Abstract 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, ) (X, d, ν), it holds true that u (x) -u (y) d (x, y) ^{s{p}} ₋㵵ₖ (ₗ ₗ, ) u ₋㵵 (ₗ, ), u (x) - u (y) d (x, y) s p L w p (X × X, ν ⊗ ν) ≈ ‖ u ‖ L p (X, ν), where s is a generalised dimension associated to X and ₋㵵ₖ · L w p is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.
Buccheri et al. (Mon,) studied this question.