Differential processes generated by two interpolators
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Differential processes generated by two interpolators Jesús M. F. Castillo1 Manuel González5
· Willian H. G. Corrêa2 · Valentin Ferenczi3,4 ·
Received: 21 May 2020 / Accepted: 5 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We study pairs of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to obtain and generalize most known results for commutators or translation operators, in particular those of Cwikel et al. (Arkiv. Mat. 33:199–216, 1995) for differential methods and those of Carro et al. (Adv. Math. 169:241–312, 2002) for compatible pairs of interpolators. We also consider stability issues extending the results in Refs. (J. Inst. Math. Jussieu (to appear). http://arxiv.org/abs/1906.06677v1, 2020; Trans. Am. Math. Soc. 369:4671–4708, 2017; Adv. Comput. Math. 5(1006):329–359, 2007) from the complex method to general differential methods. Keywords Abstract complex interpolation · Commutator theorems · Exact sequences Mathematics Subject Classification 46B70 · 46E30 · 46M18
1 Introduction We study pairs of interpolators, the derivations they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process that allow us to jump from an ordered pair (, ) of interpolators to its reverse pair (, ). Our results are flexible enough to apply to a variety of interpolation methods: those of the kinds considered in [8] and [19], translation operators and the differential methods of [19]. In particular, we will obtain in Sect. 7 some general commutator results which exhibit a kind of symmetry. In the last section we generalize stability results in [11,12,16] from the complex method to general differential methods. Our approach will be free of categorical
The research of the first author has been supported in part by Project IB16056 de la Junta de Extremadura; the research of the first and fourth authors has been supported in part by Projects MTM2016-76958 and PID2019-103961GB, Spain. The research of the second author has been supported in part by CNPq, grant 140413/2016-2, CAPES, PDSE program 88881.134107/2016-0, and São Paulo Research Foundation (FAPESP), grants 2016/25574-8 and 2018/03765-1. The research of the third author has been supported by São Paulo Research Foundation (FAPESP), grants 2013/11390-4, 2015/17216-1, 2016/25574-8, and by CNPq, grants 303034/2015-7 and 303731/2019-2. Extended author information available on the last page of the article 0123456789().: V,-vol
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elements. We also avoid for most of the paper any compatibility condition in the sense of [8]. The reason to delay compatibility assumptions until 6.1 is that, in a sense, they obscure the symmetry between the results one obtains for an ordered pair (, ) of interpolators a
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