Optimal Sobolev regularity of $\bar\partial$ on the Hartogs triangle

TL;DR

This paper constructs a solution operator for the ar problem on the Hartogs triangle that preserves Sobolev regularity, achieving optimal regularity results for data in certain Sobolev spaces.

Contribution

It introduces a solution operator ar that maintains optimal Sobolev regularity on the Hartogs triangle, extending regularity results to all positive integer orders for p>4.

Findings

The operator ar preserves Sobolev regularity for data in W^{k,p} spaces.

The solutions achieve the optimal Sobolev regularity as indicated by Kerzman-type examples.

The results hold for all positive integers k and p>4.

Abstract

In this paper, we show that for each $k\in \mathbb Z^+, p>4$, there exists a solution operator $\mathcal T_k$ to the $\bar\partial$ problem on the Hartogs triangle that maintains the same $W^{k, p}$ regularity as that of the data. According to a Kerzman-type example, this operator provides solutions with the optimal Sobolev regularity.