Bσ-grand Morrey spaces

In this paper, we define the Bσ-type grand Morrey spaces and establish the extrapolation theorem on the Bσ-type grand Morrey spaces. In the process of proving the theorem, we find that the predual spaces of these spaces are Hσ-block spaces and obtain the boundedness of the Hardy–Littlewood maximal operator on the predual spaces. By the extrapolation theory, the boundedness of the Calderón–Zygmund operator and commutators on the nonhomogeneous Bσ-type grand Morrey space is also obtained. In particular, the classical bounded mean oscillation (BMO) spaces are characterized by establishing the John–Nirenberg inequality on the nonhomogeneous Bσ type grand Morrey space.