Weighted norm inequalities on graphs

Nadine Badr (1) , Martell José Maria (2)

1 ICJ - Institut Camille Jordan
2 Instituto de Ciencias Matematicas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Cientficas

Let $(\Gamma,\mu)$ be an infinite graph endowed with a reversible Markov kernel $p$ and let $P$ be the corresponding operator. We also consider the associated discrete gradient $\nabla$. We assume that $\mu$ is doubling, a uniform lower bound for $p(x,y)$ when $p(x,y)>0$, and gaussian upper estimates for the iterates of $p$. Under these conditions (and in some cases assuming further some Poincaré inequality) we study the comparability of $(I-P)^{1/2} f$ and $\nabla f$ in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood-Paley-Stein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.

Domaines: Analyse fonctionnelle [math.FA]

Nadine Badr, Martell José Maria. Weighted norm inequalities on graphs. Journal of Geometric Analysis, 2012, pp.1173-1210. ⟨hal-00863448⟩