Continuous sparse domination and dimensionless weighted estimates for the Bakry–Riesz vector

Abstract We present a fundamentally new proof of the dimensionless <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> L^{p} boundedness of the Bakry–Riesz vector on manifolds with bounded geometry, with our proof being new even in the Euclidean setting. Our proof has the significant advantage that it allows for a much stronger conclusion than previous arguments, namely that of some new dimensionless weighted estimates with optimal exponent. Part of the importance of this task lies in the novelty of the techniques: we develop the self-similarity argument known as “sparse domination” in the setting of uniformly integrable càdlàg Hilbert space valued martingales and extend the domination to a process with infinite memory. We provide a range of optimal weighted estimates and weak type estimates for these stochastic processes. Previous geometric Riesz transform estimates relied on Bellman functions and did not provide this range of weighted estimates. The development of sparse domination in this probabilistic setting and its use for high-dimensional problems is new.