Q-convex Domain Research Articles

The $$\bar{\partial }$$-Neumann problem on the intersection of two weakly q-convex domains

We establish some functional analytic properties for the $$\bar{\partial }$$ -Neumann operator $$N_s$$ on the intersection of two bounded weakly q-convex domains with $$\mathcal {C}^2$$ -boundaries in $$\mathbb {C}^n$$ . Attention is focussed on questions of $$L^2$$ -existence and compactness of $$N_s$$ for all $$s\ge q$$ . Sobolev and boundary regularity for the $$\bar{\partial }$$ -equation are consequently achieved.

The ∂ Cauchy-Problem on Weakly q-Convex Domains in CPn

Let D be a weakly q-convex domain in the complex projective space ℂPn. In this paper, the (weighted) ∂-Cauchy problem with support conditions in D is studied. Specifically, the modified weight function method is used to study the L2 existence theorem for the ∂-Neumann problem on D. The solutions are used to study function theory on weakly q-convex domains via the ∂-Cauchy problem.

Lp and weighted estimates for barred derivatives

We generalize the basic L2 inequality for barred derivatives that holds on bounded pseudoconvex domains. The analogue in Lp-Sobolev norms is established for smooth, bounded pseudoconvex domains of finite type with comparable Levi eigenvalues. A weighted L2 version and Lp estimates with loss are obtained on any smooth bounded weakly q-convex domain, where the weights are fractional powers of the distance to the boundary.

Existence and Interior Regularity Theorems for $$\bar{\partial }$$ ∂ ¯ on Q-Convex Domains

We establish for a q-convex domain $$\Omega \subset {{\mathbb {C}}}^n $$ existence results in $$L^2_{p,k-1}(\Omega ,\text {loc})$$ and $$C_{p,k-1}^\infty (\Omega )$$ for the equation $$\bar{\partial }\textit{u}=f$$ , where f is a (p, k)-form on $$\Omega $$ of degree $$k\ge q$$ such that $$\bar{\partial }\textit{f}=0$$ .

Existence and compactness for the $${\overline{\partial}}$$ ∂ ¯ -Neumann operator on q-convex domains

The aim of this paper is to give a sufficient condition for existence and compactness of the \({\overline{\partial}}\) -Neumann operator Nq on \({L^2_{(0,q)}(\Omega)}\) in the case Ω is an arbitrary q-convex domain in \({\mathbb{C}^n}\).

Q-subharmonicity and q-convex domains in ℂn

Abstract In this paper we study q-subharmonic and q-plurisubharmonic functions in ℂn. Next as an application, we give the notion of q-convex domains in ℂn which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.

SOLUTION TO $\overline\partial$ AND $\overline\partial_{b}$ PROBLEM FOR SMOOTH FORMS AND CURRENTS ON STRICTLY q-CONVEX DOMAINS

On a strictly q-convex domain D in a Kähler manifold X, we obtain the solvability of the [Formula: see text]-problem for smooth forms and currents on boundaries of D. Moreover, we study the solvability of the [Formula: see text]-problem for extensible currents.

Solvability of the tangential Cauchy-Riemann equations on boundaries of strictly q-convex domains

We shall study the solvability of the tangential Cauchy-Riemann equations for smooth forms on boundaries of strictly q-convex domains in an n-dimensional Kahler manifold. This is done for forms of type (r, s) with values in a holomorphic line bundle satisfies some positivity condition.

Formula omitted]-Approximation of differential forms by [formula omitted]-closed ones on smooth hypersurfaces

In this paper we study approximation of ( 0 , q − 1 ) -forms with continuous coefficients in a q-convex domain in C n by ∂ ¯ -closed smooth ( 0 , q − 1 ) -forms in L 2 -norm on level sets of a smooth strictly q-subharmonic function.

Global Boundary Regularity for the $${\overline\partial}$$ -problem on Strictly q-convex and q-concave Domains

We prove the boundary global regularity of the \({\overline\partial}\)-operator on strictly q-convex and q-concave domains in Kahler manifolds. Applications to the solvability of the tangential Cauchy–Riemann equations for smooth forms on boundaries of such domains are given.

SOLUTION TO $\overline{\partial}$ PROBLEM WITH EXACT SUPPORT AND REGULARITY FOR THE $\overline{\partial}$-NEUMANN OPERATOR ON WEAKLY q-CONVEX DOMAINS

Let Ω be a weakly q-convex domain in ℂn. We establish the L2 existence theorem for the [Formula: see text]-Neumann operator N when the boundary of Ω is C1. Using this result, we study the [Formula: see text] problem with exact support on such domains. Furthermore, there exists a number ℓ0 > 0 such that the operators N, [Formula: see text] and the Bergman projection are regular in the Sobolev space Wℓ(Ω) for ℓ < ℓ0 when the boundary of Ω is C∞.

Local solvability of the -equation with boundary regularity on weakly q-convex domains

We consider weakly q-convex domains with smooth boundary and show that the Open image in new window-equation is locally solvable with regularity up to the boundary for smooth forms of degree (p,s) for s≥q.

VANISHING THEOREMS ON STRONGLY Q-CONVEX MANIFOLDS

Let X be strongly q-convex domain of an n-dimensional Kähler manifold M and E be a holomorphic vector bundle over M. Then, if E satisfies certain positivity conditions, we prove vanishing theorems for the [Formula: see text]-cohomology groups of X with values in E.

SOLUTIONS TO ${\bar\partial}$-EQUATION ON STRONGLY q-CONVEX DOMAINS WITH Lp-ESTIMATES

The purpose of this paper is to construct a solution with Lp-estimates, 1≤p≤∞, to the equation [Formula: see text] on strongly q-convex domain of Kähler manifold. This is done for forms of type (n,s), s≥ max (q,k), with values in a holomorphic vector bundle which is Nakano semi-positive of type k and for forms of type (0,s), q≤s≤n-k, with values in a holomorphic vector bundle which is Nakano semi-negative of type k.

The -neumann problem on certain piecewise smooth domains in

Let D 1 be a smoothly bounded uniformly strongly qconvex domain in D 2 smoothly bounded strongly pseudoconvex domain in . Suppose that ∂ D 1, ∂ D 2 intersect transversally_with each other. Let Then for all the ∂[pbar]Neumann operators exist and satisfy the following estimate: For ( )