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Convergence in capacity

Pham Hoang Hiep — 2008

Annales Polonici Mathematici

We prove that if ( Ω ) u j u ( Ω ) in Cₙ-capacity then l i m i n f j ( d d c u j ) n 1 u > - ( d d c u ) n . This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.

ω-pluripolar sets and subextension of ω-plurisubharmonic functions on compact Kähler manifolds

Le Mau HaiNguyen Van KhuePham Hoang Hiep — 2007

Annales Polonici Mathematici

We establish some results on ω-pluripolarity and complete ω-pluripolarity for sets in a compact Kähler manifold X with fundamental form ω. Moreover, we study subextension of ω-psh functions on a hyperconvex domain in X and prove a comparison principle for the class 𝓔(X,ω) recently introduced and investigated by Guedj-Zeriahi.

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre DemaillySławomir DinewVincent GuedjPham Hoang HiepSławomir KołodziejAhmed Zeriahi — 2014

Journal of the European Mathematical Society

Let ( X , ω ) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L p right hand side, p > 1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range ( X , ω ) of the complex Monge-Ampère operator acting on ω -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with L p -density belong to ( X , ω ) and proving that ( X , ω ) has the...

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