Third Version of Weak Orlicz–Morrey Spaces and Its In-clusion Properties

Orlicz–Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are  three  versions  of  Orlicz–Morrey  spaces.  In  this  article,  we discussed  the  third  version  of  weak  Orlicz–Morrey  space, which is an enlargement of third version of (strong) Orlicz– Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as  a  generalization  of  weak  Orlicz  spaces,  weak  Morrey spaces,  and  generalized  weak  Morrey  spaces.  This  study investigated  some  properties  of the third  version of weak Orlicz–Morrey spaces, especially the sufficient and necessary conditions for inclusion relations between two these spaces. One of the keys to get our result is to estimate the quasi- norm of characteristics function of open balls in ℝ.