Lorentz spaces of vector measures and real interpolation of operators

Abstract

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 measures play a central role in the resulting factorization theorems. We apply our results to analyze extension of operators from classical weighted Lebesgue L^p -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 defined by L^p -improving measures acting in Lebesgue L^p -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.