Dimensionless $L^p$ estimates for the Riesz vector on manifolds

TL;DR

This paper introduces a new proof for the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry, enabling stronger weighted estimates with optimal exponents.

Contribution

It provides a novel proof approach that combines weak type estimates and sparse decomposition, improving upon previous methods for Riesz vector bounds on manifolds.

Findings

Dimensionless $L^p$ boundedness established

Weighted $L^p$ estimates with optimal exponents derived

Proof technique involves sparse decomposition and weak type estimates

Abstract

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.