Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

Abstract

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh–Nguyen.

The complex Sobolev space and Hölder continuous solutions to Monge–Ampère equations

“…If 𝐾 is a subset of 𝑋 denote by 𝟏 𝐾 the characteristic function of 𝐾. The following result was obtained in [7,18], see also [19,20]. Theorem 2.2.…”

“…If $$K$$ is a subset of $$X$$ denote by $${\bf 1}_K$$ the characteristic function of $$K$$. The following result was obtained in [7, 18], see also [19, 20]. Theorem Let $$X$$ be a compact Kähler manifold of dimension $$n$$ and $$\omega$$ a Kähler form on $$X$$ normalized so that $$\int _X\omega ^n=1$$.…”

“…End of the proof of Theorem (h4) $$\Longleftrightarrow$$ (h1) . Recall from [18] that (h1) is equivalent to the following property. (h4)'$$\mu$$ belongs to $$\mathcal {H}(\alpha )$$ for some $$\alpha >0$$ and $$\mu$$ is $${\rm PSH}(X,\omega ,[-1,0])$$ ‐Hölder continuous . By Proposition 4.1, the property (h4) is equivalent to (h4)' and hence to (h1).…”

“…Assuming the 𝑊 * -Hölder continuity of 𝜇 or the Hölder continuity of 𝜇 on a suitable family of quasi-psh functions, we first show a continuity of 𝜇 with respect to some notion of capacity, see Proposition 4.1 below. This allows us to apply a result in [18] to conclude. Alternatively, we can solve the complex Monge-Ampère equation with a continuous solution 𝑢, see Theorem 3.6.…”

“…More recently, the case of continuous and Hölder continuous solutions has been intensively studied. We refer the reader to [3–5, 7, 11, 13, 14, 17–20, 22] for more details. Some of these results will be recalled later in the present article.…”