Bourgain-Morrey-Lorentz spaces and operators on them

We introduce Bourgain-Morrey-Lorentz spaces and give a description of the predual of Bourgain-Morrey-Lorentz spaces via the block spaces. As an application of duality, we obtain the boundedness of Hardy-Littlewood maximal operator, sharp maximal operator, Calder\'on-Zygmund operator, fractional integral operator, commutator on Bourgain-Morrey-Lorentz spaces. Moreover, we obtain a weak Hardy factorization terms of Calder\'on-Zygmund operator in Bourgain-Morrey-Lorentz spaces. Using this result, we obtain a characterization of functions in $\BMO$ (the functions of ``bounded mean oscillation'') via the boundedness of commutators generated by them and a homogeneous Calder\'on-Zygmund operator. In the last, we show that the commutator generated by a function $b$ and a homogeneous Calder\'on-Zygmund operator is a compact operator on Bourgain-Morrey-Lorentz spaces if and only if $b$ is the limit of compactly supported smooth functions in $\BMO$.