Consider $\mathcal{X}\subset \mathbb{S}^n$ and Laplace kernel $k(x,y)=\exp(-\|x-y\|)$. Is the RKHS $H(\mathcal{X})$ given by $k(x,y)=\exp(-\|x-y\|), x,y \in \mathbb{S}^n$ equivalent to Sobolev space $W^{(d+1)/2,2}(\mathcal{X})$?
I have known, if $\mathcal{X}$ is a subset of $\mathbb{R}^n$ then the desired claim holds.
The RKHS of the Laplace kernel on $\mathcal X\subset \mathbb{S}^n$ should be equivalent to the Sobolev space $W^{(d+1)/2,2}$ if you assume that $\mathcal X$ is a Lipschitz domain (common regularity assumption when working with Sobolev spaces).
Looking at the proof of Lemma E.1 in https://arxiv.org/pdf/2305.14077.pdf may be helpful. Since you know the desired claim for subsets of $\mathbb{R}^n$, you can map to $\mathcal X$ via a diffeomorphism, under regularity assumptions on $\mathcal X$, and if the RKHS is equivalent to a Sobolev space before applying the diffeomorphism, it will be equivalent to the Sobolev space of the same smoothness after the diffeomorphism.