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 definedWe shall show that these averages satisfy a uniform, scale free ℓ p -improving estimate for p ∈ (1, 2), that isas long as f is supported on [0, N ].We will also show that the associated maximal functionsparse founds for p ∈ (1, 2), which implies that K * is bounded on ℓ p (w) for p ∈ (1, ∞), for all weights w in the Muckenhoupt Ap class.