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.
Dimension-free $L^p$ estimates for higher order maximal Riesz transforms in terms of the Riesz transforms
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abstract
We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the vector of higher order maximal Riesz transforms in terms of the corresponding Riesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimate for the vector of maximal Riesz transforms in terms of the input function. We also give explicit estimates for the dependencies of the constants on $p$ when the order is fixed. Analogous dimension-free estimates are also obtained for single higher order Riesz transforms with an improved estimate of the constants.
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Under the stated ranges for k, d and p, the maximal operator given by the sup over r of the absolute value of r^k times the k-th derivative of the spherical mean is bounded on L^p with a bound independent of dimension d.
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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.
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Maximal inequalities for derivatives of spherical means
Under the stated ranges for k, d and p, the maximal operator given by the sup over r of the absolute value of r^k times the k-th derivative of the spherical mean is bounded on L^p with a bound independent of dimension d.