# Maximal function inequalities and a theorem of Birch

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## 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 hypersurfaces. Let p be a homogenous polynomial in for functions

*n*variables with integer coefficients of degree*d*> 1. The maximal functions we consider are defined by$${A_*}f(y) = \begin{array}{*{20}{c}}
{\sup } \\
{N \geq 1}
\end{array}|\frac{1}{{r(N)}}\sum\limits_{p(x) = 0;x \in {{[N]}^n}} {f(y - x)|} $$

*f*: ℤ^{n}→ ℂ, where [*N*] = {−*N*,−*N*+ 1, …,*N*} and*r*(*N*) represents the number of integral points on the surface defined by p(*x*) = 0 inside the*n*-cube [*N*]^{n}. It is shown here that the operators*A** are bounded on ℓ^{p}in the optimal range*p*> 1 under certain regularity assumptions on the polynomial p.## Preview

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