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.
Discrete analogues in harmonic analysis: Spherical averages
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In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular, the "circle method". The results we obtained are by necessity limited to d \ge 5, and moreover the range of p for the L^p estimates differs from its analogue in \Bbb R^d.
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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.