Maximal Function Inequalities and a Theorem of Birch

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 $\mathfrak{p}$.