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

TL;DR

This paper introduces a new proof technique for the dimensionless boundedness of the Bakry Riesz vector on manifolds, utilizing sparse domination in a probabilistic setting to achieve optimal weighted estimates.

Contribution

It develops a novel sparse domination approach for Hilbert space valued martingales, extending it to processes with infinite memory for the first time.

Findings

Established dimensionless Lp boundedness of the Bakry Riesz vector.

Derived optimal weighted estimates with the best possible exponents.

Extended sparse domination techniques to processes with infinite memory.

Abstract

We present a fundamentally new proof of the dimensionless Lp boundedness of the Bakry Riesz vector on manifolds with bounded geometry. 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 cadlag 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.