\(\ell^p\)-improving inequalities for discrete spherical averages. (English) Zbl 1449.42033

Kesler, R.; Lacey, M. T.

Anal. Math. 46, No. 1, 85-95 (2020).

For \( \lambda ^2 \in \mathbb{N} \), let \(\mathbb{S}^d_\lambda := \{ n\in \mathbb{Z}^d \;:\; | n| = \lambda\}\). For a function \(f:\mathbb{Z} ^{d} \to \mathbb{R} \), define \[A _{\lambda } f (x) = | \mathbb{S}^d_\lambda | ^{-1} \sum_{n \in \mathbb{S}^d_\lambda } f(x-n).\] The following estimate is the main result of the paper under review: \[| A _{\lambda }| _{\ell ^{p} \to \ell ^{p'}} \leq C _{d,p, \omega (\lambda ^2 )} \lambda ^{d (1-\frac{2}p)}, \tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}, d\geq 4.\] In dimension \(d=4\) this estimate proved for odd \(\lambda ^2\). Here \(\omega (\lambda ^2)\) is the number of distinct prime factors of \(\lambda^2\).
This inequality is a discrete version of a classical inequality of W. Littman [Partial diff. Equ., Berkeley 1971, Proc. Sympos. Pure Math. 23, 479–481 (1973; Zbl 0263.44006)] and R. S. Strichartz [J. Funct. Anal. 5, 218–235 (1970; Zbl 0189.40701)] on the \(L^{p}\) improving property of spherical averages on \(\mathbb{R} ^{d}\).

Reviewer: Michael Perelmuter (Kyïv)

MSC:

42B25

Maximal functions, Littlewood-Paley theory

Keywords:

discrete spherical average; maximal function

Citations:

Zbl 0263.44006; Zbl 0189.40701