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Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves (2022)
Journal Article
Dendrinos, S., Hughes, K., & Vitturi, M. (2022). Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves. Journal of Fourier Analysis and Applications, 28(4), Article 69. https://doi.org/10.1007/s00041-022-09958-y

We apply Christ’s method of refinements to the ℓ^p-improving problem for discrete averages AN along polynomial curves in Z^d. Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations,... Read More about Some Subcritical Estimates for the ℓp-Improving Problem for Discrete Curves.

On the ergodic Waring–Goldbach problem (2021)
Journal Article
Anderson, T. C., Cook, B., Hughes, K., & Kumchev, A. (2022). On the ergodic Waring–Goldbach problem. Journal of Functional Analysis, 282(5), Article 109334. https://doi.org/10.1016/j.jfa.2021.109334

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring–Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and dist... Read More about On the ergodic Waring–Goldbach problem.

Bounds for Lacunary maximal functions given by Birch–Magyar averages (2021)
Journal Article
Cook, B., & Hughes, K. (2021). Bounds for Lacunary maximal functions given by Birch–Magyar averages. Transactions of the American Mathematical Society, 374(6), 3859-3879. https://doi.org/10.1090/tran/8152

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem di... Read More about Bounds for Lacunary maximal functions given by Birch–Magyar averages.

Lp-improving for discrete spherical averages (2020)
Journal Article
Hughes, K. (2020). Lp-improving for discrete spherical averages. Annales Henri Lebesgue, 3, 959-980. https://doi.org/10.5802/ahl.50

We initiate the theory of -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove -improving estimates for the discrete spherical averages and some of their generalizations. As an applic... Read More about Lp-improving for discrete spherical averages.

Lp→Lq bounds for spherical maximal operators (2020)
Journal Article
Anderson, T., Hughes, K., Roos, J., & Seeger, A. (2021). Lp→Lq bounds for spherical maximal operators. Mathematische Zeitschrift, 297(3-4), 1057-1074. https://doi.org/10.1007/s00209-020-02546-0

Let f∈Lp(Rd), d≥3, and let Atf(x) be the average of f over the sphere with radius t centered at x. For a subset E of [1, 2] we prove close to sharp Lp→Lq estimates for the maximal function supt∈E|Atf|. A new feature is the dependence of the results o... Read More about Lp→Lq bounds for spherical maximal operators.