Reverse inequality for the riesz transforms on Riemannian manifolds

TL;DR

This paper establishes reverse inequalities for Riesz transforms on Riemannian manifolds with specific geometric conditions, introduces new Hardy inequalities, and explores analogous inequalities on fractal-like cable systems, including the Vicsek fractal.

Contribution

It generalizes previous Riesz transform inequalities to broader geometric settings and introduces novel Hardy inequalities, also analyzing similar inequalities on fractal structures.

Findings

Proved inequalities for Riesz transforms on manifolds with doubling volume and Poincaré inequalities.

Established new Hardy inequalities relevant to the analysis.

Determined the optimal range of p for inequalities on the Vicsek fractal.

Abstract

Let $M$ be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and $L^q$ scaled Poincar\'e inequalities on suitable remote balls for some $q<2$. We prove the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all $p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when $M$ has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest. The second part of this work deals with analogous questions in fractal-like cable systems. In this framework, it was already proved by Chen, Coulhon, Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ may be false for all $p\in [1,2)$. Following a recent joint work by the two authors and Yang, we examine the validity of inequalities of the form $\left\Vert \Delta^{\gamma}e^{-\Delta}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$. In the Vicsek case, we give the optimal range of $p$ for which this inequality holds.