Averages along the Square Integers: ${\ell }^p$ improving and Sparse Inequalities

Rui Han, Michael T Lacey, Fan Yang

Let $f\in {\ell }^2(\mathbb{Z})$. Define the average of $f$ over the square integers by $A_{N}f(x):=\frac{1}{N}{\sum }_{k=1}^{N}f(x+k^2)$. We show that $A_N$ satisfies a local scale-fre…