I was reading a Navier-Stokes paper by Kato, and at the end of the paper he makes the following affirmation:
Let $f_i:(0,\infty)\times\mathbb{R^2}\to\mathbb{R}$ be a smooth function such that for all $n\in\mathbb{N}, q\in(1,\infty],k\geq0$, all the norms $||\partial^{n}_{t}D^kf_i(t)||_q$ are bounded on any compact interval $[a,b]\subset(0,\infty)$ uniformly in $i$. Then, we can extract a subsequence $\{f_{i_k}\}$ that converge locally uniformly to a smooth function $f:(0,\infty)\times\mathbb{R^2}\to\mathbb{R}$.
- here $D^k$ is the fractional derivative.
The only result I know about extracting subsequence that converge locally uniformly was Arzelá-Ascoli, but I could't show that the sequence $\{f_i\}$ was equicontínuous.
You can deduce equicontinuity from a uniform bound on the derivative.
In general, let $(f_n)_n$ be a sequence of functions with $||f_n'||_\infty \le C$ for all $n$. Then for any $x,y$ and any $n \ge 1$, $|f_n(x)-f_n(y)| \le C|x-y|$, which of course in particular gives equicontinuity.
This also extends to functions of more than one variable. Indeed, if $(f_n)_n$ is a sequence of functions from $\mathbb{R}^m$ with $||\frac{\partial}{\partial x_j}f_n||_\infty \le C$ for each $1 \le j \le m$ and each $n$, then given any $(x_1,\dots,x_m),(y_1,\dots,y_m) \in \mathbb{R}^m$ and any $n \ge 1$, we can bound $|f_n(x_1,\dots,x_m)-f_n(y_1,\dots,y_m)| \le C_m C |x-y|$, where $C_m$ is just to switch between norms.
For example, for $m=3$, we do $$|f(x_1,x_2,x_3)-f(y_1,y_2,y_3)| \le$$ $$|f(x_1,x_2,x_3)-f(x_1,x_2,y_3)|+|f(x_1,x_2,y_3)-f(y_1,y_2,y_3)|$$ $$ \le C|x_3-y_3|+|f(x_1,x_2,y_3)-f(x_1,y_2,y_3)|+|f(x_1,y_2,y_3)-f(y_1,y_2,y_3)|$$ $$\le C|x_3-y_3|+C|x_2-y_2|+C|x_1-y_1| \le C_3 C |x-y|.$$