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Further Developments of the Pluripotential Theory (Survey)

  • Azimbay Sadullaev
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

Abstract

It is well known that pluripotential theory, constructed in the 1980s, is based on plurisubharmonic (psh) functions and on the Monge-Amp\(\grave{e}\)re operator \((dd^c u)^n\). In the 1990s there were many attempts to develop and expand pluripotential theory to broader classes such as the class of m-subharmonic \((m-sh)\) functions \((1\le m \le n)\). In this paper we will discuss some of the most important results of the theory of \(m-sh\) function as well as the difficulties and problems of constructing a potential theory in the class of \(m-sh\) functions.

Keywords

Pluripotential theory Plurisubharmonic functions Operator Monge-Ampere Pluripolar sets M-subharmonic functions Maximal M-subharmonic functions 

Notes

Acknowledgements

I would like to express my warm thanks to the referee of this paper for numerous corrections and for witty simplification of the proof of Lemma 1, using Newton’s inequality.

References

  1. 1.
    Abdullayev, B.I.: Subharmonic functions on complex Hyperplanes of \({\bf C}^n.\) J. Sib. Fed. Univ. Math. Phys. Krasnoyarsk 6(4), 409–416 (2013)Google Scholar
  2. 2.
    Abdullayev, B.I.: \(\cal{P}\)-measure in the class of \(m-wsh\) functions. J. Sib. Fed. Univ. Math. Phys. Krasnoyarsk 7(1), 3–9 (2014)Google Scholar
  3. 3.
    Abdullaev, B., Sadullaev, A.: Potential theory in the class of \(m\)-subharmonic functions. Proc. Steklov Inst. Math. RAN 279, 155–180 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abdullaev, B., Sadullaev, A.: Capacities and hessians in the class of \(m\)-subharmonic functions. Dokl. Math. RAN 87(1), 88–90 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aleksandrov, A.D.: Dirichlet problem for the equation \(\det \left(z_{i,j} \right)=\varphi .\) Vestnik Leningrad Univ. 13, 5–24 (1958)Google Scholar
  6. 6.
    Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (1994)CrossRefGoogle Scholar
  7. 7.
    Bedford, E.: Survey of pluripotential theory. Several Complex Variables. Mathematical Notes, vol. 38, pp. 48–95 (1993)Google Scholar
  8. 8.
    Bedford, E., Fornaess, J.E.: Counterexamples to regularity for the complex Monge-Amp\(\grave{e}\)re equation. Invent. Math. 50, 129–134 (1979)CrossRefGoogle Scholar
  9. 9.
    Bedford, E., Kalka, M.: Foliations and complex Monge-Amp\(\grave{e}\)re equation. Comm. Pure Appl. Math. XXX, 543–571 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge- Amp\(\grave{e}\)re equations. Invent. Math. 37(1), 1–44 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bedford, E., Taylor, B.A.: Variational properties of the complex Monge- Amp\(\grave{e}\)re equation. I. Dirichlet principle. Duke Math. J. 45(2), 375–403 (1978); II. Intrisic norms. Am. J. Math. 101, 1131–1166 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149(1–2), 1–40 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bishop, E.: Condition for the analyticity of certain sets. Michigan Math. J. 11, 289–304 (1964)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Błocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier, Grenoble, 55, 5, 1735–1756 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Caffarelli, L., Kohn, J.J., Nirenberg, L., Spruck, J.: The Dirichlet problem for non linear second order elliptic equations, II. Complex Monge-Amp\(\grave{e}\)re and Uniformly Elliptic equations. Commun. Pure Appl. Math. 38, 209–252 (1985)CrossRefGoogle Scholar
  16. 16.
    Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for non linear second order elliptic equations, III. Functions of eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Demailly, J.P.: Measures de Monge - Amp\(\grave{e}\)re et caracterisation des varietes algebriques affines. Memoires de la societe mathematique de France 19, 1–125 (1985)Google Scholar
  18. 18.
    Dinew, S., Kołodziej, S.: A priori estimates for the complex Hessian equation Anal. PDE, l7, 227–244 (2014)Google Scholar
  19. 19.
    Drnov\(\check{s}\)ek, B.D., Forstneri\(\check{c}\), F.: Minimal hulls of compact sets in \(\mathbf{R}^3 \). Trans. Am. Math. Soc. 368(10), 7477–7506, October 2016. http://dx.doi.org/10.1090/tran/6777. Article electronically published on December 14, 2015Google Scholar
  20. 20.
    Harvey Jr., F.R., Lawson, H.B.: Calibrated geometries. Acta Mathematica 148, 47–157 (1982)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Harvey, F.R., Jr, L.H.B.: An introduction to potential theory in calibrated geometry. Amer. J. Math. 131(4), 893–944 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Harvey, F.R., Jr, L.H.B.: Duality of positive currents and plurisubharmonic functions in calibrated geometry. Amer. J. Math. 131(5), 1211–1240 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Harvey, F.R., Lawson, H.B.: Plurisubharmonicity in a general geometric context. Geom. Anal. I, 363–401 (2010)Google Scholar
  24. 24.
    Harvey Jr., F.R., Lawson, H.B.: Geometric plurisubharmonicity and convexity - an introduction. Adv. Math. 230, 2428–2456 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ho, L.H.: \(\partial \)-problem on weakly \(q\)-convex domains. Math. Ann. 290, 3–18 (1991)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ivochkina, N.: Solution of the Dirichlet problem for a Monge-Amp\(\grave{e}\)re type equations. Math. USSR Sbornic 128(3), 403–415 (1985)Google Scholar
  27. 27.
    Ivochkina, N., Trudinger, N.S., Wang, X.-J.: The Dirichlet problem for degenerate Hessian equations. Comm. Partial Diff. Eq. 29, 219–235 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Joyce, D.: Riemannian holonomy groups and calibrated geometry. Oxford Graduate Texts in Mathematics 12, OUP (2007)Google Scholar
  29. 29.
    Khusanov, Z.: Capacity properties of \(q-\) subharmonic functions, I, Izv. Akad. Nauk. UzSSR, Ser. Fiz.-Mat. 1, 41–45 (1990)Google Scholar
  30. 30.
    Khusanov, Z.: Capacity properties of \(q-\) subharmonic fonctions, II, Izv. Akad. Nauk. UzSSR, Ser. Fiz.-Mat. 5, 28–33 (1990)Google Scholar
  31. 31.
    Klimek, M.: Pluripotential Theory. Clarendon Press, Oxford (1991)zbMATHGoogle Scholar
  32. 32.
    Kołodziej, S.: The complex Monge-Amp\(\grave{e}\)re equation and pluripotential theory. Mem. Amer. Math. Soc. 178, 64p (2005)Google Scholar
  33. 33.
    Krylov, N.V.: A bounded non-homogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR, ser. Mat. 47(1), 75–108 (1983)Google Scholar
  34. 34.
    Le Mau, H., Xuan Hong, N.: Maximal \(q-\) subharmonicity in \(\mathbf{C}^n\). Vietnam J. Math. 41, 1–10 (2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Levenberg, N., Taylor, B.A.: Comparison of capacities in \(\mathbf{C}^n\). Lect. Notes Math. 1094, 162–172 (1984)CrossRefGoogle Scholar
  36. 36.
    Li, S.Y.: On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian. Asian J. Math. 8, 87–106 (2004)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lu, HCh.: Solutions to degenerate Hessian equations. Jurnal de Mathematique Pures et Appliques 100(6), 785–805 (2013)Google Scholar
  38. 38.
    Rudin, W.: A geometric criterion for algebraic varieties. J. Math. Mech. 17, 671–683 (1968)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Sadullaev, A.: A criterion for algebraic varieties of analytic sets. Funct. Anal. Appl. 6(1), 85–86 (1972)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sadullaev, A., Defect divisors in the sense of Valiron. Russian Math. Sb. 108 (150):4, 567–580 (1979)Google Scholar
  41. 41.
    Sadullaev, A.: Locally and globally \(\cal{P}-\) regular compacta in \(\mathbf{C}^n\). Dokl. Akad. Nauk SSSR 250(6), 1324–1327 (1980)MathSciNetGoogle Scholar
  42. 42.
    Sadullaev, A.: Operator \((dd^c u)^n\) and condenser capacity. Dokl. Akad. Nauk SSSR 251(1), 44–57 (1980)MathSciNetGoogle Scholar
  43. 43.
    Sadullaev, A.: \(\cal{P}-\) regularity of sets in \(\mathbf{C}^n\). Lect. Notes. Math. 798, 402–407 (1980)CrossRefGoogle Scholar
  44. 44.
    Sadullaev, A.: Plurisubharmonic measures and capacities on complex manifolds. Russian Math. Surveys 36(4), 61–119 (1981)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Sadullaev, A.: Estimates of polynomials on analytic sets. Izv. Akad. Nauk SSSR, ser. Math. 46(3), 524–534 (1982)Google Scholar
  46. 46.
    Sadullaev, A.: Rational approximation and pluripolar sets. Math. USSR Sbornic 47(1), 91–113 (1984)CrossRefGoogle Scholar
  47. 47.
    Sadullaev, A.: Plurisubharmonic functions, Several Complex variables II. Springer, Berlin. Encyclopedia of Math. Sc. pp. 59–106 (1994)Google Scholar
  48. 48.
    Sadullaev, A.: Pluripotential Theory. Applications. Palmarium Akademic Publishing, Germany (2012)Google Scholar
  49. 49.
    Sadullaev, A., Chirka, E.M.: On continuation of functions with polar singularities. Math. USSR Sbornic 60(2), 377–384 (1988)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Siciak, J.: On some extremal functions and their applications in the theory of analytic functions of several complex variables. Trans. Amer. Math. Soc. 105(2), 322–357 (1962)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Trudinger, N.S.: On the Dirichlet problem for Hessian equation. Acta Math. 175, 151–164 (1995)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Trudinger, N.S., Wang, X.J.: Hessian measures, topological methods in nonlinear analysis. J. Juliusz Schauder Center 10, 225–239 (1997)Google Scholar
  53. 53.
    Trudinger, N.S., Wang, X.J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Verbitsky, M.: Plurisubharmonic functions in calibrated geometry and convexity. Mathematische Zeitschrift 264(4), 939–957 (2010)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Wang, X.J.: The \(k\)-Hessian equation. Lecture Notes in Math. Springer, Berlin, vol. 1977, 177–252 (2009)Google Scholar
  56. 56.
    Whiteley, J.N.: On Newton’s inequality for real polynomials. Am. Math. Mon. 76(8), 905–909 (1969)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Zakharyuta, V.P.: Extremal plurisubharmonic functions, Hilbert scales and isomorphisms of spaces of analytic functions, I,II. Theory of Functions, Functional analysis and Applications, (1974), 19, 133–157, (1974), 65–83Google Scholar
  58. 58.
    Zeriahi, A.: A criterion of algebraicity for Lelong classes and analytic sets. Acta Math. 184, 113–143 (2000)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.National University of UzbekistanTashkentUzbekistan

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