I've been interested to learn more about Lebesgue spaces, as they proved very useful in my PDE research. I am mainly interested in the functions which are $L^2(\mathbb{R}^2)$ function, whose Laplacian, $\Delta f\notin L^2(\mathbb{R}^2).$ Are there many known examples of such functions? Even in one dimension would be helpful..
I was playing around with Matlab, and I wondered if there is a useful algorithm in showing which (square integrable) functions have this property. So far, my method involves crude trial and error (direct computation of Laplacian+integration) with different two variable square integrable functions, (some Gaussian variation or some piecewise defined functions) however this is not very efficient. Thanks!
Let me define $\langle \xi\rangle=(1+|\xi|^2)^{1/2}$.
It is helpful to look at the problem on the fourier side. Since the Fourier transform preserves the $L^2$ norm, and $\widehat{\Delta f(x)}(\xi)=-|\xi|^2f(\xi)$, you essentially are asking what are the functions $f(\xi)\in L^2(\mathbb{R}^2)$ such that $|\xi|^2f(\xi)\notin L^2(\mathbb{R}^2)$ (then invert the fourier transform to go back to physical space). Well one class of such functions are as follows: $$\mathcal{A}_s=\{f=f(|\xi|):\mathbb{R}^2\to \mathbb{C}, C^{-1}\langle \xi\rangle^{-3}\le|f(\xi)|\le C \langle \xi\rangle^{-1-\epsilon}\}, 0<\epsilon<2.$$ Indeed, we have that $\langle\xi\rangle|f(\xi)|^2\le C\langle \xi\rangle^{-1-2\epsilon}$ meaning that $f\in L^2(\mathbb{R}^2)$. However, $\langle\xi\rangle |\langle\xi\rangle^2f(\xi)|^2=\langle\xi\rangle^5|f(\xi)|^2\ge C^{-1}\langle\xi\rangle^{-1}$, meaning that $|\xi|^2f\notin L^2(\mathbb{R}^2)$.
For example, let $f(\xi)=\langle\xi\rangle^{-3}$. Then $f\in L^2(\mathbb{R}^2)$, but $|\xi|^2f\notin L^2(\mathbb{R}^2)$. On the physical side, this translate to $e^{-|x|}\in L^2(\mathbb{R}^2)$, but $\Delta e^{-|x|}\notin L^2(\mathbb{R}^2)$