On Vicsek graphs the inequality ||∇f||_p ≤ C ||Δ^γ f||_p holds for γ < 1/(D+1) + (D-1)/(D+1)(1/p) and fails for larger γ < 1, providing the first super-Riesz example with γ > 1/2.
Reverse inequality for the Riesz transforms on Riemannian manifolds
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Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion
On Vicsek graphs the inequality ||∇f||_p ≤ C ||Δ^γ f||_p holds for γ < 1/(D+1) + (D-1)/(D+1)(1/p) and fails for larger γ < 1, providing the first super-Riesz example with γ > 1/2.