A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

TL;DR

This paper establishes local elliptic regularity results for differential operators with coefficients in various Sobolev-type spaces, using a novel rescaling technique applicable to low-regularity coefficients in geometric analysis.

Contribution

It introduces a unified approach to elliptic regularity for operators with Sobolev-type coefficients, employing rescaling estimates and covering multiple function space classes.

Findings

Regularity results for operators with low-regularity Sobolev-type coefficients.

A unified set of multiplication theorems for the considered function spaces.

Interior estimates and domain regularity characterizations for these operators.

Abstract

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider.