Estimates for $L^p$ variants of Gowers norms

Motivated by a log-convexity question of Bennett and Tao, we consider Gowers-type functionals defined by $L^p$ norms of multiple autocorrelations. We prove sharp bounds in terms of $L^q$ norms and characterise the near-extremisers on Euclidean spaces as well as locally compact abelian groups. We als