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

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 $$\mathbb{R} ^{d}$$.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory

### Keywords:

discrete spherical average; maximal function

### Citations:

Zbl 0263.44006; Zbl 0189.40701
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