Averaging with the divisor function: \(\ell^p\)-improving and sparse bounds. (English) Zbl 1505.42021

Giannitsi, Christina

Rocky Mt. J. Math. 52, No. 6, 2027-2039 (2022).

Summary: We study averages along the integers using the divisor function \(d(n)\), defined as
\[K_N f(x)=\frac{1}{D(N)} \sum_{n\leq N} d(n)\;f(x+n),\] 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
\[ \bigg(\frac{1}{N}\sum |K_N f|^{p^\prime} \bigg)^{1/p^\prime} \lesssim \bigg(\frac{1}{N}\sum |f|^p\bigg)^{1/p}\] as long as \(f\) is supported on \([0, N]\).
We will also show that the associated maximal function \(K^\ast f=\sup_N | K_N f |\) 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.

MSC:

42B25	Maximal functions, Littlewood-Paley theory
42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

divisor function; averages; improving bounds; sparse bounds

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