Hölder regularity for ∂̄ on the convex domains of finite strict type

Abstract

By using the Cauchy-Fantappiè machinery, the nonhomogeneous Cauchy-Riemann equation on convex domain D for (0, q) form f with ∂̄f = 0, ∂̄u = f, has a solution which is a linear combination of integrals on bD of the following differential forms (formula presented) j = 1, ⋯ , n-q-3, where A = 〈∂ζr(ζ), ζ-z〉, β = |z-ζ|2 and r is the defining function of D. In the case of finite strict type, Bruna et al. estimated 〈∂r(ζ), ζ-z〉 by the pseudometric constructed by McNeal. We can estimate the above differential forms and their derivatives. Then, by using a method of estimating integrals essentially due to McNeal and Stein, we prove the following almost sharp Hölder estimate (formula presented) for arbitary κ > 0. The constant only depends on κ, D and q.