Weak‐type Fefferman–Stein inequality and commutators on weak Orlicz–Morrey spaces

Abstract

We consider the Fefferman–Stein inequality for weak Orlicz–Morrey spaces and the commutators and on weak Orlicz–Morrey spaces, where T is a Calderón–Zygmund operator, is a generalized fractional integral operator and b is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness from of and from a weak Orlicz Morrey space to another weak Orlicz–Morrey space. We use the Fefferman–Stein inequality to prove the boundedness of the commutators. Since weak Orlicz–Morrey spaces contain the weak Lebesgue, weak Orlicz and weak Morrey spaces as special cases, our results contain the bounedness on these function spaces which are also new results.