In 1983 E. M. Stein proved that the vector of classical Riesz transforms has $L^p$ bounds on $\mathbb R^d$ which are independent of the dimension $d$. I will discuss an analogous result for the vector of maximal Riesz transforms. I will also mention generalizations to higher order Riesz transforms. A main principle of our approach is a comparison between a maximal Riesz transform and the corresponding Riesz transform. The talk is based on joint work with Maciej Kucharski and Jacek Zienkiewicz (Wrocław).

The video of this talk is available on the IISc Math Department channel.