For higher-order Riesz transforms the truncation kernel b_{k,d} is nonnegative with unit L1 norm only when k=1 or 2; for k>=3 its L1 norm tends to infinity with dimension while its Fourier transform stays bounded by 1, giving dimension-free L2 control of the truncated operator by the full one.
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On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms
For higher-order Riesz transforms the truncation kernel b_{k,d} is nonnegative with unit L1 norm only when k=1 or 2; for k>=3 its L1 norm tends to infinity with dimension while its Fourier transform stays bounded by 1, giving dimension-free L2 control of the truncated operator by the full one.