Dimension-free $L^p$ estimates for higher order maximal Riesz transforms in terms of the Riesz transforms

We prove a dimension-free $L^p({\mathbb{R}}^d)$, $1<p<\mathrm{\infty }$, 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.