Fixed-radius lattice sphere averages on Z^d satisfy the sharp l^p improving estimate down to the endpoint p=(d+2)/d for every d at least 4, with the optimal decay exponent.
$\ell^p(\mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We exhibit a range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $d\geq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $\ell^2(w)$ into $\ell^2(w)$ provided $w^{\frac{d}{d-4}+\delta}$ belongs to the Muckenhoupt class $A_2$ for some $\delta>0.$
citation-role summary
background 1
citation-polarity summary
fields
math.CA 1years
2026 1verdicts
CONDITIONAL 1roles
background 1polarities
unclear 1representative citing papers
citing papers explorer
-
Sharp $\ell^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages
Fixed-radius lattice sphere averages on Z^d satisfy the sharp l^p improving estimate down to the endpoint p=(d+2)/d for every d at least 4, with the optimal decay exponent.