A uniform estimate for
solutions to the equation ∂u = α in a weakly pseudoconvex domain is obtained,
provided that the form α vanishes near the set of degeneracy of the Levi form. Under
the additional hypothesis that the closure of the domain is holomorphically convex,
analogous estimates are obtained for solutions defined in a full neighborhood of the
closure. Applications are given to Mergelyan type approximation problems
in a weakly pseudoconvex domain D. In particular, it is shown that any
function in A(D) can be uniformly approximated by functions in A(D) which
extend holomorphically across all strongly pseudoconvex boundary points.
When D is holomorphically convex, it is shown that the Mergelyan problem
can be localized to a small neighborhood of the set on which the Levi form
degenerates.
