Closures of a class of Möbius invariant function spaces in the Bloch space are investigated in this paper. Moreover, the boundedness and compactness of composition operators from the Bloch space to closures of such Möbius invariant space in the Bloch space are characterized.

Let X (R n ) be a ball quasi-Banach function space on R n , W X (R n ) be the weak ball quasi-Banach function space on R n , H X (R n ) be the Hardy space associated with X (R n ) and W H X (R n ) be the weak Hardy space associated with X (R n ). In this paper, we obtain the boundedness of the Bochner-Riesz means and the maximal Bochner-Riesz means from H X (R n ) to W H X (R n ) or W X (R n ), which includes the critical case. Moreover, we apply these results to several examples of ball quasi-Banach function spaces, namely, weighted Lebesgue spaces, Herz spaces, Lorentz spaces, variable Lebesgue spaces and Morrey spaces. This shows that all the results obtained in this article are of wide applications, and more applications of these results are predictable.