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.
Bourgain,OnL p bounds for maximal functions associated to convex bodies inR n, Israel J
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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.