Inclusion between Morrey spaces and weak Morrey spaces

Abstract: 

For 1≤p≤q<∞1≤p≤q<∞, we define the Morrey space Mpq(Rn)Mqp(Rn) to be the set of all pp-locally integrable functions on RnRn such that

∥f∥Mpq:=supB⊂Rn|B|1q−1p(∫B|f(y)|dy)1p<∞,‖f‖Mqp:=supB⊂Rn|B|1q−1p(∫B|f(y)|dy)1p<∞,

where the supremum is taken over all balls B=B(a,r)B=B(a,r) in RnRn. By using the distribution function instead of the integral, one may also define the weak Morrey space wMpq(Rn)wMqp(Rn). In this talk, I would like to discuss about the inclusion relation among Morrey spaces, as well as between Morrey spaces and weak Morrey spaces.