Given a set $\mathcal{X}$ and RKHS $\mathcal{H}$ of functions on $\mathcal{X}$, we can recover a (pseudo)metric on $\mathcal{X}$ by $d(x,y)=||\phi_x-\phi_y||_{\mathcal{H}}$, where $\phi_x=k(x,\cdot)$.

It is straightforward to see that any function $f \in \mathcal{H}$ which has RKHS norm less than $L$ is Lipschitz (with respect to our metric above) with constant $L$:

$$|f(x)-f(y)|=|\langle f,\phi_x-\phi_y\rangle|\leq ||f||_{\mathcal{H}}d(x,y), $$ for any $x,y \in \mathcal{X}$.

I am very interested in the following question: if we have a function $f\in \mathcal{H}$ which is Lipschitz with constant $L$, is there anything we can say about it's norm $||f||_{\mathcal{H}}$?