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
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
for arbitary κ > 0. The constant only depends on κ,D and q.
