Delta-extremal functions in $\mathbb{C}^n$

Nurbek Kh. Narzillaev

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 3, pp. 389-398.

Résumé

The article is devoted to properties of a weighted Green function. We study the $(\delta,\psi)$-extremal Green function $V^{*}_{\delta}(z,K,\psi)$ defined by the class $\mathcal{L}_{\delta}=\big\{u(z)\in psh(\mathbb C^{n}):\ u(z) \leqslant C_{u}+\delta\ln^{+}|z|, \ z\in\mathbb C^{n}\big\}, \ \delta>0.$ We see that the notion of regularity of points with respect to different numbers $\delta$ differ from each other. Nevertheless, we prove that if a compact set $K\subset\mathbb{C}^{n}$ is regular, then $\delta$-extremal function is continuous in the whole space $\mathbb C^{n}.$

Détail

Keywords: plurisubharmonic function, Green function, weighted Green function, $\delta$-extremal function.

Bibliographie

[1] P. Lelong, Plurisubharmonic functions and positive differential forms, Gordon and Breach, New York, 1968

[2] V.P. Zakharyuta, “Bernstein's theorem on optimal polynomial approximations to holomorphic and harmonic functions of several variables”, All-Union Conf. Theory of functions, Khar'kov, 1971, 80–81

[3] V.P. Zakharyuta, “Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein–Walsh theorem for analytic functions of several complex variables”, Ann. Polon. Math., 33 (1976), 137–148

[4] J. Siciak, “Extremal plurisubharmonic functions in $\mathbb{C}^{n}$”, Proceedings of the First Finnish – Polish Summer School in complex Analysis at Podlesice, University of Lods, Lods, 1977, 115–152

[5] A. Sadullaev, “Plurisubharmonic measures and capacities on complex manifolds”, Russian Math. Surveys, 36:4 (1981), 53–105

[6] A. Sadullaev, Pluripotential theory. Applications, Palmarium Academic Publishing, 2012 (in Russian)

[7] A. Sadullaev, “Pluriregular compacts in $\mathbb{P}^{n}$”, Contemporary mathematics, 662, AMS, 2016, 145–156

[8] H. Whitney, “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36 (1934), 63–89

[9] A. Zeriahi, “Fonction de Green pluricomplexe a pole a l'infini sur un espace de Stein parabolique”, Math. Scand., 69 (1991), 89–126

[10] M.A. Alan, “Weighted regularity and two problems of Sadullaev on weighted regularity”, Complex Analysis and its Synergies, 8:5 (2019) | DOI

[11] M.A. Alan, N. Gökhan Göğüş, “Supports of weighted equilibrium measures: complete characterization”, Potential Analysis, 39:4 (2013), 411–415 (SCI)

[12] T. Bloom, N. Levenberg, “Weighted pluripotential theory in $\mathbb{C}^N$”, Amer. J. Math., 125:1 (2003), 57–103

[13] T. Bloom, “Weighted polynomials and weighted pluripotential theory”, Trans. Amer. Math. Soc., 361:4 (2009), 2163–2179

[14] M. Klimek, Pluripotential theory, Oxford University Press, New York, 1991

[15] N.X. Narzillaev, “About $\psi$-regular points”, Tashkent. Acta NUUz, 2:2 (2017), 173–175 (in Russian)

[16] E. Bedford, B.A. Taylor, “A new capacity for plurisubharmonic functions”, Acta Mathematica, 149 (1982), 1–41