I'm new here, nice to meet you all. I'm doing a PHD in mathematics, and I need your help. My question is this: Given a measure-preserving system X, And a bounded function f\in\L^\infty(X), the projection of f on the invariant factor of the system is given by E_n foT^n. From here, of course, we can also derive the L^2 norm of the projection.
What similar things can we say about the projection on the Kronecker factor? I know that the Kronecker factor is related to the second Gowers-Host-kra seminorm, and to the multiple ergodic average E_n (foT^n)(goT^2n). Do any of those yield a formula to compute the prjection itself, or at least its' L^2(X) norm?
Thanks!