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.
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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.Footnote 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 \(y\in K\)
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:
hence
Letting n to \(\infty \) we get
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
where \(\hat{F}(a)=\min \{F(a),5C\sqrt{1-|a|^2}\}\). By elementary calculations we get
and
where \(C'=\frac{20}{3}C<7C\). For \(0<\varepsilon <\sup _{B'}F-3C\) let
and let
By (1) and (2), we have \(\partial \Omega _2\cap \partial D=\partial \Omega \cap \partial D\). This gives us
Therefore by (2) it is clear that
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
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
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
and therefore the function v given by
is plurisubharmonic on \(\Omega \) and smaller than \(\tilde{\phi _k}\). Thus \( P_{\Omega _2}\tilde{\phi }_k=\hat{\phi }_k|_{\Omega _2}\). Let
then we have
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].Footnote 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
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
where
and
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
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
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.
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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?
Notes
Of course this notion does not depend on coordinates so we can define it on manifolds (for example compact manifolds are \(\mathcal {S}\)-domains). Theorem 1 can be also formulated on manifolds.
See also Theorem 2.6 there for elementary properties of \(\mathbf F \)-subharmonic functions needed in the proof. For the result of local continuous approximation of F-subharmonic functions see [5]
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Harvey, F.R., Lawson, Jr. H.B., Pliś, S.: Smooth approximation of plurisubharmonic functions on almost complex manifolds. Math. Ann. (to appear)
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The author was partially supported by the NCN Grant 2011/01/D/ST1/04192.
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Pliś, S. On regularization of plurisubharmonic functions near boundary points. Math. Z. 283, 381–385 (2016). https://doi.org/10.1007/s00209-015-1602-9
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DOI: https://doi.org/10.1007/s00209-015-1602-9