The Dirichlet problem for the complex Monge-Ampère operator: stability in $L^2$.

Cegrell, Urban; Persson, Leif

doi:10.1307/mmj/1029004461

Cited by 42 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For q = 2, it was proved by Cheng and Yau (see [1,5], and also [4]) and for arbitrary q > 1 by Ko lodziej [10]. The inequality (1) is not stated explicitly in [10] but it can be easily deduced from the proof of Theorem 3 in [10].…”

Section: Remarkmentioning

confidence: 99%

On the uniform estimate in the Calabi-Yau theorem, II

Błocki

2011

Sci. China Math.

View full text Add to dashboard Cite

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.Tosatti and Weinkove [15] recently proved a general L ∞ -estimate for the complex Monge-Ampère equation on compact Hermitian manifolds. This gave, using estimates proved earlier in [6,9,8,14], a generalization of the Calabi-Yau theorem [16] to the Hermitian case. Subsequently, the estimate from [15] was improved (with a different proof) by Dinew and Ko lodziej [7]. The aim of this note is to give yet another proof of this estimate. We will show that in fact a very simple modification of the proof for Kähler manifolds from [4] gives the required result.We assume that M is a compact complex manifold of complex dimension n equipped with Hermitian form ω. We will give a simple proof of the following estimate shown already in [7] (where the method from [11] was used):Main Theorem. Assume that ϕ ∈ C 2 (M ) is such that ω + dd c ϕ 0 and (ω + dd c ϕ) n = f ω n .

show abstract

Section: Remarkmentioning

confidence: 99%

On the uniform estimate in the Calabi-Yau theorem, II

Błocki

2011

Sci. China Math.

View full text Add to dashboard Cite

show abstract

“…The integral here is defined by (9). (1) The mixed product γ (u, v) ∧ w 1 ∧ · · · ∧ w k is well defined as a (2k + 2)-current, where γ (u, v) is given by (18).…”

Section: Lemma 31 (Proposition 33 and Corollary 31 In [20])mentioning

confidence: 99%

“…A lot of research has been done on the range and stability of the complex Monge-Ampère operator ( [9,10,13,14], to list just a few). To prove Theorem 1.2 and Corollary 1.1, we need some preparations.…”

Section: Range Of the Quaternionic Monge-ampère Operatormentioning

confidence: 99%

The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

Wan

2016

Math. Z.

View full text Add to dashboard Cite

In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge-Ampère operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge-Ampère measures ( u j ) n → ( u) n . We also obtain an equivalent condition of "convergence in C n−1 -capacity" by using methods from Xing (Proc Am Math Soc 124 (2): [457][458][459][460][461][462][463][464][465][466][467] 1996). As an application, the range of the quaternionic Monge-Ampère operator is discussed.

show abstract

“…It has been proved by the second author [7,8] (see also [5]) that MA(Ω, φ, f) still admits a unique continuous solution u ∈ PSH(Ω) ∩ C 0 (Ω) under the much milder assumption f ∈ L p (Ω), p > 1.…”

Section: Introductionmentioning

confidence: 99%

Hölder continuous solutions to Monge-Ampère equations

Demailly¹,

Dinew²,

Guedj³

et al. 2008

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain Ω⊂ℂn. We show, under a mild technical assumption, that the unique solution u to this problem is Hölder continuous if the boundary data ϕ is Hölder continuous and the density f belongs to Lp(Ω) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.

show abstract

The Dirichlet problem for the complex Monge-Ampère operator: stability in $L^2$.

Cited by 42 publications

References 0 publications

On the uniform estimate in the Calabi-Yau theorem, II

On the uniform estimate in the Calabi-Yau theorem, II

The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

Hölder continuous solutions to Monge-Ampère equations

Contact Info

Product

Resources

About