The l2 norm of the r-variation seminorm of spherical means on the hypercube has no dimension-free bound for any r when radii are unrestricted, but admits such bounds for r greater than 2 when radii are restricted to fixed parity.
M¨ uller,A geometric bound for maximal functions associated to convex bodies, Pacific J
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Uniform bounds on Delannoy numbers from lattice-point counts in cross-polytopes produce dimension-free estimates for discrete maximal functions over these polytopes for radii larger than order d to the 3/2.
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Dimension dependence and dimension-free $\ell^2$ estimates for variation seminorms of spherical means on the hypercube
The l2 norm of the r-variation seminorm of spherical means on the hypercube has no dimension-free bound for any r when radii are unrestricted, but admits such bounds for r greater than 2 when radii are restricted to fixed parity.
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Uniform estimates for Delannoy numbers and dimension-free estimates for discrete maximal functions over cross-polytopes
Uniform bounds on Delannoy numbers from lattice-point counts in cross-polytopes produce dimension-free estimates for discrete maximal functions over these polytopes for radii larger than order d to the 3/2.