In graphs and Riemannian manifolds where the kernel of the diffusion semigroup satisfies pointwise sub-Gaussian estimates, we study the range of parameters \( p \in (1, \infty) \) and \( γ\in [0, 1] \) for which the quantities \( \|Δ^γf\|_p \) and \( \|\nabla f\|_p \) can be compared. In particular, we prove that in such metric spaces, the Riesz transform \( \nabla Δ^{-1/2} \) is unbounded on \( L^p \) for all \( p \in (2, \infty) \), thereby demonstrating a clear departure from the behavior observed in the Euclidean setting.