Averaging with the Divisor Function: $\ell^p$-improving and Sparse Bounds

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.