Holder estimates for the solutions of the Cauchy-Riemann equations on convex domains of finite type

In this thesis we study the inhomogeneous Cauchy-Riemann equations ( 6 equations) in several complex variables on convex domains of finite complex type. We prove the following Holder estimate of order 1/ m: if D⊂⊂C2 is convex domain of finite complex type m, then there exists a solution u to the Cauchy-Riemann equation 6u=f on D such that u1/m,D≤C fLinfinity, where f is 6 closed (0, 1) form. Moreover, we will prove local Holder estimate on convex domains D of finite type in Cn (n ≥ 2) in case there is a local defining function of the form r=j=1nf jxj+g jyj ,zj=xj+iyj.