New Riemannian manifolds with $L^p$-unbounded Riesz transform for $p > 2$

TL;DR

This paper constructs a broad class of Riemannian manifolds demonstrating that the Riesz transform can be unbounded on L^p spaces for all p > 2, expanding understanding beyond fractal structures.

Contribution

It introduces new Riemannian manifolds with unbounded Riesz transforms for p > 2, showing fractal structure is not required for this property.

Findings

Riesz transform unbounded on constructed manifolds for p > 2

Extension of previous results beyond Vicsek manifolds

Fractal structure not necessary for unbounded Riesz transform

Abstract

We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $L^p(M)$ for all $p > 2$. This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.