Discrete maximal operators over surfaces of higher codimension

Abstract

Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research. Here we unite these themes to study discrete analogues of operators involving higher (intermediate) codimensional integration. We consider a maximal operator that averages over triangular configurations and prove several bounds that are close to optimal. A distinct feature of our approach is the use of multilinearity to obtain nontrivial ℓ 1 -estimates by a rather general idea that is likely to be applicable to other problems.

Multilinear maximal operators associated to simplices

“…We remark that it was independently and simultaneously established by Anderson, Kumchev and Palsson in [1] that in dimensions d ≥ 9, with ∆ being a equilateral triangle, that estimate (6) holds in the larger range r > max{32/(d + 8), (d + 4)/(d − 2)}. Their result follows as a direct corollary of ℓ p × ℓ ∞ → ℓ p bounds obtained by employing very different methods than those contained in this short note.…”