Suppose I have the following equation $$f=Pu+a\cdot u \quad \quad (1)$$ in $\mathscr{D}^\prime(\mathbb T^n)$ where $f$ is a distribution induced by a smooth function $f\in C^\infty(\mathbb T^n)$ and $$\displaystyle P=\sum_{|\alpha|\leq k} c_\alpha D^\alpha$$ is a differential operaor with constant coefficients and $a\in C^\infty(\mathbb T^n)$.
How can I solve $(1)$ it for $u$?
I know this has to do with fundamental solutions but I'm not familiar with it yet.
Thanks.
Obs. A fundamental solution of the operator $P$ is a distribution $E$ which satisfies $$PE=\delta,$$ where $\delta$ is the Dirac distribution.
So for solving my problem for $u$ it suffices finding $E$ and $F$ such that $PE=\delta-F$ where $F$ satisfies $F*f=a\cdot u$. If $E$ and $F$ exists then taking $u=E*f$ we get $$Pu=P(E*f)=PE*f=(\delta-F)*f=(\delta*f)-F*f=f-a\cdot u.$$