Using Principal Value distribution to solve DEs

$$\mathcal{P}f=-\!\!\!\!\!\!\int \frac{f(x)}{x}\,\mathrm{d}x$$

Similar to $$\mathcal{L}G=\delta,$$  

$$\boxed{\mathcal{N}J=\mathcal{P}}$$

or

$$\mathcal{N}[J]=\mathcal{P}[J]$$