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Schauder estimates for germs of distributions on smooth manifolds

Beatrice Costeri, Claudio Dappiaggi, Paolo Rinaldi, Matteo Savasta

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 Hölder-Zygmund spaces. Finally, by introducing a novel concept of $β$-regularizing kernels on Riemannian manifolds, we establish Schauder estimates for coherent and homogeneous germs in this context.

Schauder estimates for germs of distributions on smooth manifolds

Abstract

We discuss germs of distributions on 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 , . 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 Hölder-Zygmund spaces. Finally, by introducing a novel concept of -regularizing kernels on Riemannian manifolds, we establish Schauder estimates for coherent and homogeneous germs in this context.
Paper Structure (10 sections, 29 theorems, 168 equations)

This paper contains 10 sections, 29 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Germs of distributions on open sets
  3. The reconstruction theorem
  4. Schauder Estimates for Germs on Open Subsets of $\mathbb{R}^d$
  5. Germs of distributions on Riemannian manifolds
  6. Reconstruction Theorem on Riemannian manifolds
  7. Schauder Estimates for germs on Riemannian Manifolds
  8. Additional Properties of Homogeneous and Coherent Germs
  9. Uniform estimates for the normal coordinates
  10. Scaling of test functions on manifolds

Key Result

Theorem 1.1

Let $\{\Omega_n\}_{n \in \mathbb{N}}$ be an exhaustion by compact sets of $M$ as per Definition Def: Compact Exhaustion and let $\gamma, \beta > 0$. Let $F$ be an $(\boldsymbol{\alpha}, \gamma)$-coherent germ of distributions of order $\boldsymbol{r}$ and range $\boldsymbol{R}$, as per Definition De Then, there exists a germ of distributions, denoted $\mathcal{K}^{\gamma, \beta} F$, such that i.e

Theorems & Definitions (56)

  • Theorem 1.1: Main Theorem 1
  • Theorem 1.2: Main Theorem 2
  • Definition 1.3: Hölder-Zygmund spaces
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 2.1
  • Definition 2.2: Coherence
  • Remark 2.3
  • ...and 46 more