Hairer's multilevel Schauder estimates without Regularity Structures

TL;DR

This paper extends Hairer's multilevel Schauder estimates to a germ-based framework, providing a new approach to analyzing regularising properties of singular kernels without relying on Regularity Structures.

Contribution

It introduces a novel integration map acting on coherent germs and proves multilevel Schauder estimates in this germ-based setting with minimal assumptions.

Findings

Established a new integration map for germs.

Proved multilevel Schauder estimates without Regularity Structures.

Applicable to a broad class of singular kernels.

Abstract

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in $\mathbb{R}^d$. First we construct a suitable integration map which acts on general coherent germs. Then we focus on germs that can be decomposed along a basis (corresponding to the so-called modelled distributions in Regularity Structures) and we prove a version of Hairer's multilevel Schauder estimates in this setting, with minimal assumptions.