Schauder estimates for germs of distributions on smooth manifolds

TL;DR

This paper extends Schauder estimates for germs of distributions from Euclidean spaces to smooth Riemannian manifolds, introducing new concepts and proving regularity results without additional geometric assumptions.

Contribution

It develops a framework for Schauder estimates on manifolds, including a reconstruction theorem and regularity results, using the exponential map and novel regularizing kernels.

Findings

Established Schauder estimates for germs on manifolds

Proved regularity of reconstructed distributions in Hölder-Zygmund spaces

Introduced $eta$-regularizing kernels for Riemannian settings

Abstract

We discuss germs of distributions on $d-$dimensional smooth Riemannian manifolds and, in particular, we derive \emph{multi-level Schauder estimates} without making any further assumptions on the underlying geometry. As a preliminary step, we define the notions of coherence and homogeneity for germs of distributions on open subsets of $\mathbb{R}^d$, $d \ge 1$. Subsequently, we formulate both the reconstruction theorem, cf., [CZ20], and the Schauder estimates, cf., [BCZ24], in this setting. Leveraging the properties of the exponential map, we extend these results to Riemannian manifolds. Specifically, we devise a counterpart of the reconstruction theorem previously established in the literature [RS21], while additionally proving the regularity of the reconstructed distribution in suitable H\"older-Zygmund spaces. Finally, by introducing a novel concept of $\beta$-regularizing kernels on Riemannian manifolds, we establish Schauder estimates for coherent and homogeneous germs in this context.