Abstract
In this paper we firstly define the m-generalized Lelong numbers of a m-positive closed current. Secondly, we associate a capacity to a given class of m-positive closed currents. The pluripolarity and the quasicontinuity of m-subharmonic functions with respect to such a capacity are investigated. Finally, we introduce the notion of m-generalized Lelong numbers of a given class of m-positive closed currents and we prove that these numbers can be expressed in terms of the capacity previously defined.
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References
Abdullaev, B., Sadullaev, A.: Potential theory in the class of \(m\)-sh functions. Proc. Steklov Inst. Math. V279, 155–180 (2012)
Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
Benali, A., Ghiloufi, N.: Lelong numbers of m-subharmonic functions. J. Math. Anal. Appl. 466, 1373–1392 (2018)
Blocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier Grenoble 55(5), 1735–1756 (2005)
Dabbek, K., Elkhadhra, F.: Capacité associée à un courant positif fermé. Documenta Math 11, 469–486 (2006)
Demailly J.-P.: Complex analytic and differential geometry. Preprint (2009). http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf
Dinew, S., Kolodziej, S.: A priori estimates for complex Hessian equations. Anal. PDE 7(1), 227–244 (2014)
Dhouib, A., Elkhadhra, F.: \(m\)-Potential theory associated to a positive closed current in the class of \(m\)-sh functions. Complex Var. Elliptic Equ. 61(7), 875–901 (2016)
Elkhadhra, F.: Lelong-Demailly numbers in terms of capacity and weak convergence for closed positive currents. AMSCI 33B(6), 1652–1666 (2013)
Lu, H.-C.: A variational approach to complex Hessian equations in \(\mathbb{C}^n\). J. Math. Anal. Appl. 431, 228–259 (2015)
Nguyen, N.-C.: Subsolution Theorem for the complex Hessian equation. Univ. Lag. Acta. Math. Fasciculus L 69–88 (2012)
Wan, D., Wang, W.: Complex Hessian operator and Lelong number for unbounded \(m\)-subharmonic functions. Potential Anal. 44, 53–69 (2016)
Xing, Y.: Weak convergence of currents. Math. Z. 260, 253–264 (2008)
Zeriahi, A.: Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J 50(1), 671–703 (2001)
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Elkhadhra, F. m-Generalized Lelong Numbers and Capacity Associated to a Class of m-Positive Closed Currents. Results Math 74, 10 (2019). https://doi.org/10.1007/s00025-018-0933-3
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DOI: https://doi.org/10.1007/s00025-018-0933-3