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We know that if a real function is in $L^2$ then it is in $L^1$, but the reverse is not necessarily true. So what are the examples of functions that are $L^1$ but not $L^2$, especially those intrinsically unbounded ones if any? Welcome more examples, or classes of examples. Thanks.

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5 Answers 5

up vote 10 down vote accepted

I assume that you mean integrable functions in $[0,1]$.

In the sense of Baire almost every function in $L^{1}[0,1]$ is not in $L^{2}[0,1]$:

The space $L^{2}[0,1]$ is meager in $L^{1}[0,1]$ (that is to say it is a countable union of sets whose closure has empty interior in $L^{1}$).

The easiest way to see this by using the open mapping theorem: the inclusion $L^{2}[0,1] \to L^{1}[0,1]$ is continuous but not onto. More explicitly, the set $B_{n} = \{f \in L^{1}\,:\,\int |f|^{2} \leq n\}$ is easily seen to be closed and have empty interior and $L^{2}[0,1] = \bigcup_{n=1}^{\infty} B_{n}$. Similarly, if $1 \leq p < q \leq \infty$ then $L^{q}[0,1] \subset L^{p}[0,1]$ is meager.

This result is analogous to Baire's theorem saying that almost every continuous function on $[0,1]$ is nowhere differentiable, and with the same defect: If you choose a 'generic' function it won't be differentiable (or square-integrable) but from the statement you don't have a clue what such a function looks like.

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Neither containment holds in general. On $\mathbb{R}$, the function $f$ such that $f(x)=1/x$ if $x\geq1$ and $f(x)=0$ otherwise is in $L^2\setminus L^1$, and the function $g$ such that $g(x)=1/\sqrt{x}$ if $0\lt x\leq1$ and $g(x)=0$ otherwise is in $L^1\setminus L^2$.

For bounded domains, $L^2\subset L^1$, because what keeps a function from being integrable on a bounded set is being too large, and squares make large numbers larger. If $f$ is a bounded function in $L^1$, then $f$ is also in $L^2$, because what keeps a bounded function from being integrable is not going to zero fast enough, and squares make numbers go to zero faster.

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+1. I think your reply is really nice! Partially answer my question –  Tim Dec 28 '12 at 18:53

For an important example,

Check out Kolmogorov's example of a function in $L^1$ whose Fourier series diverges almost everywhere: http://books.google.com/books?id=ikN59GkYJKIC&pg=PA1

By Carleson's theorem this cannot be in $L^2$.

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But fortunately you don't need such heavy artillery to see that (non-square integrability of that example, I mean). –  t.b. Jan 21 '11 at 4:46
    
This example is really neat. –  Jonas Meyer Jan 21 '11 at 4:58
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Something for mathoverflow.net/questions/42512/… perhaps? ;) –  Hans Lundmark Jan 21 '11 at 11:36
    
@Hans: Perhaps :-) Liked the statement "FLT is not strong enough to prove $\sqrt{2}$ is irrational" :-) But Carleson's theorem is an important result and I thought it was worth mentioning. Besides OP did ask for a class of examples :-) –  Aryabhata Jan 21 '11 at 17:47

You can get an example from any two sequences of positive reals such that $\sum \alpha_n \beta_n < \infty$ but $\sum \alpha_n^2 \beta_n = \infty$, e.g. $\alpha_n = n$ and $\beta_n = 1/n^3$.

For the reverse non-containment, take $\alpha_n = 1/n$ and $\beta_n = 1$.

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If we look at the discrete group $\mathbb{Z}$, we have $L^1(\mathbb{Z})=\ell^1(\mathbb{Z})\subset \ell^2(\mathbb{Z})=L^2(\mathbb{Z})$ - see the thread How do you show that $\ell^p\subset \ell^q$ for $p≤q$?.

If we look at the compact group $\mathbb{T}$ (the unit circle), we have $L^1(\mathbb{T})\subset L^2(\mathbb{T})$, since by Hölder's inequality $$\|f\|_{L^1}=\int|f|dx=\int1\cdot|f|dx\leq\sqrt{\int1^2dx}\sqrt{\int|f|^2dx}=1\cdot\|f\|_{L^2}=\|f\|_{L^2}$$

If we look at the continuous non-compact group $\mathbb{R}$ there is no inclusion.

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