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 n variables with integer coefficients of degree d > 1. The maximal functions we consider are defined by
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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Acknowledgements
The author would like to thank the Fields Institute for being the most gracious of hosts and would also like to personally thank Ákos Magyar, Andreas Seeger, Steve Wainger and Trevor Wooley.
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The author was supported by the Fields Institute and by NSF grant DMS1147523.
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Cook, B. Maximal function inequalities and a theorem of Birch. Isr. J. Math. 231, 211–241 (2019). https://doi.org/10.1007/s11856-019-1853-y
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DOI: https://doi.org/10.1007/s11856-019-1853-y