使用除数函数求平均：$\ell^p$ -改进和稀疏边界
Averaging with the Divisor Function: $\ell^p$-improving and Sparse Bounds
摘要：
我们使用除数函数$d(n)$研究整数的平均值，定义为$$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$，其中$D(N) = \sum _{n=1} ^N d(n) $。我们将证明，这些平均值满足$p \in (1,2)$的统一的、无标度的$\ell^p$改进估计，即只要$[0,N]$支持$f$，就满足$$ \left( \frac{1}{N} \sum |K_Nf|^{p'} \right)^{1/p'} \lesssim \left(\frac{1}{N} \sum |f|^p \right)^{1/p} $$。我们还表明，相关的最大函数$K^*f = \sup_N |K_N f|$满足$(p,p)$稀疏founds的$p \in (1,2)$，这意味着$K^*$是有界的$\ell ^p (w)$为$p \in (1, \infty )$，所有的权重$w$在Muckenhoupt $A_p$类。
英文摘要：
We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$ where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$, that is $$ \left( \frac{1}{N} \sum |K_Nf|^{p'} \right)^{1/p'} \lesssim \left(\frac{1}{N} \sum |f|^p \right)^{1/p} $$ as long as $f$ is supported on $[0,N]$. We also show that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse founds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.
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