The Hartogs Theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it for $C^k$$\overline{\partial }$-closed forms at the critical degree, $0\le k\le \infty$ (Theorem 1.1). Part of Frenkel’s lemma in $C^k$ category is also...

We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for the $\bar{\partial }$-equation. The generalization consists in multiplication by a weight factor and addition of suitable lower order terms, and is found via a representation as an “oscillating integral”. As special cases we consider weights which behave like a power of the distance to the boundary, like exp-$\varphi$ with $\varphi$ convex, and weights of polynomial decrease in ${\mathbf{C}}^n$. We also briefly consider kernels with singularities on subvarieties...

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ in the complex tangent space. Their real and imaginary parts span a $2n$-dimensional subspace of the real tangent space, which has dimension $2n+1.$ If the Levi matrix of $M$ is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...

Let D be a bounded strict pseudoconvex non-smooth domain in Cn. In this paper we prove that the estimates in Lp and Lipschitz classes for the solutions of the ∂-equation with Lp-data in regular strictly pseudoconvex domains (see [2]) are also valid for D. We also give estimates of the same type for the ∂b in the regular part of the boundary of these domains.

We study Hölder continuity of solutions to the Monge-Ampère equations on compact Kähler manifolds. T. C. Dinh, V.A. Nguyen and N. Sibony have shown that the measure ${\omega }_u^n$ is moderate if $u$ is Hölder continuous. We prove a theorem which is a partial converse to this result.

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is ${}^{1,1}$ and the right hand side has a density in $L^p\left(\Omega \right)$ for some p > 1, and prove the Hölder continuity of the solution.