Let be a compact Kähler manifold and be a smooth closed form of bidegree which is nonnegative and big. We study the classes of -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class . This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori -estimate.
@article{ASNSP_2008_5_7_1_81_0, author = {Benelkourchi, Slimane and Guedj, Vincent and Zeriahi, Ahmed}, title = {A priori estimates for weak solutions of complex {Monge-Amp\`ere} equations}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {81--96}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {1}, year = {2008}, zbl = {1150.32011}, mrnumber = {2413673}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2008_5_7_1_81_0/} }
TY - JOUR AU - Benelkourchi, Slimane AU - Guedj, Vincent AU - Zeriahi, Ahmed TI - A priori estimates for weak solutions of complex Monge-Ampère equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 DA - 2008/// SP - 81 EP - 96 VL - Ser. 5, 7 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_1_81_0/ UR - https://zbmath.org/?q=an%3A1150.32011 UR - https://www.ams.org/mathscinet-getitem?mr=2413673 LA - en ID - ASNSP_2008_5_7_1_81_0 ER -
Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed. A priori estimates for weak solutions of complex Monge-Ampère equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 1, pp. 81-96. http://archive.numdam.org/item/ASNSP_2008_5_7_1_81_0/
[1] Comparison of two capacities in , Math. Z. 186 (1984), 407-417. | EuDML 173456 | MR 744831 | Zbl 0576.32029
and ,[2] Équations du type Monge-Ampère sur les variétés Kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), 63-95. | MR 494932 | Zbl 0374.53022
,[3] A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. | MR 674165 | Zbl 0547.32012
and[4] On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A 48 (2005) , suppl., 244-247. | MR 2156505 | Zbl 1128.32025
,[5] Arithmetic characteristic classes of automorphic vector bundles, Doc. Math. 10 (2005), 619-716. | EuDML 125532 | MR 2218402 | Zbl 1080.14028
, and ,[6] On Kähler manifolds with vanishing canonical class. Algebraic geometry and topology, In: “A symposium in Honor of S. Lefschetz”, Princeton Univ. Press, Princeton, N. J. (1957), 78-89. | MR 85583 | Zbl 0080.15002
,[7] Pluricomplex energy, Acta Math. 180 (1998), 187-217. | MR 1638768 | Zbl 0926.32042
,[8] Singular Kähler-Einstein metrics, preprint arxiv math. AG/0603431. | MR 2505296 | Zbl 1215.32017
, and ,[9] Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), 607-639. | MR 2203165 | Zbl 1087.32020
and ,[10] The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), 442-482. | MR 2352488 | Zbl 1143.32022
and ,[11] The range of the complex Monge-Ampère operator, Indiana Univ. Math. J. 43 (1994), 1321-1338. | MR 1322621 | Zbl 0831.31009
,[12] The complex Monge-Ampère equation, Acta Math. 180 (1998), 69-117. | MR 1618325 | Zbl 0913.35043
,[13] The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), 667-686. | MR 1986892 | Zbl 1039.32050
,[14] “The Complex Monge-Ampère Equation and Pluripotential Theory”, Mem. Amer. Math. Soc., Vol. 178, 2005. | MR 2172891 | Zbl 1084.32027
,[15] Generalized arithmetic intersection numbers, J. Reine Angew. Math. 534 (2001), 209-236. | MR 1831639 | Zbl 1084.14028
,[16] A note on the preceding paper, Duke Math. J. 36 (1969), 799-800. | MR 290114 | Zbl 0201.45801
,[17] “Potential Theory in the Complex Plane”, London Mathematical Society Student Texts, Vol. 28, Cambridge University Press, Cambridge, 1995. | MR 1334766 | Zbl 0828.31001
,[18] “Canonical Metrics in Kähler Geometry”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000. | MR 1787650 | Zbl 0978.53002
,[19] On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31 (1978), 339-411. | MR 480350 | Zbl 0369.53059
,[20] Volume and capacity of sublevel sets of a Lelong class of psh functions, Indiana Univ. Math. J. 50 (2001), 671-703. | MR 1857051 | Zbl 1138.31302
,