A metric counterpart of the Gu-Yung formula

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