I have a question about non-Lipschitz functions. Let $f_1,f_2,...f_n$ be some collection of Holder continuous non-Lipschitz scalar functions defined on a compact subset of $\mathbb{R}^n$. My questions are of the type:
When does there exist a function $h$ s.t $h>0$ and $\sum_i hf_i$ or better all $hf_i$ are Lipschitz or all $hf_i$ are Lipschitz with respect to some of the chosen variables (in case $n>1$)
Any keywords regarding such issues topics or any branches that you know of which studies such things is welcome. Thanks
To make a single function $f$ Lipschitz, one can multiply if by $h(x) = \operatorname{dist}(x,f^{-1}(0)) / f(x)$. (Define $h=1$ on the zero set of $f$. The result is just a signed distance function to the zero set of $f$, which is $1$-Lipschitz.
I would not expect to be able to do this for two or more functions at once, unless they are tightly related in some way. It would take some very specific assumptions. It's better to just work with the functions you have than try to come up with a set of conditions.
Finally, I would not expect there to be articles on "how to make a non-Lipschitz function Lipschitz by multiplication". This just does not sound like a topic of a research paper. As a general reference for Lipschitz functions, I find Lipschitz Analysis by Heinonen useful.