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I want to find a continuous unbounded function $f\colon \mathbb R ^2 \to \mathbb R$ with $\iint _{\mathbb R^2} |f|^2\, dx\,dy < \infty$ .

At first I thought it would be easy, but the continuity makes it harder. And the more I am thinking , the more I am convinced it is not possible. And I cannot prove either that it is not possible. Any idea or help would be appreciated.

Przemysław Scherwentke
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anumosh
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  • Do you know how to do this when $f$ is a map $\mathbb R\rightarrow \mathbb R$? – Potato May 07 '15 at 06:30
  • @AlexR That's not true. – Potato May 07 '15 at 06:34
  • @Potato Oh, I see it now myself. Didn't think about summing bumps ;) – AlexR May 07 '15 at 06:36
  • Some idea are given in the 1-dimensional case here, which you can easily adapt to the 2-dimension version: http://math.stackexchange.com/questions/482934/continuous-unbounded-but-integrable-functions – Potato May 07 '15 at 06:37
  • https://math.stackexchange.com/questions/2527941/if-f-in-l1-bbb-r-dx-then-prove-that-for-almost-every-x-in-bbb-r-lim-lim – Guy Fsone Jun 10 '22 at 15:46

3 Answers3

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Consider the function $f:[0,+\infty)\to\mathbb R$ whose graph is the polyngonal line with vertices \begin{align} &(0,0),\\ &(1-\tfrac1{2},0),(1,1),(1+\tfrac1{2},0),\\ &(2-\tfrac1{2^2},0),(2,2),(2+\tfrac1{2^2},0),\\ &(3-\tfrac1{2^3},0),(3,3),(3+\tfrac1{2^3},0),\\ \dots \end{align} This is unbounded, continuous, and square-integrable.

Now consider $g:\mathbb R^2\to\mathbb R$ such that $g(x)=f(||x||)$.

Mariano Suárez-Álvarez
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HINT: Consider points $a_n=(0,n)$, where $n\in\mathbb{N}$. Make $f^2$ in the following way. Take at $a_n$ a cone of height $n$ and volume $1/2^n$ and $f^2$ is equal to 0 otherwise. Let $f$ be a nonegatitive square root of $f^2$. It is continuous (why?) and the integral is bounded (why?).

Przemysław Scherwentke
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If you have an example for $f:\mathbb R\to\mathbb R$ you can extend this by $$f(x,y) := f(x)\phi(y)$$ Where $\phi(y) \in L^2(\mathbb R)$ is a bump (such as the gaussian $\phi(y) = \exp(-y^2)$).
For one dimension, think of a sequence of compactly supported bumps with increasing height but decreasing with. Take a bump function $\psi\in L^2([0,1]) \cap C_0^0([0,1])$ with $\|\psi\|_{L^2} = 1$ and let $f$ be a sum of shifted and scaled copies of $\psi$ $$f = \sum_{k=0}^{\infty} a_k \sqrt k\psi(kx-1)$$ Such that $\{a_k\}\in \ell^2(\mathbb N_0)$. Then $f$ satisfies the conditions and $$\|f\|^2_{L^2} \le \sum_{k=0}^\infty a_k^2 \|\psi\|^2_{L^2} = \sum_{k=0}^\infty a_k^2 = \|a_k\|_{\ell^2} < \infty$$ A concrete $a_k$ can be $a_k = \frac1k$ and a concrete $\psi$ can be $$\psi(x) = \begin{cases} \sqrt{ 4x} & x\in[0,\frac12]\\ \sqrt{1-4x} & x\in(\frac12, 1]\\0&x\notin[0,1]\end{cases}$$

AlexR
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