Capacity associated to a positive closed current

  • Dabbek Khalifa

    Département de Math Département de Math Faculé de sciences de Gab`es Faculé de sciences Monastir 6071 Gab`es Tunisie 5000 Monastir Tunisie
  • Elkhadhra Fredj

    Département de Math Département de Math Faculé de sciences de Gab`es Faculé de sciences Monastir 6071 Gab`es Tunisie 5000 Monastir Tunisie
Capacity associated to a positive closed current cover
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Abstract

Let Ω\Omega be an open set of Cn\mathbb{C}^n and TT be a positive closed current of dimension p1p\geq 1 on Ω\Omega, we define a capacity associated to TT by: CT(K,Ω)=CT(K)=sup{\dsKT(ddcv)p, vpsh(Ω), 0<v<1}C_T(K,\Omega)=C_T(K)={sup} \left\{\ds\int_K{T\wedge(dd^c v)^p,\ v\in {psh}(\Omega),\ 0<v<1}\right\} where KK is a compact set of Ω\Omega. We prove, in the same way as Bedford-Taylor, that a locally bounded plurisubharmonic function is quasi-continuous with respect to CTC_T. In the second part we define the convergence relatively to CTC_T and we prove that if (uj)(u_j) is a family of locally uniformly bounded plurisubharmonic functions and uu is a locally bounded plurisubharmonic function such that ujuu_j \rightarrow u relatively to CTC_T then T(ddcuj)pT(ddcu)pT\wedge (dd^cu_j)^p\rightarrow T\wedge (dd^cu)^p in the current sense.

Cite this article

Dabbek Khalifa, Elkhadhra Fredj, Capacity associated to a positive closed current. Doc. Math. 11 (2006), pp. 469–486

DOI 10.4171/DM/219