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 \(\left\| \Delta ^{1/2}f\right\| _p\lesssim \left\| \nabla f\right\| _p\) 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 \(\left\| \Delta ^{\gamma }e^{-\Delta }f\right\| _p\lesssim \left\| \nabla f\right\| _p\) , in the Vicsek case, 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 case. It remains an open question to investigate reverse quasi-Riesz for other cable systems.
