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

Abstract

We study averages along the integers using the divisor function $d(n)$, and defined as $$K_{N}f(x)=\frac{1}{D(N)}\sum _{n\le N}d(n)f(x+n),$$ 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 $${\left(\frac{1}{N}\sum |K_{N}f{|}^{p^{\prime }}\right)}^{1/p^{\prime }}\lesssim {\left(\frac{1}{N}\sum |f{|}^p\right)}^{1/p}$$ as long as $f$ is supported on $[0,N]$. We also show that the associated maximal function $K^{\ast }f={\sup }_N|K_{N}f|$ satisfies $(p,p)$ sparse founds for $p\in (1,2)$, which implies that $K^{\ast }$ is bounded on ${\ell }^p(w)$ for $p\in (1,\infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

arXiv comment13 pages. Updated version: some typos fixed and some references added as per reviewer recommendations. To appear in Rocky Mountain Journal of Mathematics