We establish the comparison principle in the class ${}_p\left(f\right)$. The result obtained is applied to the Dirichlet problem in ${}_p\left(f\right)$.

We prove some existence results for the complex Monge-Ampère equation $\left(d{d}^{c}u\right)ⁿ=gd\lambda$ in ℂⁿ in a certain class of homogeneous functions in ℂⁿ, i.e. we show that for some nonnegative complex homogeneous functions g there exists a plurisubharmonic complex homogeneous solution u of the complex Monge-Ampère equation.

Viene studiata l'equazione $\overline{\partial }u=f$ per le forme regolari sulla chiusura dell'intersezione di $k$ domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme ${𝐂}^k$.

We give a short proof of the extension theorem of Ohsawa-Takegoshi. The same method also gives a generalization of the $\bar{\partial }$-theorem of Donnelly and Fefferman for the case of $(n,1)$-forms.

We define and study the domain of definition for the complex Monge-Ampère operator. This domain is the most general if we require the operator to be continuous under decreasing limits. The domain is given in terms of approximation by certain " test"-plurisubharmonic functions. We prove estimates, study of decomposition theorem for positive measures and solve a Dirichlet problem.

Let $G/K$ be an irreducible Hermitian symmetric space of noncompact type. We study a $G$- invariant system of differential operators on $G/K$ called the Hua system. It was proved by K. Johnson and A. Korányi that if $G/K$ is a Hermitian symmetric space of tube type, then the space of Poisson-Szegö integrals is precisely the space of zeros of the Hua system. N. Berline and M. Vergne raised the question about the nature of the common solutions of the Hua system for Hermitian symmetric spaces of nontube type. In...

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973).

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...