Holder regularity for partial derivative on the convex domains of finite strict type

By using the Cauchy-Fantappie machinery, the nonhomogeneous Cauchy-Riemann equation on convex domain D for (0, q) form f with partial derivativef = 0, partial derivativeu = f, has a solution which is a linear combination of integrals on bD of the following differential forms [GRAPHICS] j = 1, . . ., n-q-3, where A = [partial derivative (zeta)r(zeta), zeta - z], beta = \z -zeta\(2) and r is the defining function of D. In the case of finite strict type, Bruna et al. estimated [partial derivativer(zeta), zeta - z], by the pseudometric constructed by McNeal. We can e stimate 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 Holder estimate [GRAPHICS] for arbitary kappa > 0. The constant only depends on kappa, D and q.