AVERAGING WITH THE DIVISOR FUNCTION: ℓp-IMPROVING AND SPARSE BOUNDS

Abstract

We study averages along the integers using the divisor function $d(n)$, defined as

$$K_{N}f(x)=\frac{1}{D(N)}{\displaystyle \sum _{n\le N}d(n)f(x+n)},$$

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

as long as $f$ is supported on $[0,N]$.

We will also show that the associated maximal function $K^{\ast }f={\text{sup}}_N|K_{N}f|$ satisfies $(p,p)$ sparse bounds 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.