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

We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the vector of maximal Riesz transforms of odd order 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 Riesz transforms of odd orders with an improved estimate of the constants. These results are a dimension-free extension of the work of J. Mateu, J. Orobitg, C. P\'erez, and J. Verdera. Our proof consists of factorization and averaging procedures, followed by a non-obvious application of the method of rotations.