Han, Rui; Krause, Ben; Lacey, Michael T.; Yang, Fan Averages along the primes: improving and sparse bounds. (English) Zbl 1436.42013 Concr. Oper. 7, 45-54 (2020). Summary: Consider averages along the prime integers \(\mathbb{P}\) given by \[\mathcal{A}_{N}f(x) = N^{ - 1}\sum\limits_{p \in \mathbb{P}:\ p \le N} (\log p)f(x - p).\] These averages satisfy a uniform scale-free \(\ell^p \)-improving estimate. For all \(1 < p < 2\), there is a constant \(C_p\) so that for all integer \(N\) and functions \(f\) supported on \([0, N]\), there holds \[N^{ - 1/p^\prime}\| \mathcal{A}_N f \|_{\ell p^\prime} \leq {C_p}N^{ - 1/p} \| f \|_{\ell p}.\] The maximal function \(\mathcal{A}^\ast f = \operatorname{sup}_N | \mathcal{A}_{N} f |\) satisfies \((p, p)\) sparse bounds for all \(1 < p < 2\). The latter are the natural variants of the scale-free bounds. As a corollary, \( \mathcal{A}^\ast\) is bounded on \(\ell^p (w)\), for all weights \(w\) in the Muckenhoupt \(\mathcal{A}_p\) class. 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