Integral kernels and Hölder estimates for\bar \partial on pseudoconvex domains of finite type in C2

Dedicated to Professor Hans Grauert on his 60th birthday     

The \bar \partial -problem on q-pseudoconvex domains with applications

For a q-pseudoconvex domain Ω in ? ⁿ , 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $ -problem with exact support in Ω. Moreover, we solve the $\bar \partial $ -problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.     

Sharp estimates for\bar \partial on convex domains of finite typeon convex domains of finite type

Let Ω be a bounded convex domain in C _n , with smooth boundary of finite typem. The equation is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL ^p(Ω),p≥1. We solve the -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.     

Global analytic hypoellipticity of the\bar \partial-Neumann problem on circular domains

In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic.     

Homotopy formulas and\bar \partial -equation on localq-convex domains in Stein manifolds-equation on localq-convex domains in Stein manifolds

The homotopy formulas of (r, s) differential forms and the solution of $\bar \partial $ -equation of type (r, s) on localq-convex domains in Stein manifolds are obtained. The homotopy formulas on localq-convex domains have important applications in uniform estimates of $\bar \partial $ -equation and holomorphic extension of CR-manifolds.     

On Hölder and BMO estimates for \bar \partial on convex domains in C2on convex domains in C2

We prove that on convex domains in C² a suitable integral solution operator for the Cauchy-Riemann equations preserves exact Hölder regularity, and that it maps bounded (0,1) forms into BMO with respect to volume measure.     

Non-isotropic weighted Lp estimates for the $$ \bar \partial $$-equation on piecewise smooth strictly pseudoconvex domains

In this paper we obtain non-isotropic weighted L ^p estimates with the boundary distance weight function for the -equation on piecewise smooth strictly pseudoconvex domains under a hypothesis of complex transversality in ℂⁿ using the explicit formula of solutions by Berndtsson-Andersson. This work was supported by the Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2005-070-C00007)