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A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate

Masato Hoshino

TL;DR

This work develops a semigroup-based framework to prove the reconstruction theorem and the multilevel Schauder estimate within regularity structures. By defining Besov spaces associated with a G-type semigroup and introducing regularity-integrability structures, it yields short, self-contained proofs that rely only on semigroup properties and kernel upper bounds, while accommodating inhomogeneous kernels. The paper then establishes existence, uniqueness, and stability for reconstructions, and lifts integral operators via abstract integration to obtain multilevel Schauder estimates with explicit bounds and Lipschitz continuity across models. These results are applicable to quasilinear SPDEs and the inductive convergence of random models, offering a flexible analytic toolbox for problems combining regularity and integrability scales. The introduced framework broadens the scope of Besov-type regularity analysis beyond classical homogeneous settings, enabling more fine-grained control in Malliavin-type contexts and beyond.

Abstract

The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures [17]. Inspired by [26], we provide elementary proofs for them by using the semigroup of operators. Essentially, we use only the semigroup property and the upper estimates of kernels. Moreover, we refine the several types of Besov reconstruction theorems [18, 7] and introduce the new framework "regularity-integrability structures". The analytic theorems in this paper are applied to the study of quasilinear SPDEs [5] and an inductive proof of the convergence of random models [4].

A semigroup approach to the reconstruction theorem and the multilevel Schauder estimate

TL;DR

This work develops a semigroup-based framework to prove the reconstruction theorem and the multilevel Schauder estimate within regularity structures. By defining Besov spaces associated with a G-type semigroup and introducing regularity-integrability structures, it yields short, self-contained proofs that rely only on semigroup properties and kernel upper bounds, while accommodating inhomogeneous kernels. The paper then establishes existence, uniqueness, and stability for reconstructions, and lifts integral operators via abstract integration to obtain multilevel Schauder estimates with explicit bounds and Lipschitz continuity across models. These results are applicable to quasilinear SPDEs and the inductive convergence of random models, offering a flexible analytic toolbox for problems combining regularity and integrability scales. The introduced framework broadens the scope of Besov-type regularity analysis beyond classical homogeneous settings, enabling more fine-grained control in Malliavin-type contexts and beyond.

Abstract

The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures [17]. Inspired by [26], we provide elementary proofs for them by using the semigroup of operators. Essentially, we use only the semigroup property and the upper estimates of kernels. Moreover, we refine the several types of Besov reconstruction theorems [18, 7] and introduce the new framework "regularity-integrability structures". The analytic theorems in this paper are applied to the study of quasilinear SPDEs [5] and an inductive proof of the convergence of random models [4].
Paper Structure (15 sections, 14 theorems, 112 equations)

This paper contains 15 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Besov spaces associated with the semigroup of operators
  3. Notations
  4. Weighted $L^p$ spaces
  5. Semigroup of operators
  6. Besov spaces associated with semigroup
  7. Basic notions of regularity-integrability structures
  8. Regularity-integrability structures
  9. Models
  10. Modelled distributions
  11. Reconstruction theorem
  12. Multilevel Schauder estimate
  13. Regularizing kernels
  14. Abstract integrations and compatible models
  15. Multilevel Schauder estimate in regularity-integrability structures

Key Result

Lemma 2.3

Let $p,q,r\in[1,\infty]$ be such that $1/r=1/p+1/q$. For any weights $w_1,w_2$ and functions $f\in L^p(w_1)$ and $g\in L^q(w_2)$, we have

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: MW17
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • ...and 42 more