The concept of ball Quasi-Banach function (BQBF) spaces was introduced in 2017 by Y. Sawano, K.-P. Ho, D. Yang and S. Yang. It is well known that some well-known function spaces, such as Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, and Orlicz-slice spaces, are ball Quasi-Banach function spaces, but not Quasi-Banach function spaces. In this talk, we will introduce some recent developments of the real-variable theory of function spaces associated with ball Banach function spaces, including the boundedness and the compactness of commutators on ball Banach function spaces, (weak) Hardy spaces associated with ball Banach function spaces, and Sobolev spaces associated with ball Banach function spaces. In particular, we will introduce some methods on how to overcome the difficulties caused by the deficiency of the explicit expression of the quasi-norm of BQBF spaces.

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