On regularization of plurisubharmonic functions near boundary points

Abstract

We prove in an elementary way that for a Lipschitz domain $D\subset \mathbb {C}^n$, all plurisubharmonic functions on D can be regularized near any boundary point.

1 Introduction

Let $D\subset \mathbb {C}^n$. Using a local convolution, for any plurisubharmonic function u on D, one can find a sequence $u_k$ of smooth plurisubharmonic functions, which decreases to u on compact subsets of D. The purpose of this note is to show that (for regular enough D) it is possible to choose such a sequence near any point on the boundary.

A domain $D\subset \mathbb {C}^n$ will be called an $\mathcal {S}$-domain if for any plurisubharmonic function u on D, one can find a sequence $u_k$ of smooth plurisubharmonic functions on D, which decreases to u.¹ Note that pseudoconvex domains, tube domains and Riendhard domains are $\mathcal {S}$-domains (see [3]).

Theorem 1

Let $D\subset \mathbb {C}^n$ be a domain with Lipschitz boundary. Then for any boundary point P there is a neighbourhood U of P such that $D\cap U$ is an $\mathcal {S}$-domain with Lipschitz boundary.

In Sect. 3, by a slight modification of an example from [2], we show the necessity of the assumption on the boundary.

Because of the fact that not every smooth domain is an $\mathcal {S}$-domain (see [1]) we have the following (surprising for the author) corollary:

Corollary 2

Being an $\mathcal {S}$-domain is not a local property of the boundary.

2 Proof

We need the following lemma:

Lemma 3

Let $D\subset \mathbb {R}^m$ be an open set. Let u be a subharmonic function on D and let $P\in D$. Let $a,b,R,C>0$, $B=\{x\in \mathbb {R}^m:|x-P|<R\}$, $K=\{(x',x_m)\in \mathbb {R}^{m-1}\times \mathbb {R}=\mathbb {R}^m:-a< x_m<-b|x'|\}$ and $B+K=\{x+y:x\in B, y\in K\}\subset D$. Assume that for any $x\in B$ and $y\in K$

$$\begin{aligned} u(x+y)\le u(x)+\delta (|y|), \end{aligned}$$

where $\delta :(0,+\infty )\rightarrow (0,+\infty )$ is increasing and such that $\lim _{t\rightarrow 0^+}\delta (t)=0$. Then u is continuous at P.

Proof

Let $(x_n)$ be any sequence in D which converges to P. Let $S_n=\{x\in \mathbb {R}:|x-P|=2|x_n-P|\}$, $A_n=S_n\cap (\{x_n\}+K)$ and $B_n=S_n{\setminus } A_n$. For n large enough $x_n\in B$ and $|x_n-P|\le \frac{a}{3}$. Hence there is a constant $\alpha >0$ (which depends only on b) such that $\alpha _n=\frac{\sigma (A_n)}{\sigma (S_n)}\ge \alpha$ where $\sigma$ is the standard measure on a sphere. Let $M_n=sup_{S_n}u$. We can estimate using the assumptions:

$$\begin{aligned}&u(P)\le \sigma (S_n)^{-1}\int _{A_n}ud\sigma +\sigma (S_n)^{-1}\int _{B_n}ud\sigma \\&\quad \le \alpha _n(u(x_n)+\delta (3|x_n-P|))+(1-\alpha _n)M_n, \end{aligned}$$

hence

$$\begin{aligned} u(x_n)\ge u(P)+\frac{1-\alpha _n}{\alpha _n}(u(P)-M_n)-\delta (3|x_n-P|). \end{aligned}$$

Letting n to $\infty$ we get

$$\begin{aligned} \varliminf _{n\rightarrow \infty }u(x_n)\ge u(P). \end{aligned}$$

Since u is upper semicontinuous the proof is completed. $\square$

The function $P_Df:=\sup \{u\in \mathcal {PSH}(D):u\le f\}$, where $D\subset \mathbb {C}^n$ and f is a (real) function on D, is called a plurisubharmonic envelope of f.

Proof of Theorem 1

We use the following notation $\mathbb {C}^n\ni z=(a,x)\in (\mathbb {C}^{n-1}\times \mathbb {R})\times \mathbb {R}$. We put $B=\{a\in \mathbb {C}^{n-1}\times \mathbb {R}:|a|<1\}$. After an affine change of coordinates we can assume that there is a constant $C>0$ and a function $F:B\rightarrow (3C,4C)$ such that:

• (i) $F(a)-F(b)\le C|a-b|$ for $a,b\in B$,

• (ii) $\partial D\cap B\times [-5C,5C]=\{(a,F(a)):a\in B\}$ and $P=(0,F(0))$,

• (iii) $0\in D$.

Let $U=\{(a,x)\in B\times \mathbb {R}:|a|^2+\left( \frac{x}{5C}\right) ^2<1\}$. We will show that $\Omega =D\cap U$ is an $\mathcal {S}$-domain. Observe that

$$\begin{aligned} \partial \Omega \cap \left( B\times [0,5C]\right) =\{(a,\hat{F}(a)):a\in B\}, \end{aligned}$$

where $\hat{F}(a)=\min \{F(a),5C\sqrt{1-|a|^2}\}$. By elementary calculations we get

$$\begin{aligned} \{\hat{F}=F\}\subset B'=\left\{ a\in \mathbb {C}^{n-1}\times \mathbb {R}:|a|<\frac{4}{5}\right\} \end{aligned}$$

(1)

and

$$\begin{aligned} \hat{F}(a)-\hat{F}(b)\le C'|a-b| \quad \hbox {for } a,b\in B', \end{aligned}$$

(2)

where $C'=\frac{20}{3}C<7C$. For $0<\varepsilon <\sup _{B'}F-3C$ let

$$\begin{aligned} K=K(\varepsilon )=\{(a,x):-\varepsilon <x<-7C|a|\} \end{aligned}$$

and let

$$\begin{aligned} \Omega _k=\Omega _k(\varepsilon )=\{z\in \Omega :z+kw\in \Omega \hbox { for any } w\in \bar{K}\},\hbox { for }k=1,2. \end{aligned}$$

By (1) and (2), we have $\partial \Omega _2\cap \partial D=\partial \Omega \cap \partial D$. This gives us

$$\begin{aligned} \partial \Omega _2=\left( \partial \Omega _2\cap \partial D\right) \cup \left( \partial \Omega _2\cap \partial U\right) \cup \left( \partial \Omega _2\cap \Omega _1\right) . \end{aligned}$$

Therefore by (2) it is clear that

$$\begin{aligned} z+w\in \Omega \end{aligned}$$

for $z\in \partial \Omega _2\cap \partial D$ and $w\in \bar{K}$. The same inequality holds on $\partial \Omega _2\cap \partial U$ because of the convexity of U. Thus we get

$$\begin{aligned} L=L(\varepsilon ):=\{z\in \bar{\Omega }_2:\mathrm{dist}(z,\partial \Omega )\ge \mathrm{dist}(z+w,\partial \Omega )\hbox { for some } w\in \bar{K}\}\Subset \Omega _1. \end{aligned}$$

Let $d=-\log (\mathrm{dist}(\cdot ,\partial \Omega ))$ and $d'=-\log (\mathrm{dist}(\cdot ,\partial U))$. We only know that the second function is plurisubharmonic, but by the construction of $\Omega$ and $\Omega _2$ (decreasing $\varepsilon$ if necessary) we have $d=d'$ on $\Omega {\setminus }\Omega _2$, hence on this set, the function d is plurisubharmonic too.

Let $u\in \mathcal {PSH}(\Omega )$ and let $\phi _k$ be a sequence of continuous functions on $\Omega$ which decreases to u. We can choose an increasing convex function $p:\mathbb {R}\rightarrow \mathbb {R}$ such that for a function $\rho =p\circ d$ we have $\lim _{z\rightarrow \partial \Omega }\rho -\phi _1=+\infty$ (see claim 3.5 in [5]). Put $\tilde{\phi }_k=\max \{\phi _k,\rho -k\}$. Observe that functions $\hat{\phi }_k=P_\Omega \tilde{\phi }_k$ are plurisubharmonic and they decrease to u.

Fix k. Because $\tilde{\phi }_k=\rho -k$ outside of a compact set, for $\varepsilon >0$ small enough, we have

$$\begin{aligned} cl_\Omega (\Omega {\setminus }\Omega _2)\subset S, \end{aligned}$$

where $S=S(\varepsilon )=int\{\tilde{\phi }_k=\rho -k\}$.

The function $\rho ':=p\circ d'$ is plurisubharmonic and $\rho '\le \tilde{\phi }_k$. Thus on $\Omega _2$ we have

$$\begin{aligned} \tilde{\phi }_k\ge P_{\Omega _2}\tilde{\phi }_k\ge \hat{\phi }_k\ge \rho '-k \end{aligned}$$

and therefore the function v given by

$$\begin{aligned} v(z)=\left\{ \begin{array}{ll} P_{\Omega _2}\tilde{\phi }_k(z) &{} \quad \hbox {for} \quad z\in \Omega _2\\ \rho (z)'-k(=\tilde{\phi }_k) &{} \quad \hbox {on} \quad \Omega {\setminus }\Omega _2, \end{array}\right. \end{aligned}$$

is plurisubharmonic on $\Omega$ and smaller than $\tilde{\phi _k}$. Thus $P_{\Omega _2}\tilde{\phi }_k=\hat{\phi }_k|_{\Omega _2}$. Let

$$\begin{aligned} L'=L'(\varepsilon ):=\{z\in \bar{\Omega }_2:\rho (z)\ge \sup _{\Omega \setminus S}\phi _k\}\Subset \Omega _1, \end{aligned}$$

then we have

$$\begin{aligned} N=N(\varepsilon ):=(L\cup L')+\bar{K}\Subset \Omega . \end{aligned}$$

Observe that for $z\in \Omega _2$ and $w\in K$ we have $\tilde{\phi }_k(z)\ge \tilde{\phi }_k(z+w)-\omega (|w|)$, where $\omega$ is the modulus of continuity of the function $\tilde{\phi }_k|_N$. Therefore, $\hat{\phi }_k(z)\ge \hat{\phi }_k(z+w)-\omega (|w|)$. By Lemma 3 the function $\hat{\phi }_k|_{\Omega _2}$ is continuous.

Because any $z\in \Omega$ is in $\Omega _2(\varepsilon )$ for some $\varepsilon$ as above, we obtain that the function $\hat{\phi }_k$ is continuous on $\Omega$. Using the Richberg theorem we can modify the sequence $\hat{\phi }_k$ to a sequence $u_k$ of smooth plurisubharmonic functions which decreases to u. $\square$

The approximation by continuous functions, can be proved in the same way in a much more general situation.

Theorem 4

Let $\mathbf F$ be a constant coefficient subequation such that all $\mathbf F$-subharmonic functions are subharmonic and all convex functions are $\mathbf F$-subharmonic. Let $D\subset \mathbb {R}^n$ be a domain with Lipschitz boundary. Then for any point $P\in \bar{D}$ there is a neighbourhood U of P such that for any function $u\in \mathbf F (D\cap U)$ there is a sequence $(u_k)\subset \mathbf F (D\cap U)$ of continuous functions decreasing to u.

Here we use terminology from [4].²

3 Example

Similarly as in Lecture 14 in [2] we construct a domain $\Omega \subset \mathbb {C}^n$ and a plurisubharmonic function u on $\Omega$ which can not be regularized. Let $A=\{\frac{1}{k}:k\in \mathbb {N}\}$ and a sequence $(x_k)\subset (0,1){\setminus } A$ is such that its limit set is equal $\bar{A}$. Put

$$\begin{aligned} \lambda (z)=\sum _{k=1}^\infty c_k\log |z-x_k| \quad \hbox {for}\,\, z\in \mathbb {C}, \end{aligned}$$

where $c_k$ is a such sequence of numbers rapidly decreasing to 0, such that

• (i) $\lambda$ is a subharmonic function on $\mathbb {C}$ and

• (ii) $\lambda |_A\ge -\frac{1}{2}$.

For $k\in \mathbb {N}$ let $D_k$ be a disc with center $x_k$ such that $\lambda |_{D_k}<-1$. Now, we can define

$$\begin{aligned} \Omega =\{(z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|^2+|z_n-1|^2<1\}{\setminus } K, \end{aligned}$$

where

$$\begin{aligned} K=\{(z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|=|z_n| \hbox { and } z_n\notin \cup _{k=1}^\infty D_k\}, \end{aligned}$$

and

$$\begin{aligned} u(z',z_n)=\left\{ \begin{array}{ll} -1 &{} \quad \hbox {for} \quad z\in \Omega \cap D\\ \max \{\lambda {(z_n)},-1\} &{} \quad \hbox {on} \quad \Omega {\setminus } D, \end{array}\right. \end{aligned}$$

where $D=\{(z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|>|z_n| \}$.

Let U be any neighbourhood of 0. We can choose numbers $k\in \mathbb {N}$, $0<r<\frac{1}{k}$ such that

$$\begin{aligned} \Omega _U:=\left\{ (z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|\le \frac{2}{k},|z_n-\frac{1}{k}|\le r\right\} {\setminus } K\subset \Omega \cap U. \end{aligned}$$

Let $y_p$ be a subsequence of $x_p$ such that $\lim _{p\rightarrow \infty }y_p=\frac{1}{k}$ and for all p we have $|y_p-\frac{1}{k}|<r$. Now, we can repeat the argument from [2]. If $u_q$ is a sequence of smooth plurisubharmonic functions decreasing to u on $\Omega _U$, then for q sufficiently large $u_q\le -\frac{3}{4}$ on the set

$$\begin{aligned} \left\{ (z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|=\frac{2}{k},|z_n-\frac{1}{k}|\le r\right\} \subset \partial \Omega _U. \end{aligned}$$

By the maximum principle (on sets $\{(z',z_n)\in \mathbb {C}^{n-1}\times \mathbb {C}:|z'|\le \frac{2}{k},z_n=y_p\}\subset \mathbb {C}^{n-1}\times \{y_p\}$) we also have $u_q(0,y_p)\le -\frac{3}{4}$ and by continuity of $u_q$ we get $u_q(0,\frac{1}{k})\le -\frac{3}{4}<u(0,\frac{1}{k})$. This is a contradiction.

Note that we have not only just proved that for the above $\Omega$ Theorem 1 does not hold but we also have the following stronger result:

Proposition 5

Let $\Omega$ and u be as above. Then the function u can not be smoothed on $U\cap \Omega$ for any neighbourhood U of $0\in \partial \Omega$.

4 Questions

In this last section we state some open questions related to the content of the note.

• 1. Is it possible to characterize $\mathcal {S}$-domains? In view of Corollary 2, even in the class of smooth domains, it seems to be a challenging problem.

• 2. Let D be an $\mathcal {S}$-domain and let f be a continuous function which is bounded from below. Is the plurisubharmonic envelope of f continuous? Assume in addition that f is smooth. What is the optimal regularity of $P_Df$? The author does not know answers even in the case of the ball in $\mathbb {C}^n$. Note that if D is not an $\mathcal {S}$-domain, then there is a smooth function f bounded from below such that $P_Df$ is discontinuous.

• 3. What is the optimal assumption about the regularity of the boundary of D in the Theorem 1? Is it enough to assume that $D=int\bar{D}$?

• 4. Let M be a real (smooth or Lipschitz) hypersurface in $\mathbb {C}^n$. Is it true that for any $P\in M$ there exists a smooth pseudoconvex neighbourhood $U\subset B$ such that M divides U into two $\mathcal {S}$-domains?