  In this paper, we define for the first time the local Morrey-type space
associated with ball quasi-Banach function spaces and show the related series
of properties. In addition, Hardy-Littlewood maximal operator's boundedness is
proved. We investigate nonsmooth decomposition of the local Morrey-type space
associated with ball quasi-Banach function spaces via the Hardy local
Morrey-type spaces associated with ball quasi-Banach function spaces. And we
consider Hardy operator's boundedness.
