I am reading these notes by Terence Tao, and I just proved Exercise 17 in Section 1. I've copied it's statement below:
Let ${0 \leq \alpha < \beta \leq 1}$. Show that any bounded sequence
of functions ${f_n \in C^{0,\beta}({\bf R}^d)}$ that are all supported
in the same compact subset of ${{\bf R}^n}$ will have a subsequence
that converges in ${C^{0,\alpha}({\bf R}^d)}$.
Note that here, $C^{0,\beta}$ denotes the Holder Space with parameter $\beta$.
I have been trying to find an example of a sequence $f_n$ as above, such that convergence holds for $\alpha < \beta$ (as per the statement of the theorem) but fails when $\alpha = \beta$.
Does anyone have a good suggestion?
Best Answer
Take $d=1$, $\beta = 1$ and $f_n(x) = \frac{1}{n} \max\{0, \min\{ 1, n x \} \}$. This is bounded in $C^{0,1}$ and converges uniformly to $0$, but the optimal Lipschitz constant of $f_n$ is $1$ (since $f_n(0)=0$, but $f_n(1/n) = 1/n$), so that you don't have convergence in $C^{0,1}$.
It should not be too hard to generalize this to $d > 1$, and probably also to $\beta < 1$.