I'm preparing for an exam and here's a practice question:
Suppose $f_n$ is a sequence of differentiable functions on $\mathbb R$ such that $$\int_{-\infty}^{+\infty}|f_n(x)| + |f_n'(x)| + |f_n''(x)| \leq M$$ for all $n \in \mathbb N$.
(a) Show that the sequence is equicontinuous.
(b) Must $\{f_n\}$ be uniformly bounded?
For part (a) I'm trying to utilize Holder's inequality fact that if $\int_{\infty}^{+\infty}|f_n'(x)|^pdx \leq M$ for some $p > 1$ and $M < \infty$, it follows from Holder's inequality that $|f_n(x) - f_n(y)| \leq M^{1/p}|x - y|^{1 - 1/p}$. But I'm not sure how to proceed. Any ideas?
Could someone also give any ideas for part (b)?
We have $$f_n(x_2)-f_n(x_1)=\int\limits_{x_1}^{x_2}f_n'(t)\,dt$$ and $$f_n'(x_2)-f_n'(x_1)=\int\limits_{x_1}^{x_2}f_n''(t)\,dt$$ The first formula implies that $f_n$ has a limit at $- \infty.$ Since the integral of $|f_n(x)|$ is convergent the limit must be $0.$ Therefore $$f_n(x)=\int\limits_{-\infty}^xf_n'(t)\,dt$$ Hence $$|f_n(x)|\le M$$ Similarly $$|f_n'(x)|\le M$$Thus the first formula gives$$|f_n(x_2)-f_n(x_1)|\le M|x_2-x_1|$$