Htykuut

I was reading a Navier-Stokes paper by Kato, and at the end of the paper he makes the following affirmation:

Let $f_i:(0,infty)timesmathbb{R^2}tomathbb{R}$ be a smooth function such that for all $ninmathbb{N}, qin(1,infty],kgeq0$, 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 $f:(0,infty)timesmathbb{R^2}tomathbb{R}$.

• 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 $$|f(x_1,x_2,x_3)-f(y_1,y_2,y_3)| le$$ $$|f(x_1,x_2,x_3)-f(x_1,x_2,y_3)|+|f(x_1,x_2,y_3)-f(y_1,y_2,y_3)|$$ $$ le C|x_3-y_3|+|f(x_1,x_2,y_3)-f(x_1,y_2,y_3)|+|f(x_1,y_2,y_3)-f(y_1,y_2,y_3)|$$ $$le C|x_3-y_3|+C|x_2-y_2|+C|x_1-y_1| le C_3 C |x-y|.$$