Let $(M,g)$ be a Riemannian manifold, $X$ a smooth vector field on $M$, and $f$ a smooth function on $M$. And we define a second-order elliptic operator $L$ by $$L(u)=-\Delta u+\langle X,\nabla u\rangle+fu,$$ where $\Delta$ is the Laplacian.
Suppose we have proved that if $L(u)$ is smooth, then so is $u$. This is a statement sometimes called a regularity theorem, and I was wondering its usage. To be more specific, let's say that $\Delta v\equiv0$ for some $v$. Then, can we conclude that $v$ is a smooth function?
I think the answer is positive, because if we choose $X$ and $f$ to be identically zero, then $L(v)$ will be the zero function, which is certainly a smooth function, making the regularity statement work very well.
Does my reasoning above make sense? Thank you.