A metric counterpart of the Gu-Yung formula

In this note we consider a generalisation to the metric setting of the recent work (Gu and Yung in J Funct Anal 281:109075, 2021). In particular, we show that under relatively weak conditions on a metric measure space $(X,d,\nu )$ , it holds true that $[\frac{u\left(x\right)-u\left(y\right)}{d{(x,y)}^{\frac{s}{p}}}{]}_{L_w^p(X\times X,\nu \otimes \nu )}\approx {\Vert u\Vert }_{L^p(X,\nu )},$ where s is a generalised dimension associated to X and ${[·]}_{L_w^p}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.