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.
Some subcritical estimates for the $\ell^p$-improving problem for discrete curves
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abstract
We apply Christ's method of refinements to the $\ell^p$-improving problem for discrete averages $\mathcal{A}_N$ along polynomial curves in $\mathbb{Z}^d$. Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type $p \to p'$ estimates for the averages $\mathcal{A}_N$ in the subcritical regime. The dependence on $N$ of the constants here obtained is sharp, except maybe for an $\epsilon$-loss.
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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.