Hölder and $L^p$ estimates for solutions of $\overline \partial u = f$ in strongly pseudoconvex domains

Holomorphic approximation on compact pseudoconvex complex manifolds

“…One cannot obtain the Mergelyan approximation property in several complex variables only with the simple topological hypothesis. Surprisingly, Diederich and Fornaess [8] Theorem (Kerzman [12], Henkin [11], Lieb [13] In the weakly pseudoconvex case, Fornaess and Nagel [9] got the same result when D C 2 is a pseudoconvex domain with real analytic boundary. Other generalizations can be found in [1,15].…”

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

“…This example is a modification of an example of Stein (see Kerzman [11] and Krantz [15]). Let D be the complex ellipsoid in en defined by D = {z E en II Z / + ... + I Z n_ 1 1 2 + Iznl2m < I}.…”

The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

“…To deal with variable domains, we will use the Grauert bumps to extend ∂-closed forms to a continuous family of larger domains, keeping the forms ∂-closed. For a fixed domain, the extension technique is well known through the works of Kerzman [15] and others. To apply the extension for a continuous family of strongly pseudoconvex domains, we will obtain precise regularity results first for a smooth family of strictly convex domains by using the Lieb-Range solution operator [20].…”

“…In section 4, we study the regularity of ∂-solutions on variable domains first for strictly convex case and then for strictly pseudoconvex case. The Lieb-Range solution operator and Kerzman's extension method [15] for ∂-closed forms are used in section 4 where Theorem 1.1 (i) is proved in Theorem 4.10.…”