Dimension-free Lp estimates for higher-order maximal Riesz transforms in terms of the Riesz transforms

We prove a dimension-free L p ‫ޒ(‬ d ) estimate, 1 < p < ∞, for the vector of higher-order maximal Riesz transforms in terms of the corresponding Riesz transforms.This implies a dimension-free L p ‫ޒ(‬ 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.Maciej Kucharski and Błażej Wróbel were supported by the National Science Centre (NCN), Poland research project Preludium Bis 2019/35/O/ST1/00083.The present paper is a merger of our previous two preprints in which we treated separately the cases of even and odd orders k.MSC2020: 42B15, 42B20, 42B25.