# Weighted Sobolev $L^{p}$ estimates for homotopy operators on strictly pseudoconvex domains with $C^{2}$ boundary

## TL;DR

This paper establishes weighted Sobolev estimates for homotopy operators on strictly pseudoconvex domains with $C^2$ boundary, demonstrating near-half derivative gain for solutions to the $ar{ ext{d}}$-problem in complex analysis.

## Contribution

It provides new weighted Sobolev estimates for homotopy operators on $C^2$ boundary domains, enabling solutions with near-half derivative gain for $ar{ ext{d}}$-closed forms.

## Key findings

- Solutions gain almost $rac{1}{2}$-derivative in weighted Sobolev spaces.
- Estimates hold for a wide range of $p$ and $k$, including $k=1$ with any $p$.
- The results extend regularity theory for the $ar{ ext{d}}$-problem on pseudoconvex domains.

## Abstract

We derive estimates in a weighted Sobolev space $W^{k,p}_{\mu}(D)$ for a homotopy operator on a bounded strictly pseudoconvex domain $D$ of $C^2$ boundary in ${\C}^n$. As a result, we show that given any $2n < p < \infty$, $k > 1$, $q \geq 1$, and a $\dbar$-closed $(0,q)$ form $\var$ of class $W^{k,p}(D)$, there exist a solution $u$ to $\dbar u = \var$ such that $u \in W^{k,p}_{\yh-\ve}(D)$ for any $\ve > 0$. If $k=1$, then we can take $p$ to be any value between $1$ and $\infty$. In other words, the solution gains almost $\yh$-derivative in a suitable sense.