8. Operator-Adapted Spaces

Operator-adapted Hardy–Sobolev spaces are our main tool in this monograph and will be essential for understanding most of the following chapters. They have been developed in various references starting with semigroup generators in Auscher et al. (J Geom Anal 18(1):192–248, 2008), Duong and Li (J Funct Anal 264(6), 1409–1437, 2013), Hofmann and Mayboroda (Math Ann 344(1): 37–116, 2009) and Hofmann et al. (Ann Sci Éc Norm Supér (4) 44(5): 723–800, 2011) up to the recent monographs focusing on bisectorial operators (Amenta and Auscher, Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018; Auscher and Stahlhut, Mém Soc Math Fr (N.S.)(144):vii+164, 2016). Still we need some unrevealed features and we take this opportunity to correct some inexact arguments from the literature. For general properties of adapted Hardy spaces we closely follow Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018, Sec.3), where the authors develop an abstract framework of two-parameter operator families that provides a unified approach to sectorial and bisectorial operators. The application to bisectorial operators with firstorder scaling has been detailed in Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018, Sect. 4) and we review their results in Sect. 8.1. Section 8.2 provides all necessary details in order to apply the framework to sectorial operators with second-order scaling, and we summarize the results that are relevant to us. This will justify using parts of Amenta and Auscher (Elliptic Boundary Value Problems with Fractional Regularity Data. American Mathematical Society, Providence, 2018) for sectorial operators in the further course. The abstract framework allows us to treat operator-adapted Besov spaces simultaneously without any additional effort.