Sobolev regularity of the \overline{\partial }-equation on the Hartogs triangle

The regularity of the $\overline{\partial }$ -problem on the domain $\{\left|{z_1}\right|\!<\!\left|{z_2}\right|\!<\!1\}$ in $\mathbb C ^2$ is studied using $L^2$ -methods. Estimates are obtained for the canonical solution in weighted $L^2$ -Sobolev spaces with a weight that is singular at the point $(0,0)$ . In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.     

On the \partial\overline{\partial}-Lemma and Bott-Chern cohomology

On a compact complex manifold X, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the $\partial\overline{\partial}$ -Lemma.     

The $$\overline \partial $$-Neumann operator on Lipschitz q-pseudoconvex domains

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

L^2-properties of the \overline{\partial } and the \overline{\partial }-Neumann operator on spaces with isolated singularities

Let \(X\) be a Hermitian complex space of pure dimension with only isolated singularities and \(\pi : M\rightarrow X\) a resolution of singularities. Let \(\Omega \subset \subset X\) be a domain with no singularities in the boundary, \(\Omega ^*=\Omega {\setminus }\!{{\mathrm{Sing}}}X\) and \(\Omega '=\pi ^{-1}(\Omega )\) . We relate \(L^2\) -properties of the \(\overline{\partial }\) and the \(\overline{\partial }\) -Neumann operator on \(\Omega ^*\) to properties of the corresponding operators on \(\Omega '\) (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the \(\overline{\partial }\) -equation on \(\Omega ^*\) exactly if there are such operators on the resolution \(\Omega '\) , and the \(\overline{\partial }\) -Neumann operator is compact on \(\Omega ^*\) exactly if it is compact on \(\Omega '\) .     

L 2-estimates with Levi-singular weight, and existence for\overline \partial

We give a symplectic proof of the link between pseudoconvexity of domains ofC ⁿ and of their boundaries (cf. [7, Th. 2.6.12]). Our approach also allows us to treat boundaries of codimension >1. We then extend the estimates by Hörmander in [7, Ch. 4, 5] and [6] toL ²-norms which haveC ¹ but notC ² weights and under a less restrictive assumption of weakq-pseudoconvexity. (A special trick is needed as a substitute for the method of thelowest positive eigenvalue of [6].)     

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.     

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 K?hler manifolds. Applications to the solvability of the tangential Cauchy?CRiemann equations for smooth forms on boundaries of such domains are given.     

L 2-\overline{\partial}-Cohomology groups of some singular complex spaces

Let X be a pure n-dimensional (where n≥2) complex analytic subset in ? ^N with an isolated singularity at 0. In this paper we express the L ²-(0,q)- $\overline{\partial}$ -cohomology groups for all q with 1≤q≤n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data. We also obtain identifications of the L ²-(0,q)- $\overline{\partial}$ -cohomology groups of the smooth points of X, in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary regularity for the\bar \partial _b - Neumann problem, part 1problem, part 1

We establish sharp regularity and Fredholm theorems for the operator on domains satisfying some nongeneric geometric conditions. We use these domains to construct explicit examples of bad behavior of the Kohn Laplacian: It is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.