H\"older regularity of the $\bar\partial-$equation on the polydisc

TL;DR

This paper constructs an optimal solution operator for the ar-equation on the polydisc that preserves Hf6lder regularity, using Henkin's formula, highlighting its non-improvement of regularity as optimal.

Contribution

It introduces a solution operator for the ar-equation on the polydisc that preserves Hf6lder regularity, demonstrating its optimality.

Findings

Solution operator preserves Hf6lder regularity

Operator constructed via Henkin's formula

Operator is optimal in regularity preservation

Abstract

In this note, we show the existence of a solution operator to the $\bar\partial-$equation in the polydisc that preserves H\"older 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 H\"older regularity. Hence, this solution operator is optimal in that regard.