沿着素数的近似：改进和稀疏边界
Averages Along the Primes: Improving and Sparse Bounds
摘要：
考虑由\begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*}给出的素数$ \mathbb P $的平均值。这些平均值满足统一的无标度$ \ell ^{p}$改进估计。对于所有$ 1< p < 2$，存在常数$ C_p$，使得对于$ [0,N]$上支持的所有整数$ N$和函数$ f$，存在保持\begin{equation*} N ^{-1/p' }\lVert \mathcal{A}_N f\rVert_{\ell^{p'}} \leq C_p N ^{- 1/p} \lVert f\rVert_{\ell^p}. \end{equation*}最大函数$ \mathcal{A}^{\ast} f =\sup_{N} \lvert \mathcal{A}_N f \rvert$对于所有$ 1< p < 2$满足$ (p,p)$稀疏边界。后者是无标度边界的自然变体。作为推论，对于Muckenhoupt $A_p$类中的所有权重$ w$，$ \mathcal{A}^{\ast} $在$ \ell ^{p} (w)$上有界。$ \mathcal{A}^{\ast} $的先验加权不等式未知。
英文摘要：
Consider averages along the prime integers $ \mathbb P $ given by \begin{equation*} \mathcal{A}_N f (x) = N ^{-1} \sum_{ p \in \mathbb P \;:\; p\leq N} (\log p) f (x-p). \end{equation*} 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 \begin{equation*} N ^{-1/p' }\lVert \mathcal{A}_N f\rVert_{\ell^{p'}} \leq C_p N ^{- 1/p} \lVert f\rVert_{\ell^p}. \end{equation*} The maximal function $ \mathcal{A}^{\ast} f =\sup_{N} \lvert \mathcal{A}_N f \rvert$ 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 $A_p$ class. No prior weighted inequalities for $ \mathcal{A}^{\ast} $ were known.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
