2024/03/05 by Devyver, Baptiste, Russ, Emmanuel
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.02779
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 \Vert Δ1/2f\Vertp\lesssim \Vert ∇ f\Vertp is false for all p∈ [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 \Vert Δγe-Δf\Vertp\lesssim \Vert ∇ f\Vertp, in the (unbounded) Vicsek cable system, for p∈ (1,+∞) and γ>0. These reverse inequalities are strongly related to the problem of Lp boundedness of the operators ∇ e-ΔΔ-ε, 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 γ∈ (0,1) and p∈ (1,+∞) 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.