Maximal Function Inequalities and a Theorem of Birch

TL;DR

This paper extends maximal function inequalities to algebraic surfaces defined by homogeneous polynomials, proving boundedness on  spaces for a broad class of such surfaces, generalizing previous spherical results.

Contribution

It introduces a maximal function associated with homogeneous algebraic surfaces and proves its boundedness on  spaces, generalizing discrete spherical maximal theorems.

Findings

Maximal functions are bounded on  for p>1.

Results apply to surfaces defined by homogeneous polynomials.

Provides a new framework for analyzing algebraic surface averages.

Abstract

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous polynomial in $n$ variables with integer coefficients of degree $d>1$. The maximal functions we consider are defined by \[ A_*f(y)=\sup_{N\geq1}\left|\frac{1}{r(N)}\sum_{\mathfrak{p}(x)=0;\,x\in[N]^n}f(y-x)\right|\] for functions $f:\mathbb{Z}^n\to\mathbb{C}$, where $[N]=\{-N,-N+1,...,N\}$ and $r(N)$ represents the number of integral points on the surface defined by $\mathfrak{p}(x)=0$ inside the $n$-cube $[N]^n.$ It is shown here that the operators $A_*$ are bounded on $\ell^p$ in the optimal range $p>1$ under certain regularity assumptions on the polynomial $\mathfrak{p}$.