A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds

Amenta, Alex; Tolomeo, Leonardo

doi:10.1090/proc/14730

A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds
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Proc. Amer. Math. Soc. 147 (2019), 4797-4803 Request permission

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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