Improving Estimates for Discrete Polynomial Averages

Sharp $l^p$-improving estimates for the discrete paraboloid

“…In [6] the authors study averaging operators along a discrete moment curve. More precisely, they consider…”

“…We remark that the ℓ p -improving estimates in the discrete setting have been studied extensively in the recent years, see e.g. [1,6,7,8,9].…”

Subcritical paucity and $\ell^p$-improving estimates for finite-type polynomial curves

“…Previous results in the subcritical range were known when γpT q " pT 2 q and γpT q " pT, φ 2 pT qq where φ 2 pT q is a quadratic polynomial. Indeed, Conjecture 1.2 was proved in these cases; see [HKLMY19] for more information. 1.2.…”

Some subcritical estimates for the $\ell^p$-improving problem for discrete curves

“…A first incarnation of the problem considered here has been the problem of studying the ℓ p → ℓ q mapping properties of discrete fractional integrals along discrete varieties (see [6] for why the two are quite related). This issue has attracted a great deal of attention over the years -see [10,12,13,14,15,16,17] for a selection of works in the area.…”

“…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…”