On the structure of tensor norms related to $(p, \sigma)$-absolutely continuous operators

We define an interpolation norm on tensor products of $p$-integrable function spaces and Banach spaces which satisfies intermediate properties between the Bochner norm and the injective norm. We obtain substitutes of the Chevet - Persson - Saphar inequalities for this case. We also use the calculus of traced tensor norms in order to obtain a tensor product description of the tensor norm associated to the interpolated ideal of $(p,\sigma)$-absolutely continuous operators defined by Jarchow and Matter. As an application we find the largest tensor norm less than or equal to our interpolation norm.