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

The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures by Hairer (2014). Inspired by Otto and Weber’s work (2019), 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 considered by Hairer–Labbé (2017) and Broux–Lee (2022) and introduce the new framework of “regularity-integrability structures”. The analytic theorems in this paper are applied to the study of quasilinear SPDEs by Bailleul–Hoshino–Kusuoka (2022+) and an inductive proof of the convergence of random models by Bailleul–Hoshino (2023+).