We exhibit a smoothly bounded domain $\Omega$ with the property that for
suitable $K\subset\partial \Omega$ and $z\in \Omega$ the "Sadullaev boundary
relative extremal functions" satisfy the inequality
$\omega_1(z,K,\Omega)<\omega_2(z,K,\Omega)\le \omega(z,K,\Omega)$.Comment: 3 page

Wiegerinck, Jan

English

arXiv

We exhibit a smoothly bounded domain Ω with the property that for suitable K⊂∂Ω and z∈Ω the Sadullaev boundary relative extremal functions satisfy ω1(z,K,Ω)<ω2(z,K,Ω)≤ω(z,K,Ω)

Wiegerinck, J.J.O.O.

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An example concerning Sadullaev’s boundary relative extremal functions

We exhibit a smoothly bounded domain Ω with the property that for suitable K⊂∂Ω and z∈Ω the Sadullaev boundary relative extremal functions satisfy ω1(z,K,Ω)&lt;ω2(z,K,Ω)≤ω(z,K,Ω)

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)An example concerning Sadullaev’s boundary relative extremal functionsWiegerinck, J.J.O.O.DOI10.4064/ap180604-19-6Publication date2019Document VersionSubmitted manuscriptPublished inAnnales Polonici mathematiciLink to publicationCitation for published version (APA):Wiegerinck, J. J. O. O. (2019). An example concerning Sadullaev’s boundary relativeextremal functions. Annales Polonici mathematici, 123, 505-508.https://doi.org/10.4064/ap180604-19-6General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:26 Jul 2022arXiv:1805.05329v1  [math.CV]  14 May 2018AN EXAMPLE CONCERNING SADULLAEV’S BOUNDARY RELATIVEEXTREMAL FUNCTIONSJAN WIEGERINCKIn memory of Józef SiciakAbstract. We exhibit a smoothly bounded domain Ω with the property that for suitableK ⊂ ∂Ω and z ∈ Ω the Sadullaev boundary relative extremal functions satisfy the inequalityω1(z,K,Ω) < ω2(z,K,Ω) 6 ω(z,K,Ω).1. IntroductionIn [5] Sadullaev introduced several so-called boundary relative extremal functions for compactsets K in the boundary of domains D ⊂ Cn, and asked whether their regularizations are perhapsalways equal. Recently Djire and the author [1, 2] gave a positive answer in certain cases whereD and K are particularly nice.In this note we show that in general equality does not hold. The example is formed by a suitablecompact set in the boundary of the domain Ω that was constructed by Fornæss and the author[3] as an example of a domain D where bounded plurisubharmonic functions that are continuouson D cannot be approximated by plurisubharmonic functions that are continuous on D. We startby briefly recalling the definitions of boundary relative extremal functions and the construction ofthe domain Ω.1.1. Boundary relative extremal functions. We follow Sadullaev [5, Section 27]. Let D bea domain with smooth boundary in Cn, ξ ∈ ∂D, and Aα(ξ) = {z ∈ D; |z − ξ| < αδξ(z)}, whereα > 1 and δξ(z) is the distance from z to the tangent plane at ξ to ∂D. For a function u definedon D, putũ(ξ) = supα>1lim supz→ξz∈Aα(ξ)u(z), ξ ∈ ∂D.Definition 1.1. Let PSH(D) denote the plurisubharmonic functions on D and let K ⊂ ∂D becompact. We define the following boundary relative extremal functions(1)ω(z,K,D) = sup{u(z) : u ∈ PSH(D), u 6 0, ũ|K 6 −1};(2)ω1(z,K,D) = sup{u(z) : u ∈ PSH(D) ∩ C(D), u 6 0, u|K 6 −1};(3)ω2(z,K,D) = sup{u(z) : u ∈ PSH(D), u 6 0, lim supz→ξz∈D6 −1, for all ξ ∈ K}.The upper semi-continuous regularization u∗ of a function u on a domain D is defined asu∗(z) = lim supw→z{u(w)}.The functions ω∗, ω∗1 , ω∗2 are plurisubharmonic. Observing that ω1(z,K,D) 6 ω2(z,K,D) 6ω(z,K,D), Sadullaev’s question is for what j is ω∗(z,K,D) ≡ ω∗j (z,K,D)?2000 Mathematics Subject Classification. 32U05,32U20.Key words and phrases. plurisubharmonic function; boundary relative extremal function.12 JAN WIEGERINCK1.2. The domain Ω. We briefly recall the construction and properties of the domain Ω from [3].(1.1) Ω = {(z, w) ∈ C2; |w − eiϕ(|z|)|2 < r(|z|)}.Here r and ϕ are in C∞(R) with the following properties: −1 6 r 6 2; r(t) 6 0 for t 6 1 and fort > 17; r(t) ≡ 1 for 3 6 t 6 8 and for 10 6 t 6 15; r(t) takes its maximum value = 2 precisely att = 2, 9, and 16. Moreover, r′(t) > 0 on 1 6 t < 2, 8 < t < 9 and 15 < t < 16, while f ′(t) < 0 on2 < t < 3, 9 < t < 10, and 16 < t 6 17. Next ϕ satisfies ϕ(t) < −π/2 for t 6 4 and for t > 14;ϕ(t) > π/2+ 100 for 5 6 t 6 6 and for 12 6 t 6 13 and ϕ(t) < −π/2+ 100 for 7 < t < 10, and wedemand in addition that ϕ 6 108.From [3] we recall that Ω is a Hartogs domain with smooth boundary, and that the annulus(1.2) A = {(z, w);w = 0, 2 6 |z| 6 15}is contained in Ω.2. Negative answer to Sadullaev’s questionTheorem 2.1. Let K = {(z, w ∈ ∂Ω; |z| = 2 or |z| = 16}. Thenω1((z, w),K,Ω) < ω2((z, w),K,Ω)for (z, w) in an open neighborhood of {w = 0, |z| = 9}.Proof. Let u ∈ PSH(Ω) ∩ C(Ω), u 6 0, u|K 6 −1. Then by the maximum principle, |u| 6 −1 onthe discs |w − eiϕ(|z|)| 6 2, where z is fixed and satisfies |z| = 2 or |z| = 16, and in particular onthe circles C1(w) = {(z, w) : |z| = 2} and C2(w) = {(z, w) : |z| = 16}, where |w| < 1. Because Ωis a smoothly bounded domain, it follows from [3, Theorem 1] (see also [4] for recent extensions ofthis theorem), that u can be approximated uniformly on Ω by smooth plurisubharmonic functionsv defined on shrinking neighborhoods of Ω.Let Ωδ = {ζ ∈ C2; d(ζ,Ω) < δ}. Then given ε > 0, there exist δ > 0 and v ∈ PSH(Ωδ), suchthat |u − v| < ε on Ω. For |w| < δ the annulus Aw = {(z, w) : 2 6 |z| 6 16} is contained inΩδ. On its boundary, which equals C1(w) ∪ C2(w), we have that v < −1 + ε, hence this alsoholds on Aw. It follows that u < −1 + 2ε on Aw ∩ Ω, in particular u < −1 + 2ε on the open setV = {(z, w) : 8 < |z| < 10, |w| < δ, |w| < r(|z|)−1} ⊂ Ω. It follows that ω1((z, w),K,Ω) 6 −1+2εon V , and therefore also ω∗1((z, w),K,Ω) 6 −1 + 2ε on V .Next we will construct a plurisubharmonic function in the family that determines ω2. Theconstruction is as in [3, Section 2]. On Ω ∩ ({3 < |z| < 8} ∪ {10 < |z| < 15} there exists acontinuous branch of argw, denoted by h(z, w), such thatϕ(z)− π/2 6 h(z, w) 6 ϕ(z) + π/2.In [3] we constructed the following plurisubharmonic function.(2.1) f(z, w) =0 if |z| < 4 or if |z| > 14max{0, h(z, w)} if 3 < |z| < 6 or if 12 < |z| < 14max{100, h(z, w)} if 5 < |z| < 8 or if 10 < |z| < 13100 if 7 < |z| < 11.It satisfies f 6 110 on Ω, f ≡ 0 on {|z| 6 3} and on {|z| > 14}, hence f extends continuously by 0to Ω∩ ({|z| 6 3}∪ {|z| > 14}), and f = 100 on V . The plurisubharmonic function g on Ω definedbyg(ζ) =f(ζ)− 110110, (ζ = (z, w))is negative, identically equal to −1 on Ω ∩ ({|z| 6 3} ∪ {|z| > 14}), and equal to −10/11 on V .Hence also ω∗2((z, w),K,Ω) > ω2((z, w),K,Ω) > −10/11 on V . Choosing ε < 1/10 completes theproof. AN EXAMPLE CONCERNING SADULLAEV’S BOUNDARY RELATIVE EXTREMAL FUNCTIONS 3References[1] Djire, I. K., Wiegerinck, J. Characterizations of boundary pluripolar hulls, Comp. Var. and Ellip. Equat. 61(2016) no. 8, 1133–1144.[2] Djire, I. K., Wiegerinck, J. On a question of Sadullaev concerning boundary relative extremal functions,arXiv:1611.01132v3 [math.CV]. To appear in Math. Scand.[3] Fornæss, J.E., Wiegerinck, J. Approximation of plurisubharmonic functions. Ark. Mat. 27 (1989), 257—272.[4] Persson, H., Wiegerinck, J. A note on approximation of plurisubharmonic functions. Ark. Mat. 55 (2017), no.1, 229—241.[5] Sadullaev, A. Plurisubharmonic measures and capacities on complex manifolds, (Russian) Uspekhi Mat. Nauk36 (1981) no. 4, 53–105. English translation in Russian Math. Surveys, 36 (1981), no. 4, 61–119.KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, P.O. box 94248,1090 GE Amsterdam, The NetherlandsE-mail address: j.j.o.o.wiegerinck@uva.nl

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)An example concerning Sadullaev’s boundary relative extremal functionsWiegerinck, J.J.O.O.DOI10.4064/ap180604-19-6Publication date2019Document VersionSubmitted manuscriptPublished inAnnales Polonici mathematiciLink to publicationCitation for published version (APA):Wiegerinck, J. J. O. O. (2019). An example concerning Sadullaev’s boundary relativeextremal functions. Annales Polonici mathematici, 123, 505-508.https://doi.org/10.4064/ap180604-19-6General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:09 Mar 2023arXiv:1805.05329v1  [math.CV]  14 May 2018AN EXAMPLE CONCERNING SADULLAEV’S BOUNDARY RELATIVEEXTREMAL FUNCTIONSJAN WIEGERINCKIn memory of Józef SiciakAbstract. We exhibit a smoothly bounded domain Ω with the property that for suitableK ⊂ ∂Ω and z ∈ Ω the Sadullaev boundary relative extremal functions satisfy the inequalityω1(z,K,Ω) < ω2(z,K,Ω) 6 ω(z,K,Ω).1. IntroductionIn [5] Sadullaev introduced several so-called boundary relative extremal functions for compactsets K in the boundary of domains D ⊂ Cn, and asked whether their regularizations are perhapsalways equal. Recently Djire and the author [1, 2] gave a positive answer in certain cases whereD and K are particularly nice.In this note we show that in general equality does not hold. The example is formed by a suitablecompact set in the boundary of the domain Ω that was constructed by Fornæss and the author[3] as an example of a domain D where bounded plurisubharmonic functions that are continuouson D cannot be approximated by plurisubharmonic functions that are continuous on D. We startby briefly recalling the definitions of boundary relative extremal functions and the construction ofthe domain Ω.1.1. Boundary relative extremal functions. We follow Sadullaev [5, Section 27]. Let D bea domain with smooth boundary in Cn, ξ ∈ ∂D, and Aα(ξ) = {z ∈ D; |z − ξ| < αδξ(z)}, whereα > 1 and δξ(z) is the distance from z to the tangent plane at ξ to ∂D. For a function u definedon D, putũ(ξ) = supα>1lim supz→ξz∈Aα(ξ)u(z), ξ ∈ ∂D.Definition 1.1. Let PSH(D) denote the plurisubharmonic functions on D and let K ⊂ ∂D becompact. We define the following boundary relative extremal functions(1)ω(z,K,D) = sup{u(z) : u ∈ PSH(D), u 6 0, ũ|K 6 −1};(2)ω1(z,K,D) = sup{u(z) : u ∈ PSH(D) ∩ C(D), u 6 0, u|K 6 −1};(3)ω2(z,K,D) = sup{u(z) : u ∈ PSH(D), u 6 0, lim supz→ξz∈D6 −1, for all ξ ∈ K}.The upper semi-continuous regularization u∗ of a function u on a domain D is defined asu∗(z) = lim supw→z{u(w)}.The functions ω∗, ω∗1 , ω∗2 are plurisubharmonic. Observing that ω1(z,K,D) 6 ω2(z,K,D) 6ω(z,K,D), Sadullaev’s question is for what j is ω∗(z,K,D) ≡ ω∗j (z,K,D)?2000 Mathematics Subject Classification. 32U05,32U20.Key words and phrases. plurisubharmonic function; boundary relative extremal function.12 JAN WIEGERINCK1.2. The domain Ω. We briefly recall the construction and properties of the domain Ω from [3].(1.1) Ω = {(z, w) ∈ C2; |w − eiϕ(|z|)|2 < r(|z|)}.Here r and ϕ are in C∞(R) with the following properties: −1 6 r 6 2; r(t) 6 0 for t 6 1 and fort > 17; r(t) ≡ 1 for 3 6 t 6 8 and for 10 6 t 6 15; r(t) takes its maximum value = 2 precisely att = 2, 9, and 16. Moreover, r′(t) > 0 on 1 6 t < 2, 8 < t < 9 and 15 < t < 16, while f ′(t) < 0 on2 < t < 3, 9 < t < 10, and 16 < t 6 17. Next ϕ satisfies ϕ(t) < −π/2 for t 6 4 and for t > 14;ϕ(t) > π/2+ 100 for 5 6 t 6 6 and for 12 6 t 6 13 and ϕ(t) < −π/2+ 100 for 7 < t < 10, and wedemand in addition that ϕ 6 108.From [3] we recall that Ω is a Hartogs domain with smooth boundary, and that the annulus(1.2) A = {(z, w);w = 0, 2 6 |z| 6 15}is contained in Ω.2. Negative answer to Sadullaev’s questionTheorem 2.1. Let K = {(z, w ∈ ∂Ω; |z| = 2 or |z| = 16}. Thenω1((z, w),K,Ω) < ω2((z, w),K,Ω)for (z, w) in an open neighborhood of {w = 0, |z| = 9}.Proof. Let u ∈ PSH(Ω) ∩ C(Ω), u 6 0, u|K 6 −1. Then by the maximum principle, |u| 6 −1 onthe discs |w − eiϕ(|z|)| 6 2, where z is fixed and satisfies |z| = 2 or |z| = 16, and in particular onthe circles C1(w) = {(z, w) : |z| = 2} and C2(w) = {(z, w) : |z| = 16}, where |w| < 1. Because Ωis a smoothly bounded domain, it follows from [3, Theorem 1] (see also [4] for recent extensions ofthis theorem), that u can be approximated uniformly on Ω by smooth plurisubharmonic functionsv defined on shrinking neighborhoods of Ω.Let Ωδ = {ζ ∈ C2; d(ζ,Ω) < δ}. Then given ε > 0, there exist δ > 0 and v ∈ PSH(Ωδ), suchthat |u − v| < ε on Ω. For |w| < δ the annulus Aw = {(z, w) : 2 6 |z| 6 16} is contained inΩδ. On its boundary, which equals C1(w) ∪ C2(w), we have that v < −1 + ε, hence this alsoholds on Aw. It follows that u < −1 + 2ε on Aw ∩ Ω, in particular u < −1 + 2ε on the open setV = {(z, w) : 8 < |z| < 10, |w| < δ, |w| < r(|z|)−1} ⊂ Ω. It follows that ω1((z, w),K,Ω) 6 −1+2εon V , and therefore also ω∗1((z, w),K,Ω) 6 −1 + 2ε on V .Next we will construct a plurisubharmonic function in the family that determines ω2. Theconstruction is as in [3, Section 2]. On Ω ∩ ({3 < |z| < 8} ∪ {10 < |z| < 15} there exists acontinuous branch of argw, denoted by h(z, w), such thatϕ(z)− π/2 6 h(z, w) 6 ϕ(z) + π/2.In [3] we constructed the following plurisubharmonic function.(2.1) f(z, w) =0 if |z| < 4 or if |z| > 14max{0, h(z, w)} if 3 < |z| < 6 or if 12 < |z| < 14max{100, h(z, w)} if 5 < |z| < 8 or if 10 < |z| < 13100 if 7 < |z| < 11.It satisfies f 6 110 on Ω, f ≡ 0 on {|z| 6 3} and on {|z| > 14}, hence f extends continuously by 0to Ω∩ ({|z| 6 3}∪ {|z| > 14}), and f = 100 on V . The plurisubharmonic function g on Ω definedbyg(ζ) =f(ζ)− 110110, (ζ = (z, w))is negative, identically equal to −1 on Ω ∩ ({|z| 6 3} ∪ {|z| > 14}), and equal to −10/11 on V .Hence also ω∗2((z, w),K,Ω) > ω2((z, w),K,Ω) > −10/11 on V . Choosing ε < 1/10 completes theproof. AN EXAMPLE CONCERNING SADULLAEV’S BOUNDARY RELATIVE EXTREMAL FUNCTIONS 3References[1] Djire, I. K., Wiegerinck, J. Characterizations of boundary pluripolar hulls, Comp. Var. and Ellip. Equat. 61(2016) no. 8, 1133–1144.[2] Djire, I. K., Wiegerinck, J. On a question of Sadullaev concerning boundary relative extremal functions,arXiv:1611.01132v3 [math.CV]. To appear in Math. Scand.[3] Fornæss, J.E., Wiegerinck, J. Approximation of plurisubharmonic functions. Ark. Mat. 27 (1989), 257—272.[4] Persson, H., Wiegerinck, J. A note on approximation of plurisubharmonic functions. Ark. Mat. 55 (2017), no.1, 229—241.[5] Sadullaev, A. Plurisubharmonic measures and capacities on complex manifolds, (Russian) Uspekhi Mat. Nauk36 (1981) no. 4, 53–105. English translation in Russian Math. Surveys, 36 (1981), no. 4, 61–119.KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, P.O. box 94248,1090 GE Amsterdam, The NetherlandsE-mail address: j.j.o.o.wiegerinck@uva.nl

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An example concerning Sadullaev's boundary relative extremal functions

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