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

TL;DR

This paper offers elementary proofs of the reconstruction theorem and multilevel Schauder estimate using semigroup methods, introduces regularity-integrability structures, and applies these results to quasilinear SPDEs and convergence of random models.

Contribution

It provides a new semigroup-based approach to key theorems in regularity structures and introduces the framework of regularity-integrability structures.

Findings

Elementary proofs of the reconstruction theorem and Schauder estimate using semigroup methods

Refinement of Besov reconstruction theorems

Introduction of regularity-integrability structures

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].