Hölder regularity of the $\bar{\partial }-$equation on the polydisc

Abstract

In this note, we show the existence of a solution operator to the $\bar{\partial }-$equation in the polydisc that preserves Hölder regularity. This solution operator is constructed using Henkin's formula. It is a well-known fact that solution operators to the $\bar{\partial }-$equation on product domains do not improve Hölder regularity. Hence, this solution operator is optimal in that regard.