On $L^\infty$ estimates for complex Monge-Ampère equations

Abstract

A PDE proof is provided for the sharp L ∞ estimates for the complex Monge-Ampère equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.

On the modulus of continuity of solutions to complex Monge-Ampère equations

“…As in [7,8,9], we aim to compare ψ s,k with the solution u to (1.1). Consider the following test function…”

“…for some uniform constant C > 0. Given (3.7), we can apply the generalized Young's inequality as in [7] to conclude that…”

“…Next we apply a De Giorgitype iteration argument (c.f. [11,7,8]). Choose a δ 0 > 0 small which depends only on n, p, ω 0 , e F L 1 ( log L) p such that for all δ ∈ (0, δ 0 ], it follows from Lemma 3…”

“…The complex Monge-Ampère equation has been extensively studied ever since Yau's seminal work on the solution of the Calabi conjecture [17]. Notably, assuming that the right hand side is in some Orlicz space, Kolodziej [11] showed using pluripotential theory that the solution must be in L ∞ (see also [7] for a recent PDE-based proof), and in fact continuous. When the right hand side is in L q for some q > 1, it is known that the solution must be Hölder continuous [13,4].…”

“…|u(x)−u(y)| ≤ Cd(x, y) a for some a ∈ (0, 1). Our proof of Theorem 1 can be easily modified to give a new and PDE-based proof of the Hölder continuity of u, in the spirit of the proof of L ∞ estimates developed in [7]. We can readily show in this manner that, if e F ∈ L q for some q > 1 and q * = q q−1 is the conjugate exponent of q, then |u(x) − u(y)| ≤ Cd(x, y) α 0 for α 0 = 2 1+(n+1)q * , for some constant C > 0 depending only on n, q, ω 0 and e F L q .…”