Let $L$ be an operator on $C^2(\mathbb R)$, defined by
$$L \phi (x) = \int\_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$
for a measure $\nu(dy) = |y|^{-2} dy$. It is evident that $L\phi$ is well-defined for $\phi\in C^2$ since$$|\phi(x+y) - \phi(x) - \phi'(x) \ y| \le K\_x y^2$$in $B\_1(x)$. Note that the definition of $L$ is explicitly dependent up to the first derivative of $\phi$.


**Question.** Is there a function $\phi \in C^1 \setminus C^2$, such that $L \phi $ is not well-defined?



Domain of Laplacian
$L\phi$ is a convolution with a distribution with compact support, thus it is "well defined" at least as a distribution, for any $\phi\in\cal D'$. A closer look reveals $L$ maps $H^s$ to $H^{s-1}$, $\forall s\in\mathbb R$. Indeed, this operator is a truncated version of the (positive) **square root** of (minus) the Laplacian, that multiplies $\hat\phi(\xi)$ with a constant times $|\xi|$. (It **doesn't** map $C^1$ to $C^0$, however).


