Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 1 (2014), 227-244.

A priori estimates for complex Hessian equations

Sławomir Dinew and Sławomir Kołodziej

Full-text: Open access


We prove some L a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of n and on compact Kähler manifolds. We also show optimal Lp integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Błocki. Finally we obtain a local regularity result for W2,p solutions of the real and complex Hessian equations under suitable regularity assumptions on the right-hand side. In the real case the method of this proof improves a result of Urbas.

Article information

Anal. PDE, Volume 7, Number 1 (2014), 227-244.

Received: 6 February 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32U15: General pluripotential theory
Secondary: 32U05: Plurisubharmonic functions and generalizations [See also 31C10]

Hessian equation a priori estimate pluripotential theory


Dinew, Sławomir; Kołodziej, Sławomir. A priori estimates for complex Hessian equations. Anal. PDE 7 (2014), no. 1, 227--244. doi:10.2140/apde.2014.7.227.

Export citation


  • E. Bedford and B. A. Taylor, “The Dirichlet problem for a complex Monge–Ampère equation”, Invent. Math. 37:1 (1976), 1–44.
  • E. Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions”, Acta Math. 149:1-2 (1982), 1–40.
  • Z. Błocki, “Interior regularity of the degenerate Monge–Ampère equation”, Bull. Austral. Math. Soc. 68:1 (2003), 81–92.
  • Z. Błocki, “Weak solutions to the complex Hessian equation”, Ann. Inst. Fourier $($Grenoble$)$ 55:5 (2005), 1735–1756.
  • Z. Błocki, “Defining nonlinear elliptic operators for non-smooth functions”, pp. 111–124 in Complex analysis and digital geometry (Uppsala, 2006), edited by M. Passare, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. 86, Uppsala Universitet, Uppsala, 2009.
  • Z. Błocki and S. Dinew, “A local regularity of the complex Monge–Ampère equation”, Math. Ann. 351:2 (2011), 411–416.
  • L. Caffarelli, L. Nirenberg, and J. Spruck, “The Dirichlet problem for nonlinear second-order elliptic equations, III: Functions of the eigenvalues of the Hessian”, Acta Math. 155:3-4 (1985), 261–301.
  • K.-S. Chou and X.-J. Wang, “A variational theory of the Hessian equation”, Comm. Pure Appl. Math. 54:9 (2001), 1029–1064.
  • S. Dinew, “An inequality for mixed Monge–Ampère measures”, Math. Z. 262:1 (2009), 1–15.
  • S. Dinew and S. Kołodziej, “Liouville and Calabi–Yau type theorems for complex Hessian equations”, preprint, 2012.
  • L. Gårding, “An inequality for hyperbolic polynomials”, J. Math. Mech. 8:6 (1959), 957–965.
  • Z. Hou, “Complex Hessian equation on Kähler manifold”, Int. Math. Res. Not. 2009:16 (2009), 3098–3111.
  • Z. Hou, X.-N. Ma, and D. Wu, “A second order estimate for complex Hessian equations on a compact Kähler manifold”, Math. Res. Lett. 17:3 (2010), 547–561.
  • N. Ivochkina, N. Trudinger, and X.-J. Wang, “The Dirichlet problem for degenerate Hessian equations”, Comm. Partial Differential Equations 29:1-2 (2004), 219–235.
  • A. Jbilou, “Équations Hessiennes complexes sur des variétés Kählériennes compactes”, C. R. Math. Acad. Sci. Paris 348:1-2 (2010), 41–46.
  • V. N. Kokarev, “\cyr Smeshannye formy ob'ema i kompleksnoe uravnenie tipa Monzha–Ampera na Ke1lerovykh mnogoobraziyakh polozhitel'noĭ krivizny”, Izv. Ross. Akad. Nauk Ser. Mat. 74:3 (2010), 65–78. Translated as “Mixed volume forms and a complex equation of Monge–Ampère type on Kähler manifolds of positive curvature” in Izv. Math. 74:3 (2010), 501–514.
  • S. Kołodziej, “Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge–Ampère operator”, Ann. Polon. Math. 65:1 (1996), 11–21.
  • S. Kołodziej, “The complex Monge–Ampère equation”, Acta Math. 180:1 (1998), 69–117.
  • S. Kołodziej, “Equicontinuity of families of plurisubharmonic functions with bounds on their Monge–Ampère masses”, Math. Z. 240:4 (2002), 835–847.
  • S. Kołodziej, “The Monge–Ampère equation on compact Kähler manifolds”, Indiana Univ. Math. J. 52:3 (2003), 667–686.
  • S. Kołodziej, “The complex Monge–Ampère equation and pluripotential theory”, Mem. Amer. Math. Soc. 178:840 (2005).
  • N. V. Krylov, “On the general notion of fully nonlinear second-order elliptic equations”, Trans. Amer. Math. Soc. 347:3 (1995), 857–895.
  • D. A. Labutin, “Potential estimates for a class of fully nonlinear elliptic equations”, Duke Math. J. 111:1 (2002), 1–49.
  • S.-Y. Li, “On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian”, Asian J. Math. 8:1 (2004), 87–106.
  • N. S. Trudinger, “Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations”, Invent. Math. 61:1 (1980), 67–79.
  • N. S. Trudinger, “On the Dirichlet problem for Hessian equations”, Acta Math. 175:2 (1995), 151–164.
  • N. S. Trudinger and X.-J. Wang, “Hessian measures, II”, Ann. of Math. $(2)$ 150:2 (1999), 579–604.
  • J. Urbas, “An interior second derivative bound for solutions of Hessian equations”, Calc. Var. Partial Differential Equations 12:4 (2001), 417–431.
  • X.-J. Wang, “The $k$-Hessian equation”, pp. 177–252 in Geometric analysis and PDEs (Cetraro, 2007), edited by M. J. Gursky et al., Lecture Notes in Math. 1977, Springer, Dordrecht, 2009.