Grand Triebel–Lizorkin–Bourgain–Morrey spaces: Nontriviality, embeddings, and boundedness of certain operators

Let [Formula: see text], [Formula: see text], and [Formula: see text] denote the Triebel–Lizorkin–Bourgain–Morrey space, whose special case was originally introduced by Bourgain. In this paper, motivated by the structure of grand Lebesgue spaces, we introduce a new grand variant of [Formula: see text] via adding an additional index [Formula: see text], called the grand Triebel–Lizorkin–Bourgain–Morrey space [Formula: see text]. We find the sufficient and necessary condition for its nontriviality and obtain the proper embeddings with grand Besov–Bourgain–Morrey and Orlicz–Morrey-type spaces; this further leads to the diversity of [Formula: see text]. We also establish the boundedness on both [Formula: see text] and its associate space, with sharp indices, of the Hardy–Littlewood maximal operator, fractional type operators, and rough homogeneous singular integrals, whose proofs strongly depend on the extrapolation theorem, also obtained in this paper, and some sparse domination principles for oscillations and certain operators, obtained by Lerner et al.; as an application, we characterize the boundedness and the compactness of commutators on [Formula: see text].