Endpoint $\ell^r$ improving estimates for prime averages

Let $\Lambda$ denote von Mangoldt’s function, and consider the averages\[A_N f(x) = \frac{1}{N} \sum_{1 \leq n \leq N} f(x-n) \Lambda (n).\]We prove sharp $\ell^p$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $F,G \subset [0,N]$ there holds\[N^{-1} \langle A_N 1_F , 1_G \rangle \ll\dfrac{\lvert F \rvert \cdot \lvert G \rvert }{N^2}{\left( \operatorname{Log} \dfrac{\lvert F \rvert \cdot \lvert G \rvert }{N^2} \right)}^t \; \textrm{,}\]where $t = 2$, or assuming the Generalized Riemann Hypothesis, $t = 1$. The corresponding sparse bound is proved for the maximal function $\sup_N A_N 1_F$. The inequalities for $t = 1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.

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One of us (MTL) is a 2020 Simons Fellow and their research is supported in part by the Australian Research Council (ARC) through the research grant DP170101060. Research of all authors supported in part by grant from the US National Science Foundation, DMS-1949206.

Received 25 January 2021

Accepted 10 June 2021

Published 4 May 2023