Previous |  Up |  Next

Article

Keywords:
$k$-convex function; $k$-Hessian operator; $k$-Hessian measure; $k$-Green function
Summary:
The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations $F_{k}[u]=0$, where $F_{k}[u]$ is the elementary symmetric function of order $k$, $1\leq k\leq n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_{1}[u]$ is the Laplacian $\Delta u$ and $F_{n}[u]$ is the real Monge-Ampère operator det $D^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give several estimates for the mixed $k$-Hessian operator. Applications of these estimates to the $k$-Green functions are also established.
References:
[1] Błocki, Z.: Estimates for the complex Monge-Ampère operator. Bull. Pol. Acad. Sci., Math. 41 (1993), 151-157. MR 1414762 | Zbl 0795.32003
[2] Carlehed, M., Cegrell, U., Wikström, F.: Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function. Ann. Pol. Math. 71 (1999), 87-103. MR 1684047 | Zbl 0955.32034
[3] Cegrell, U.: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier 54 (2004), 159-179. DOI 10.5802/aif.2014 | MR 2069125 | Zbl 1065.32020
[4] Cegrell, U., Persson, L.: An energy estimate for the complex Monge-Ampère operator. Ann. Pol. Math. 67 (1997), 95-102. MR 1455430 | Zbl 0892.32012
[5] Czyż, R.: Convergence in capacity of the pluricomplex Green function. Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. 1285, Acta Math. 43 (2005), 41-44. MR 2331471 | Zbl 1109.32029
[6] Demailly, J. P.: Monge-Ampère operators, Lelong numbers and intersection theory. Complex Analysis and Geometry Ancona, Vincenzo The University Series in Mathematics, Plenum Press, New York 115-193 (1993). MR 1211880 | Zbl 0792.32006
[7] Gårding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8 (1959), 957-965. MR 0113978 | Zbl 0090.01603
[8] Labutin, D. A.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111 (2002), 1-49. DOI 10.1215/S0012-7094-02-11111-9 | MR 1876440 | Zbl 1100.35036
[9] Trudinger, N. S., Wang, X. J.: Hessian measures. II. Ann. Math. (2) 150 (1999), 579-604. MR 1726702 | Zbl 0947.35055
[10] Trudinger, N. S., Wang, X. J.: Hessian measures. III. J. Funct. Anal. 193 (2002), 1-23. DOI 10.1006/jfan.2001.3925 | MR 1923626 | Zbl 1119.35325
[11] Walsh, J. B.: Continuity of envelopes of plurisubharmonic functions. J. Math. Mech. 18 (1968), 143-148. MR 0227465 | Zbl 0159.16002
[12] Wan, D., Wang, W.: Lelong-Jensen type formula, $k$-Hessian boundary measure and Lelong number for $k$-convex functions. J. Math. Pures Appl. 99 (2013), 635-654. DOI 10.1016/j.matpur.2012.10.002 | MR 3055211
[13] Wikström, F.: Jensen measures and boundary values of plurisubharmonic functions. Ark. Mat. 39 (2001), 181-200. DOI 10.1007/BF02388798 | MR 1821089 | Zbl 1021.32014
Partner of
EuDML logo