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arXiv · 1808.06383

A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds

Alex Amenta·Leonardo Tolomeo
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Abstract

Given a sequence of complete Riemannian manifolds $(M_n)$ of the same dimension, we construct a complete Riemannian manifold $M$ such that for all $p \in (1,\infty)$ the $L^p$-norm of the Riesz transform on $M$ dominates the $L^p$-norm of the Riesz transform on $M_n$ for all $n$. Thus we establish the following dichotomy: given $p$ and $d$, either there is a uniform $L^p$ bound on the Riesz transform over all complete $d$-dimensional Riemannian manifolds, or there exists a complete Riemannian manifold with Riesz transform unbounded on $L^p$.

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Alex Amenta, Leonardo Tolomeo. 2018-08-20. A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds. https://doi.org/10.1090/proc/14730

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