4
$\begingroup$

Suppose $f_n: [0,1] \mapsto \mathbb{R} $ are continuous functions. I'm interested in knowing under what conditions does $f_n \stackrel{L^2[0,1]}{\to}f$ imply $\sup_{x \in [0,1]} |f_n(x)-f(x)| \to 0$. Clearly one example that implies this is if each $f_n$ is uniformly bounded, and has uniformly bounded first and second derivatives (the result follows from Arzela-Ascoli in this case, Does Lp-convergence and uniform boundedness in $C^2$, imply $C^{1}$ convergence?). Is this basically necessary and sufficient? Or are there weaker conditions under which convergence in mean-square implies convergence uniformly?

$\endgroup$

2 Answers 2

2
$\begingroup$

The theorem of Ascoli-Arzelá is necessary and sufficient. By this I mean that, if the sequence of continuous functions $f_n$ converges uniformly to some continuous function, then $f_n$ is uniformly bounded and equicontinuous. This is not hard to prove.

However, the a priori knowledge that $f_n\to f$ in $L^2(0,1)$ does give an important piece of information: if $f_n$ is uniformly bounded and equicontinuous, then $f_n\to f$ uniformly. Indeed, by the theorem of Ascoli-Arzelá, $f_n$ and each of its subsequences have a subsequence that converges uniformly to some continuous function. Such function must be $f$, because uniform convergence implies $L^2$ convergence on bounded intervals. And so we can conclude that $f_n\to f$ uniformly, as claimed.

The point is that Ascoli-Arzelá gives uniform convergence of a subsequence to some continuous function, which is completely unknown a priori. The $L^2$ convergence allows us to conclude that the whole sequence $f_n$, not just a subsequence, converges to $f$, not just to some function.

$\endgroup$
1
$\begingroup$

Yes there are weaker conditions. It will work as soon as you have an interpolation inequality of this kind for some fixed constant $t\in (0,1]$ $$ \|u\|_{L^\infty} \leq \|u\|_{L^2}^t \|u\|_{X}^{1-t} $$ and that you have uniform bounds in the $X$ norm, since then replacing $u$ by $f_n-f$ and using the fact that $\|f_n-f\|_{X} \leq \|f_n\|_{X} + \|f\|_{X} \leq C$, you obtain $$ \|f_n-f\|_{L^\infty} \leq C^{1-t} \|f_n-f\|_{L^2}^t $$

Now, there are a lot of interpolation inequalities of this kind, due to the theory of interpolation of spaces and of Sobolev type embeddings. Examples of spaces $X$ are the Hölder spaces $C^s$ for any $s>0$ (in particular $C^1$, but also $C^{0,1/2}$ for example), the Sobolev spaces $W^{s,p}$ with $s > 1/p$ (see e.g. Gagliardo-Nirenberg Sobolev's inequalities here) so for example $H^1 = W^{1,2}$ ...

$\endgroup$
3
  • $\begingroup$ Thanks so much! Very useful information $\endgroup$ Commented Sep 28, 2020 at 23:02
  • $\begingroup$ I found Agmon's inequality, which is a version of this bounding the esssup norm by the produce of $L^2$ norm and a Sobolev norm, en.wikipedia.org/wiki/Agmon%27s_inequality , but do you have a reference for this inequality for the Holder space case? $\endgroup$ Commented Sep 29, 2020 at 15:17
  • $\begingroup$ It should follows from interpolation of spaces and Sobolev embeddings. The first allows to control the $W^{s,p}$ norm by the product of $C^{,\alpha}$ with $L^2$ norm (look at interpolation of Besov and Triebel Lizorkin spaces). I will look for references if I have time. $\endgroup$
    – LL 3.14
    Commented Sep 29, 2020 at 16:35

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .