What is the native Hilbert space associated with the kernel $\frac{\sum \min{(x_i,y_i)}}{\sum \max{(x_i,y_i)}}$?

Question

In this answer on MSE it is shown that the function $$ K:(\mathbb{R}^{>0})^n\times (\mathbb{R}^{>0})^n\rightarrow\mathbb{R}\,\quad K(x,y)=\frac{\sum_{i=1}^n\min{(x_i,y_i)}}{\sum_{i=1}^n\max{(x_i,y_i)}}$$ is positive semi-definite, i.e. a kernel function.

I would like to know the native space, i.e. the associated reproducing kernel Hilbert-space, of this kernel. I could determine that the space in case $n=1$ is the pull back by $\log$ of the Sobolev-Space $W^{1,2}(\mathbb{R})$ (see this answer). But this is no longer true for $n>1.$