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Schauder Estimates for Germs by Scaling

Jonas Sauer, Scott A. Smith

TL;DR

This paper develops a germ-based Schauder theory for singular SPDEs by importing Simon's blow-up method into the framework of germ semi-norms and Liouville-type rigidity. The authors establish continuum and discrete Schauder estimates for germs under Laplace, heat, and general constant-coefficient elliptic operators, using scaling and Liouville arguments to translate operator bounds into higher germ regularity. They extend the theory to parabolic settings with initial-time data, to locally uniform norms, and to discrete elliptic operators, providing robust, accessible proofs that complement reconstruction and integration approaches in regularity structures and paracontrolled calculus. The results unify continuous and discrete analyses of germs and offer a versatile tool for pathwise treatments of SPDEs, with potential applications in localized Schauder bounds and fractional/parabolic contexts.

Abstract

In this expository note, we show that the blow-up arguments of L. Simon adapt well to the corresponding Schauder theory of germs used in the study of singular SPDEs. We illustrate this through some representative examples. As in the classical PDE framework, the argument relies only on the scaling properties of the germ semi-norms and the Liouville principle for the operator.

Schauder Estimates for Germs by Scaling

TL;DR

This paper develops a germ-based Schauder theory for singular SPDEs by importing Simon's blow-up method into the framework of germ semi-norms and Liouville-type rigidity. The authors establish continuum and discrete Schauder estimates for germs under Laplace, heat, and general constant-coefficient elliptic operators, using scaling and Liouville arguments to translate operator bounds into higher germ regularity. They extend the theory to parabolic settings with initial-time data, to locally uniform norms, and to discrete elliptic operators, providing robust, accessible proofs that complement reconstruction and integration approaches in regularity structures and paracontrolled calculus. The results unify continuous and discrete analyses of germs and offer a versatile tool for pathwise treatments of SPDEs, with potential applications in localized Schauder bounds and fractional/parabolic contexts.

Abstract

In this expository note, we show that the blow-up arguments of L. Simon adapt well to the corresponding Schauder theory of germs used in the study of singular SPDEs. We illustrate this through some representative examples. As in the classical PDE framework, the argument relies only on the scaling properties of the germ semi-norms and the Liouville principle for the operator.

Paper Structure

This paper contains 13 sections, 18 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. An Illustration of the Method
  4. From Germ Bounds to Hölder Bounds
  5. The Laplacian on $\mathbb{R}^{d}$
  6. The Heat Operator on $(0,T) \times \mathbb{R}^{d}$
  7. The Discrete Laplacian on $\epsilon \mathbb{Z}^{d}$
  8. Some Further Extensions
  9. Elliptic Operators with Constant Coefficients
  10. Locally Uniform Norms
  11. Discrete Elliptic Operators
  12. Acknowledgments
  13. Liouville-type Theorems

Key Result

Lemma 1

Let $R>0$ and $w \in \mathbb{R}^{d}$. For a subset $D \subseteq \mathbb{R}^{d}$ and $0<\alpha<\eta$ the following identities hold for all germs $U$ over $D$ Furthermore, if $D$ is open and $\gamma<0$, then for all distributional germs $V$ over $D$ it holds

Theorems & Definitions (43)

  • Definition 1: Germs
  • Remark 1
  • Definition 2: Distribution-Valued Germs
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 2
  • proof
  • Lemma 3
  • proof
  • ...and 33 more