Abstract. Let Ω be a bounded, pseudoconvex domain of C n satisfying the "f -Property". The f -Property is a consequence of the geometric "type" of the boundary; it holds for all pseudoconvex domains of finite type but may also occur for many relevant classes of domains of infinite type. In this paper, we prove the existence, uniqueness and "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equationThe idea of our proof goes back to Bedford and Taylor's [BT76]. However, the basic geometrical ingredient is based on a recent result by Khanh [Kha13].
We prove Lp estimates, 1 ≤ p ≤ ∞, for solutions to the Cauchy–Riemann equations [Formula: see text] on a class of infinite type domains in ℂ2. The domains under consideration are a class of convex ellipsoids, and we show that if ϕ is a [Formula: see text]-closed (0, 1)-form with coefficients in Lp and u is the Henkin kernel solution to [Formula: see text], then ‖u‖p ≤ C‖ϕ‖p where the constant C is independent of ϕ. In particular, we prove L1 estimates and obtain Lp estimates by interpolation.
Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$. In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$-norm, for $1\leq p<\infty$. For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.
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