My question is to find function $f\in L^2(0,2)$ such that $f\in D(|A|^{1/2})$ but $f\notin D(A)$ where $(Af)(x)=\ln(x)f(x)$ i.e. $$ \int_0^2 |\ln (x)| |f(x)|^2 dx <\infty = \int_0^2 |ln(x)|^2 |f(x)|^2 dx$$
I tried common functions that we could think of such as $\frac{1}{\ln(x)}$ , also tried to use to fact that symmetry of ln(x) along x=y line which gives $e^y$ $$ \int_0^{-\infty} e^x = \int_0^1 \ln(x)$$ Corresponding function for $\ln(x)^2$ should be (I'm not sure) $e^{-\sqrt{x}}$. Then i tried to do things on these functions then go back to $\ln(x)$ but couldn't manage to do so. Maybe i missed something pretty simple but it took my lot of time because i felt like i should be able to find. As you can see i couldn't...
