Let ${\cal F}_k$ be the RKHS of functions on an open set $\cal X\subseteq {\bf R}^n$ with kernel $k$. For which $k$ can ${\cal F}_k$ be embedded in the Sobolev space $W_\text{loc}^{\beta,2}(\cal X)$ (for given $\beta>0$)?
Gemini claims that, for all $\beta>0$, $$\tag{1} {\cal F}_k \subseteq W_\text{loc}^{\beta,2}({\cal X}) \quad \Leftrightarrow\quad k\in W_\text{loc}^{\beta,2}(\cal X\times X). $$ Is this correct, and if so, how can it be proven, or is there a good reference for this result? (Gemini claims it is well-known, but its references are not immediately helpful or wrong. GPT-5 gets hopelessly confused with the question.)
Some background. It is fairly easy to prove that ${\cal F}_k$ can be embedded in a Holder-$\beta$ ($0<\beta\le 1$) space if and only if there exists a $c>0$ such that, for all $x,x'\in\cal X$, $$\tag{2} \sqrt{k(x,x)+k(x',x')-2k(x,x')}\le c\|x-x'\|_{\cal X}^\beta.$$ But Holder smoothness in some sense understates the smoothness. For example, with ${\cal X}=[0,1]$, the kernel $k$ on $[0,1]^2$ defined by $k(x,x')=\min(x,x')$ (Brownian motion kernel on $[0,1]$) satisfies (2), so ${\cal F}_k$ is Holder-1/2, as is well-known. It is also well-known that essentially ${\cal F}_k=W^{1,2}([0,1])$ (i.e., smoother than Holder). If the above claim is true, this also follows from (1), since $k\in W_\text{loc}^{1,2}([0,1]^2)$.
(See also Compact embeddings RKHSs into Sobolev Spaces. )
