Endpoint ${\ell }^r$ improving estimates for Prime averages

Michael T. Lacey, Hamed Mousavi, Yaghoub Rahimi

Let $\Lambda$ denote von Mangoldt's function, and consider the averages \begin{align*} A_N f (x) &=\frac{1}{N}\sum_{1\leq n \leq N}f(x-n)Λ(n) . \end{align*} We prove sharp ${\ell }^p$-…