Reverse inequality for the riesz transforms on Riemannian manifolds

2022/09/12 by Russ, Emmanuel, Devyver, Baptiste · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.05083

Abstract

Let M be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and L^q scaled Poincaré inequalities on suitable remote balls for some q<2. We prove the inequality \Vert Δ^1/2f\Vert_p\lesssim \Vert ∇ f\Vert_p for all p∈ (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 \Vert Δ^1/2f\Vert_p\lesssim \Vert ∇ f\Vert_p may be false for all p∈ [1,2). Following a recent joint work by the two authors and Yang, we examine the validity of inequalities of the form \Vert Δ^γe^-Δf\Vert_p\lesssim \Vert ∇ f\Vert_p. In the Vicsek case, we give the optimal range of p for which this inequality holds.

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• Isoperimetric Inequality on Manifolds with Quadratically Decaying Curvature

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