Endpoint Riesz transforms on normalized graphs

Abstract

For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian $\Delta$, we prove weak type $(1,1)$ estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carr\'e-du-champ transform has constant $H_1+1$ whenever $H_1=\sup_x m(B(x,1))/m(x)<\infty$. These endpoint constants apply to real-valued inputs and yield strong $\ell^p$ bounds for $1<p\le2$. The proof uses a discrete obstacle decomposition obtained by a contraction mapping, followed by elementary support estimates for the two gradients.