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 J. Bourgain. In this article, 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 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 article, and some sparse domination principles for oscillations and certain operators, obtained by A. K. Lerner et al.; as an application, we characterize the boundedness and the compactness of commutators on Formula: see text.