On a dimension-free control of maximal truncated Riesz transforms in terms of Riesz transforms

Abstract:

In a recent joint work with Maciej Kucharski and Jacek Zienkiewicz (Wrocław) we proved a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for (arbitrary) higher order maximal truncated Riesz transforms in terms of the corresponding Riesz transforms. Similar estimates were also obtained for vectors of maximal Riesz transforms. Our results are a dimension-free extension of the work of J. Mateu, J. Orobitg, C. Pérez, and J. Verdera. During the talk I will mostly focus on first order (classical) Riesz transforms on $\mathbb{R}^d$. I will also highlight the extra steps needed to treat higher order Riesz transforms.