Abstract
We initiate the theory of $\ell^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove $\ell^p$-improving estimates for the discrete spherical averages and some of their generalizations. As an application of our $\ell^p$-improving inequalities for the dyadic discrete spherical maximal function, we give a new estimate for the full discrete spherical maximal function in four dimensions. Our proofs are analogous to Littman's result on Euclidean spherical averages. One key aspect of our proof is a Littlewood--Paley decomposition in both the arithmetic and analytic aspects. In the arithmetic aspect this is a major arc-minor arc decomposition of the circle method.
Highlights
The motivation for this paper is Littman’s Lp(Rd)-improving result for spherical averages from [Lit73]
In this note we will be interested in estimates for the discrete spherical averages which are analogous to (1)
Cp depending on p such that for all Λ ∈ N, we have the p-improving inequality for the Λ-dyadic discrete spherical maximal function sup |Aλf |
Summary
The motivation for this paper is Littman’s Lp(Rd)-improving result for spherical averages from [Lit73]. For dimensions d 2 and functions f : Rd → C define the spherical average (over the unit sphere) by. Af (x) := f (x − y) dσ(y) Sd−1 where dσ is the Euclidean surface measure on the unit sphere Sd−1 in Rd. In this note we will be interested in estimates for the discrete spherical averages which are analogous to (1). For λ ∈ N and functions f : Zd → C, define the discrete spherical averages. We are motivated by Lee’s work [Lee03] which proved that the dyadic spherical maximal function variant of Littman’s theorem holds. Cp depending on p such that for all Λ ∈ N, we have the p-improving inequality for the Λ-dyadic discrete spherical maximal function (1.2). This clearly implies the same is true for a single average and we may remove the
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