Some new estimates for the complex Monge-Ampère equation
The authors study regularity estimates for solutions of complex Monge-Ampère equations on an \(n\)-dimensional compact Kähler manifold \((X, \omega)\). More precisely, they look at equations of the type \[ (\omega+i\partial\bar{\partial} \varphi )^n= e^F \omega^n. \] The question is then to understand the regularity of \(\varphi\) depending on the regularity of \(F\). Several results are already known. Among these: if \(F\in C^\infty\), then \(\varphi\in C^\infty\) [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)]; if \(e^F\in L^p\), \(p>1\) then \(\varphi\in C^\alpha\), for some \(\alpha\in(0,1)\) [\textit{S. Kołodziej}, Acta Math. 180, No. 1, 69--117 (1998; Zbl 0913.35043)]; if \(F\) is Lipschitz, then there is a bound on \(\nabla \varphi\) (given a bound on \( \|\varphi\|_{C^0}\)) [\textit{Z. Błocki}, Math. Ann. 344, No. 2, 317--327 (2009; Zbl 1167.32023)]. The paper makes a contribution in this direction: the authors prove that if \(F\) is merely continuous then \(\nabla \varphi\in L^p\) for all \(p\in (0, +\infty)\), giving a very precise bound on the \(L^p\)-norm of \(\nabla \varphi\) depending on \(\omega\), on \(|F|\) and on the \textit{modulus of continuity} of \(F\). Moreover, when \(n=2\) they are able to get rid of the dependence on the latter quantity. It is worth observing that this (integral) gradient estimate does not require any bound on the derivatives of \(F\). As a corollary they also obtain pointwise gradient estimates: if \(F\in C^\alpha\) for some \(\alpha\in(0,1)\), then \(|\nabla \varphi|<C \) pointwise. The authors also generalise all of the above results to the case of manifolds with boundary (this is done in Section 3). In the last Section they show an integrability estimate for \(e^{-\varphi}\) assuming an extra condition on \(F\) (e.g., \(F\) with \textit{finite entropy}). The latter result can be obtained as a corollary of some key facts in pluripotential theory (that hold under much weaker conditions on \(F\)). The novelty here is in the proof that does not use pluripotential methods but only uses explicit estimates.
scientific article; zbMATH DE number 7137523
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