“…The ℓ p -improving problem for discrete averages A N proper is more recent but has seen a certain degree of activity lately. In particular, in [7] Han, Lacey and Yang have studied the ℓ p -improving properties of the averages A γ N along the polynomial γ(n) = n 2 , while in [6] the same authors together with Kovač and Madrid have studied the case of γ = Γ the moment curve. In the latter, using the celebrated solution to the Vinogradov Mean Value conjecture of Wooley [21] for d = 3 and of Bourgain, Demeter and Guth [2] for arbitrary d, they have shown for the moment curve the optimal supercritical estimates (2) for exponents p, q such 1/p 1/q…”