This question is on the same line as this older one metric entropy for Lipschitz functions.
I would like to know if there is a reference for bounds of the covering numbers $\mathcal N(\mathsf{Lip}_L(X,Y), \epsilon, ||\cdot||_\infty)$, where $(X,d_X), (Y,d_Y)$ are metric spaces.
I'm using the usual notation in which $\mathsf{Lip}_L$ denotes the space of $L$-Lipschitz functions, and $||f-g||_\infty:=\sup_{x\in X} d_Y(f(x),g(x))$.
I'd like to know an upper bound for the above covering number, in terms of covering numbers $\mathcal N(X,L/\epsilon, d_X)$ and $\mathcal N(Y, \epsilon, d_Y)$ or some related quantities. This is due to Kolmogorov-Tikhomirov for the case $X$ is centralizable and $Y$ is an interval of the reals, but what about other nice doubling $Y$'s?
The proof of this type of bounds would not be that much different than the original Kolmogorov-Tikhomirov proof, but how could it be that nobody wrote it so far? I could not find any paper discussing it, maybe I'm using the wrong keywords in Google Scholar.
What would be good keywords? Or is there a problem, due to which the case of general metric space targets $Y$ is very complicated? Any indications would be highly appreciated!
Edit
The definition Kolmogorov-Tikhomirov use for covering is "covering by sets of diameter at most 2r" and the centralizability condition allows to pass to "covering by r-balls", and then most of the proof strategy proceeds by using r-balls. Thus when using coverings by balls, the original proof is sufficient. I'm guessing that since most application are based on r-ball covers, the covering using general sets is out of the radar of most of the community.
