Is this family of functions equicontinuous?

Question

Given a sequence $f_n:[0,1]\to[0,1]$ of continuous functions, define $g_n:[0,1]\to{\Bbb R}$ by setting $$ g_n(x)=\int_0^1\frac{f_n(t)}{(t-x)^{1/3}}dt,\quad x\in[0,1]. $$ Show that $(g_n)_{n\in{\Bbb N}}$ has a uniformly convergent subsequence.

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Since $K=[0,1]$ is compact, by Arzela-Ascoli, if one can show

1. $g_n\in C(K)$;
2. $\{g_n\}$ is pointwise bounded;
3. $\{g_n\}$ is equicontinuous on $K$;

then we are done.

I have difficulty in 3. I messed up by only using the definition of equicontinuity. Is there any other theorem which might be useful here?

Answer 1

It should be useful to estimate this integral: $$\int_0^1\left|\frac{1}{(t-x_2)^{1/3}}-\frac{1}{(t-x_1)^{1/3}}\right|\,dt$$ assuming $x_1<x_2$ for simplicity, you can calculate the integral directly.

There is a theorem you can use instead, if you know it: Translation acts continuosly on $L^1$ functions. Apply it to the function $x\mapsto x^{-1/3}$. The above suggestion is bascially just the corresponding direct calculation.