New Riemannian manifolds with \(L^p\)-unbounded Riesz transform for \(p > 2\) (Q2223502)

The author constructs a special case of Riemannian manifolds with interesting property. Key point for constructing such manifolds is the Riesz transform \(\nabla\varDelta^{-1/2}\) on manifold \(M\), where \(\nabla\) is gradient and \(\Delta\) is the Laplace-Beltrami operator. The Riesz operator boundedly acts from \(L^2(M) \) into the space of square integrable vector fields \(L^2(M:TM)\). The question is as follows: can one obtain the same property for \(p\neq 2\). In other words, the following property \[ |||\nabla f|||_p\lesssim||\Delta^{1/2}f||_p,~~~\forall f\in C^{\infty}_c, \tag{\(1\)} \] is valid or not? Some manifolds for which the property fails for some \(p > 2\) are known [\textit{L. Chen} et al., J. Geom. Anal. 27, No. 2, 1489--1514 (2017; Zbl 1371.58015)]. In this article, the author constructs a class of manifolds of arbitrary dimension for which (1) fails for all \(p > 2\). These manifolds are thickenings of so-called spinal graphs, satisfying generalized dimension conditions defined in terms of the spinal structure along with a polynomial volume lower bound. As author notes, previous constructions use a fractal nature of special Vicsek manifolds and he describes a large class of non-fractal spinal graphs with the desired dimension conditions and volume lower bounds; these manifolds of arbitrary dimension have no fractal structure and the property (1) fails for all \(p > 2\).