\(\ell^p\)-improving inequalities for discrete spherical averages
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 \(\lambda^2\). \par This inequality is a discrete version of a classical inequality of \textit{W. Littman} [Partial diff. Equ., Berkeley 1971, Proc. Sympos. Pure Math. 23, 479--481 (1973; Zbl 0263.44006)] and \textit{R. S. Strichartz} [J. Funct. Anal. 5, 218--235 (1970; Zbl 0189.40701)] on the \(L^{p}\) improving property of spherical averages on \(\mathbb{R} ^{d}\).
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