Reverse inequalities for quasi-Riesz transform on the Vicsek cable system

This work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ is false for all $p\in [1,2)$. Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form $\left\Vert \Delta^{\gamma}e^{-\Delta}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$, in the (unbounded) Vicsek cable system, for $p\in (1,+\infty)$ and $\gamma>0$. These reverse inequalities are strongly related to the problem of $L^p$ boundedness of the operators $\nabla e^{-\Delta}\Delta^{-\varepsilon}$, the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of $\gamma\in (0,1)$ and $p\in (1,+\infty)$ such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these.