Maximal function inequalities and a theorem of Birch

  • Brian CookEmail author


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 n variables with integer coefficients of degree d > 1. The maximal functions we consider are defined by
$${A_*}f(y) = \begin{array}{*{20}{c}} {\sup } \\ {N \geq 1} \end{array}|\frac{1}{{r(N)}}\sum\limits_{p(x) = 0;x \in {{[N]}^n}} {f(y - x)|} $$
for functions f: ℤn → 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 ℓp in the optimal range p > 1 under certain regularity assumptions on the polynomial p.


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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsKent State UniversityKentUSA

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