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We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for thevector of higher order maximal Riesz transforms in terms of the correspondingRiesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimatefor the vector of maximal Riesz transforms in terms of the input function. Wealso give explicit estimates for the dependencies of the constants on $p$ whenthe order is fixed. Analogous dimension-free estimates are also obtained forsingle higher order Riesz transforms with an improved estimate of theconstants.

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