Classic and exotic Besov spaces induced by good grids

In a previous work we introduced Besov spaces ${\mathcal{B}}_{p,q}^s$ defined on a measure spaces with a good grid, with $p\in [1,\mathrm{\infty })$, $q\in [1,\mathrm{\infty }]$ and $0<s<1\mathrm{/}p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces.