Sobolev estimates for the Lewy operator on weakly pseudo-convex boundaries

Optimal Hölder and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$

“…First we consider Po < p < 00. and define (zl -l)b to be the branch with 0 < Arg(zl -l)b < 2n for all ZI ¢. [1,00 …”

“…In this paper we study the optimal regularity of the solution for the equation (1) where f is a (0,1) form on the boundary aD in Holder and L P classes on the boundaries of real ellipsoids in en, n ~ 2. These domains are a special class of weakly pseudo-convex domain of finite type.…”

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

“…We also note that there are well-known non-solvability results of tangential Cauchy-Riemann equation for n = 2 [18] and for q = n − 1 [11]. Note, however, that the local∂-closed extension problem and the local solvability of∂ b equation are closely related [19,3]. Using the results of Theorem 1.1, we solve the local ∂ b -equation in Sobolev spaces.…”

“…Analogous result was obtained by Henkin and Leiterer [13] using kernel methods. For the case when Ω is a bounded pseudoconvex domain in C n , Shaw and Boas [19,3] constructed a two-sided∂-closed extension for∂ b -closed forms near bΩ using the L 2 -Cauchy problem for∂, and solved∂ b -problem on the boundary.…”

\overline∂_{𝑏}-equations on certain unbounded weakly pseudo-convex domains

“…First of all, the existence of a smooth solution for a smooth data was proved by J.-P. Rosay and the existence and regularity in L and Sobolev spaces were achieved by H. Boas, J. J. Kohn, and M. Shaw [BS,K2,R,S]. Quite recently, an optimal Holder estimate of the solution when n = 2 was established by C. Fefferman and J. J. Kohn, and independently by M. Christ under the assumption that db has a closed range [C, FK].…”