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,ν ) ( X , d , ν ) , it holds true that [ u(x)-u(y)d(x,y)s/p ]Lᵖw(X × X, ν ⊗ ν ) ≈ u Lᵖ(X,ν ), [ 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 [· ]Lᵖw [ · ] L w p is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal.