Use of abstract Hardy spaces, real interpolation and applications to bilinear operators

This paper can be considered as the sequel of Bernicot and Zhao (J Func Anal 255:1761–1796, 2008), where the authors have proposed an abstract construction of Hardy spaces H1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderón-Zygmund operators in a far more abstract framework as in the Euclidean case.