Lorentz spaces of vector measures and real interpolation of operators

Using the representation of the real interpolation of spaces of p-integrable functions with respect to a vector measure, we show new factorization  theorems for p-th power factorable operators acting in interpolation couples of Banach function spaces. The recently introduced Lorentz spaces of the semivariation of vector mea- sures play a central role in the resulting factorization theorems. We apply our results to analyze extension of operators from classical weighted Lebesgue Lp-spaces | in general with different weights | that can be extended to their q-th powers. This is the case, for example, of the convolution operators dened by Lp-improving measures acting in Lebesgue Lp-spaces or Lorentz spaces. A new  representation theorem for Banach lattices with a special lattice geometric property, as a space of vector measure integrable functions, is also proved. Key words: Banach function space, vector measure, real interpolation, factorable operator, bidual concave operator, improving measures