Let $\Omega$ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty }$ boundary, and let $\square =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\overline{\partial }$ be the self-adjoint $\overline{\partial }$-Neumann operator on the space $L_{0,q}^2\left(\Omega \right)$ of forms of type $(0,q)$. If the Levi form of $\partial \Omega$ has everywhere at least $n-q$ positive or at least $q+1$ negative eigenvalues, it is well known that $\mathrm{Ker}$$\square$ has finite dimension and that the range of $\square$ is the orthogonal complement. In...

We consider the solution operator $S{ℱ}_{\mu ,(p,q)}\to L^2{\left(\mu \right)}_{(p,q)}$ to the $\overline{\partial }$-operator restricted to forms with coefficients in ${ℱ}_{\mu }=\left\{ff\text{is}\text{entire}\text{and}{\int }_{{ℂ}^n}{\left|f\left(z\right)\right|}^2\mathrm{d}\mu \left(z\right)<\infty \right\}$. Here ${ℱ}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in ${ℱ}_{\mu }$, $L^2\left(\mu \right)$ is the corresponding $L^2$-space and $\mu$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\overline{\partial }$. This solution operator will have the property $Sv\perp {ℱ}_{(p,q)}\forall v\in {ℱ}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators...

Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...

We prove that compactness of the canonical solution operator to $\overline{\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫,\overline{M}]$ defined on the whole $L_{(0,1)}^2\left(\Omega \right),$ where $\overline{M}$ is the multiplication by $\overline{z}$ and $𝒫$ is the orthogonal projection of $L_{(0,1)}^2\left(\Omega \right)$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\overline{\partial }$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications...

On a bounded $q$-pseudoconvex domain $\Omega$ in ${ℂ}^n$ with a Lipschitz boundary, we prove that the $\overline{\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega$ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.

Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in ${ℂ}^n$ where the fibre is nontrivial, has to exceed $n$. This is shown not to be the case.

We study the $\bar{\partial }$-equation with Hölder estimates in $q$-convex wedges of ${ℂ}^n$ by means of integral formulas. If $D\subset {ℂ}^n$ is defined by some inequalities $\{{\rho }_i\le 0\}$, where the real hypersurfaces $\{{\rho }_i=0\}$ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the ${\rho }_i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\bar{\partial }f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q\le r\le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^{\beta }g$ is bounded, then for all $ϵ\&gt;0$,...

We derive formulas for the values in the interior of the L2-minimal solutions of the ∂∂-equation in the unit ball of Cn. These formulas generalize previously known formulas for the boundary values of the same solutions. We estimate the solution and obtain a (known) result concerning weighted Nevanlinna classes.

We prove some existence results for equations of complex Monge-Ampère type in strictly pseudoconvex domains and on Kähler manifolds.