$B_{\sigma }$-grand Morrey spaces

In this paper, we define the $B_{\sigma }$-type grand Morrey spaces and establish the extrapolation theorem on the $B_{\sigma }$-type grand Morrey spaces. In the process of proving the theorem, we find that the predual spaces of these spaces are $H_{\sigma }$-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 $B_{\sigma }$-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 $B_{\sigma }$ type grand Morrey space.