Averaging with the Divisor Function: ℓ^p-improving and Sparse Bounds

2021/02/02 by Giannitsi, Christina
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.01778

Abstract

We study averages along the integers using the divisor function d(n), and defined as K_N f (x) = (1)/(D(N)) ∑ _{n ≤ N} d(n) f(x+n) , where D(N) = ∑ _n=1 ^N d(n) . We shall show that these averages satisfy a uniform, scale free ℓ^p-improving estimate for p ∈ (1,2), that is ( (1)/(N) ∑ |K_Nf|^p' )^1/p' \lesssim ((1)/(N) ∑ |f|^p )^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 ∈ (1,2), which implies that K^* is bounded on ℓ ^p (w) for p ∈ (1, ∞ ), for all weights w in the Muckenhoupt A_p class.

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