Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion

Abstract

In the $D$-dimensional Vicsek graph, we prove that the Riesz-like inequality $ \|\nabla f\|_p \leq C \|\Delta^\gamma f\|_p $ holds for every $p\in(1,\infty)$ and every $ 0<\gamma<\gamma^*(p):=\frac{1}{D+1}+\frac{D-1}{D+1}\,\frac{1}{p}, $ while it fails whenever $p\in(1,\infty)$ and $\gamma^*(p)<\gamma<1$. Thus, the validity of the inequality remains open only at the critical exponent $\gamma=\gamma^*(p)$. This provides the first example of an $L^p$-bounded ``super-Riesz transform'', namely an operator of the form $\nabla \Delta^{-\gamma}$ with $\gamma$ strictly larger than the Euclidean threshold $\frac12$. To achieve this, we establish a more general result linking the diffusion escape rate and a Poincar\'e inequality on balls to the validity of the reverse Riesz-like inequality $\|\Delta^\gamma f\|_p \leq C \|\nabla f\|_p.$