset U_{z_j}\) be such that \(\partial \Omega \subset \bigcup _{j=1}^NV_{j}\). For \(x<0\), we then define the following continuous functions

*j*, take a smooth function \(\varphi _j\) such that \({\text {supp}}(\varphi _j)\subset V_j\), \(0\le \varphi _j\le 1\), and \(\varphi _j=1\) on a neighborhood of \({{\bar{V}}}_j'\). Furthermore, there are constants \(M_1,M_2>0\) such that \(|z|^2-M_1\le 0\) on \(\Omega \), and such that the functions \(\varphi _j+M_2(|z|^2-M_1)\) are

*m*-subharmonic for \(j=1,\dots ,N\). Let us define

*c*such that

*m*-subharmonic function defined on \(\Omega \). Finally note that, for \(c<-\varepsilon \), the following function

*m*-subharmonic, and \(\psi _{\varepsilon }\le 0\) on \(\Omega \). For \(z\in \partial \Omega \cap V_j'\), we have that

### Remark

*m*-hyperconvex domain, and \(E\Subset \Omega \) is an open subset such that \(\Omega \setminus {{\bar{E}}}\) is a regular domain in the sense of Theorem A. Then we have that

- (1)
\(\displaystyle {\mathbf{S }_E^c(z)=\mathbf{S }_{\mathbf{H }_E}^c(z)}\), and

- (2)
\(\displaystyle {\mathbf{S }_E=\mathbf{S }_E^c=\mathbf{S }_{\mathbf{H }_E}^c=\mathbf{S }_{\mathbf{H }_E}}\).

*m*-hyperconvex domain it can be proved as in part (3) of Theorem 3.5 that \(\mathbf{S }_{\mathbf{H }_E}^c\in \mathcal {SH}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\). Therefore, \(\mathbf{S }_E^c(z)\ge \mathbf{S }_{\mathbf{H }_E}^c(z)\). Property (2) follows from Theorem 2.5 together with the remark after Definition 2.4.

## 4 The Geometry of *m*-Regular Domains

In this section, we shall investigate the geometry of the corresponding notions of *B*-regular and hyperconvex domains within the Caffarelli–Nirenberg–Spruck model. More precisely, in Theorem 4.3, we prove what degenerates into Theorem B when \(m=n\), and in Theorem 4.1, we prove what is Theorem C in the case \(m=n\).

### Theorem 4.1

- (1)
\(\Omega \) is

*m*-hyperconvex in the sense of Definition 3.1; - (2)
\(\partial \Omega \) has a weak barrier at every point that is

*m*-subharmonic; - (3)
\(\Omega \) admits an exhaustion function that is negative, smooth and strictly

*m*-subharmonic; and - (4)
For every \(z\in \partial \Omega \), and every \(\mu \in {\mathcal {J}}_z^m\), we have that \({\text {supp}} (\mu ) \subseteq \partial \Omega \).

### Proof

The implications \((1)\Rightarrow (2)\), and \((3)\Rightarrow (1)\) are trivial. The implication \((1)\Rightarrow (3)\) is postponed to Theorem 5.4 in Sect. 5.

*z*that is

*m*-subharmonic. Since there exists a constant \(M_z>0\) such that

*m*-hyperconvex, and that \(u \in {\mathcal {SH}}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\) is an exhaustion function for \(\Omega \). If \(z \in \partial \Omega \), and \(\mu \in {\mathcal {J}}_z^m\), then

*z*, and

*m*-hyperconvex. \(\square \)

Before we can start with the proof of Theorem 4.3, we need the following corollary.

### Corollary 4.2

*m*-hyperconvex domain in \({\mathbb {C}}^n\), and let \(f \in {\mathcal {C}}(\partial \Omega )\). Then there exists a function \(u \in {\mathcal {SH}}_m(\Omega ) \cap {\mathcal {C}}({\bar{\Omega }})\) such that \(u=f\) on \(\partial \Omega \) if, and only if,

### Proof

*f*to a continuous function on \({\bar{\Omega }}\) (for instance one can take \({\text {PWB}}_f\), which was defined in Theorem A in the introduction) and for simplicity denote it also by

*f*. Since \(\Omega \) is a

*m*-hyperconvex domain then by Theorem 4.1 for any \(z\in \partial \Omega \) and any \(\mu \in {\mathcal {J}}_z^m\) it holds that \({\text {supp}} (\mu ) \subseteq \partial \Omega \), so we have

*j*, we have that

### Remark

*m*-hyperconvex, then we have a similar result as in Corollary 4.2 namely that there exists a function \(u \in {\mathcal {SH}}_m(\Omega ) \cap {\mathcal {C}}({\bar{\Omega }})\) such that \(u=f\) on \({\bar{\Omega }}\) if, and only if, there exists a continuous extension \(\varphi \) of

*f*to \({\bar{\Omega }}\) such that

We end this section by proving Theorem 4.3, and its immediate consequence. We have decided to deviate in Theorem 4.3 the notation from Definition 2.3. This to simplify the comparison with Theorem B in the introduction.

### Theorem 4.3

- (1)\(\partial \Omega \) is \(B_m\)-regular at every boundary point \(z_0\in \partial \Omega \), in the sense thatfor each continuous function \(f:\partial \Omega \rightarrow {\mathbb {R}}\). Here$$\begin{aligned} \lim _{\begin{array}{c} z\rightarrow z_0\\ z\in \Omega \end{array}} {\text {PB}}^m_f(z) =f(z_0), \end{aligned}$$$$\begin{aligned} {\text {PB}}^m_f(z)=\sup \Bigg \{v(z): v\in {\mathcal {SH}}_m(\Omega ),\; \varlimsup _{\begin{array}{c} \zeta \rightarrow \xi \\ \zeta \in \Omega \end{array}}v(\zeta )\le f(\xi ), \;\; \forall \xi \in \partial \Omega \Bigg \}. \end{aligned}$$
- (2)
\(\partial \Omega \) has a strong barrier at every point that is

*m*-subharmonic; - (3)\(\Omega \) admits an exhaustion function \(\varphi \) that is negative, smooth,
*m*-subharmonic, and such that$$\begin{aligned} \left( \varphi (z)-|z|^2\right) \in {\mathcal {SH}}_m(\Omega )\, ; and \end{aligned}$$ - (4)
\(\partial \Omega =\partial _{{\mathcal {J}}_z^m}\) in the sense of Definition 2.7.

### Proof

\((1)\Rightarrow (2):\) Fix \(z\in \partial \Omega \), and let *f* be a continuous function on \(\partial \Omega \) such that \(f(z)=0\) and \(f(\xi )<0\) for \(\xi \ne z\). Then \({\text {PB}}^m_f\) is a strong barrier at *z*.

*m*-subharmonic, and by the generalized Walsh theorem (Theorem 2.5) it is sufficient to show that

*w*that is

*m*-subharmonic. Then there exists a constant \(M>0\) such that

*f*be a continuous function on \(\partial \Omega \) such that \(f(z)=0\) and \(f(\xi )<0\) for \(\xi \ne z\). Then \({\text {PB}}^m_f\in \mathcal {SH}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\), and \({\text {PB}}^m_f=f\) on \(\partial \Omega \). Let \(\mu \in {\mathcal {J}}_z^m\) then, since \(\mu \) is a probability measure on \({\bar{\Omega }}\), we have that

\((4)\Rightarrow (1):\) This follows from Corollary 4.2.

*v*that is

*m*-subharmonic and

*g*defined on a neighborhood of \({\bar{\Omega }}\) such that

An immediate consequence of Theorem 4.3 is the following corollary.

### Corollary 4.4

Let \(\Omega \) be a bounded domain in \({\mathbb {C}}^n\) such that for every \(z\in \partial \Omega \) there exists a neighborhood \(U_z\) such that \(\Omega \cap U_z\) is \(B_m\)-regular, then \(\Omega \) is \(B_m\)-regular.

### Proof

*z*, and let \(u_z\) be a strong barrier at

*z*, that is

*m*-subharmonic, and defined in some neighborhood of \({{\bar{U}}}_z\cap \Omega \). Now let \(\delta >0\), be such that \(u_z<-\delta \) on \(\partial U_z\cap \Omega \). Then we can define a (global) strong barrier at

*z*, that is

*m*-subharmonic:

## 5 The Existence of Smooth Exhaustion Functions

The purpose of this section is to prove the implication \((1)\Rightarrow (3)\) in Theorem 4.1. That we shall do in Theorem 5.4. This section is based on the work of Cegrell [19], and therefore shall need a few additional preliminaries.

### Definition 5.1

*m-Hessian measure*of

*u*is defined by

### Remark

The *m*-Hessian measure is well-defined for much more general functions than needed in this section. For further information see e.g., [9].

*m*-hyperconvex domain in \({\mathbb {C}}^n\), we shall use the following notation

*m*-hyperconvex domain admits an exhaustion function that is smooth, and strictly

*m*-subharmonic. Our method is that of approximation. Therefore, we first need to prove a suitable approximation theorem. Theorem 5.2 was first proved in the case \(m=n\) by Cegrell [19]. If the approximating sequence \(\{\psi _j\}\) is assumed to be only continuous on \(\Omega \), then the corresponding result was proved by Cegrell [18, Theorem 2.1] in the case \(m=n\), and Lu [46, Theorem 1.7.1] for general

*m*. In connection with Theorem 5.2, we would like to make a remark on Theorem 6.1 in a recent paper by Harvey et al. [34]. There they prove a similar approximation theorem, but there is an essential difference. They assume that the underlying space should admit a negative exhaustion function that is \(\mathcal {C}^2\)-smooth, and strictly

*m*-subharmonic. Thereafter, they prove that approximation is possible. Whereas we prove that smooth approximation is always possible on an

*m*-hyperconvex domain, i.e., there should only exist a negative exhaustion function. Thereafter, we prove the existence of a negative and smooth exhaustion function that is strictly

*m*-subharmonic, and has bounded

*m*-Hessian measure. We believe that Theorem 5.2 is of interest in its own right.

### Theorem 5.2

Assume that \(\Omega \) is a bounded *m*-hyperconvex domain in \({\mathbb {C}}^n\). Then, for any negative *m*-subharmonic function *u* defined on \(\Omega \), there exists a decreasing sequence \(\{\psi _j\}\subset {\mathcal {E}}_m^0(\Omega )\cap {\mathcal {C}}^{\infty }(\Omega )\) such that \(\psi _j\rightarrow u\), as \(j\rightarrow \infty \).

Before proving Theorem 5.2, we need the following lemma. The proof is as in [19], and therefore it is omitted.

### Lemma 5.3

Let *u*, *v* be smooth *m*-subharmonic functions in \(\Omega \) and let \(\omega \) be a neighborhood of the set \(\{u=v\}\). Then there exists a smooth *m*-subharmonic function \(\varphi \) such that \(\varphi \ge \max \{u,v\}\) on \(\Omega \) and \(\varphi =\max \{u,v\}\) on \(\Omega \setminus \omega \).

Now to the proof of Theorem 5.2.

### *Proof of Theorem *5.2

By Theorem 3.5, property (3), we can always find a continuous and negative exhaustion function \(\alpha \) for \(\Omega \) that is strictly *m*-subharmonic.

*Step 1.*Fix a constant \(s<0\) such that

*m*-subharmonic functions decreasing to

*bu*. Out of this sequence pick one function, \(\varphi _0'\), that is smooth in a neighborhood of the set \(\{2u\le c\}\), and such that \(\varphi _0'<u\) on \(\bar{\Omega }_0\). Next, define

*h*is the unique harmonic function on \(\Omega \setminus \Omega _0\) that is continuous up to the boundary, \(h=\varphi _0\) on \(\partial \Omega _0\) and \(h=0\) on \(\partial \Omega \). In fact the function

*h*can be obtain as (see [4])

*Step 2.*Let \(\Omega _0'\) be a given domain such that \(\Omega _0\Subset \Omega _0'\Subset \Omega \). We shall construct functions \(\varphi _1\), \({\tilde{\varphi }}_1\), and a domain \(\Omega _1\) with the following properties;

- (1)
\(\Omega _0'\Subset \Omega _1\Subset \Omega \) and \(\Omega _1=\{\alpha <s_1\}\), for some \(s_1<0\);

- (2)
\(\varphi _1,{\tilde{\varphi }}_1\in {\mathcal {E}}_m^0(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\);

- (3)
\(\varphi _0=\varphi _1\) on \(\Omega _0\);

- (4)
\(au<{\tilde{\varphi }}_1<u\) on \(\Omega \);

- (5)
\(\varphi _1={\tilde{\varphi }}_1\) on \(\Omega _1\);

- (6)
\(\{au\le \varphi _1\}\Subset \Omega \); and

- (7)
\(\varphi _1\) is smooth in a neighborhood of \(\{au\le \varphi _1\}\).

*m*-subharmonic functions decreasing to \(bu+d\). Take one function from this sequence, call it \(\varphi _1''\), such that it is smooth in a neighborhood of \(\{2u\le d\}\), and

*W*such that

*W*and \(\varphi _1=\varphi _0\) on \(\Omega _0\). It also follows that \(\varphi _1\) is smooth near \(\{au\le \varphi _1\}\) which contains \({\bar{\Omega }}_1\), since \(\varphi _1=\varphi _1'\) if \(\varphi _0<au\le \varphi _1\). Both functions \(\varphi _0\), and \(\varphi _1'\), are smooth near

*Step 1*, it follows that \({\tilde{\varphi }}_1\in \mathcal {SH}_m(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\). The constructions \(\varphi _1\), \({\tilde{\varphi }}_1\) and \(\Omega _1\) satisfy all the Conditions (1)–(7).

*Step 3*. Now if \(\Omega _j\nearrow \Omega \), then the function

*u*is a negative

*m*-subharmonic function defined on \(\Omega \). Theorem 1.7.1 in [46] implies that there exists a decreasing sequence \(\{u_j\}\subset {\mathcal {E}}_m^0(\Omega )\cap {\mathcal {C}}({\bar{\Omega }})\), \({\text {supp}}({\text {H}}_m(u_j))\Subset \Omega \), such that \(u_j\rightarrow u\), as \(j\rightarrow \infty \). Then by (5.1), there exists a sequence \(\psi _j\in {\mathcal {E}}_m^0(\Omega )\cap {\mathcal {C}}^{\infty }(\Omega )\) with

We shall end this paper by proving the implication \((1)\Rightarrow (3)\) in Theorem 4.1.

### Theorem 5.4

Assume that \(\Omega \) is a *m*-hyperconvex domain in \({\mathbb {C}}^n\), \(n\ge 2\), \(1\le m\le n\). Then \(\Omega \) admits an exhaustion function that is negative, smooth, strictly *m*-subharmonic, and has bounded *m*-Hessian measure.

### Proof

*m*-subharmonic, and belongs to \({\mathcal {E}}_m^0(\Omega )\). It is sufficient to take

*n*, we arrive at

*m*-subharmonic. Let \(\Omega '\Subset \Omega \), then there exists an index \(j_\omega \) such that on \(\Omega '\) we have that

## Notes

### Acknowledgements

R. Czyż was partially supported by NCN Grant DEC-2013/08/A/ST1/00312.

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