A geometry-independent endpoint estimate for the Littlewood--Paley $g$-function generated by the Poisson semigroup

We prove a weak $(1,1)$ estimate for the full Poisson Littlewood--Paley $g$-function on every complete Riemannian manifold without boundary, with an explicit universal constant approximately equal to $3.33606$. The estimate requires no curvature restriction, volume doubling, heat kernel bound, or st