When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral

$$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$
where $P$ is a linear partial differential operator and $\phi$ is a compactly support smooth function. In order for $t$ to be a root of the Bernstein-Sato polynomial of $f$ which is larger than $-lct(f)$  it would have to be the case that
\begin{equation}\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz=0\quad \text{for all }\phi\quad (1).\end{equation}
Kollar in [this][1] document says that this seems strange, but cannot be excluded without knowing more about $P$. This makes me wonder, is $(1)$ really possible if $P\not=0$? Or does $P\not=0$ imply that $(1)$ is not possible?
 

  [1]: https://www.uni-due.de/~mat903/sem/ws0708/pairs.pdf