On spaces of integrable functions associated to vector measures and limiting real interpolation

We investigate which spaces are obtained when considering the limiting class of real interpolation spaces (0, q ; J ) for ordered Banach couples of spaces of (scalar) integrable functions with respect to a vector measure m, defined on a sigma- algebra, with values in a Banach space. If m is in particular a finite positive scalar measure, previous known results are derived from ours. Furthermore, we study the interpolation of p- th power factorable operators by the extreme real interpolation method (1, q ; K ). We also deduce interpolation results for the (1, q ; K )-method that apply to other related classes of operators to p- th power factorable operators, such as bidual ( p, q )-power-concave operators and q- concave operators.