I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$

$$A=\sum a_\alpha\partial^\alpha$$

where 
* $\alpha=(\alpha_1,\cdots,\alpha_n)\in\mathbb{Z}^n$ is a multiindex such that $\partial^\alpha$ is defined to be $\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}$
* $a_\alpha\in\mathbb{R}$ are constants with depending only on the multiindex $\alpha$. 

**My questions** 

1. if $f\in C^{\infty}(B_1)$ satisfies
$$
A f=0\;\text{ in } B_1,
$$
when $f$ is analytic in $B_{1/2}$?<br>
I know that if $A$ is $\Delta$ then the result is true, as it happens also if $A=\Delta+1$. The regularity can even apply to distribution solutions. Is the same result true also for other operators? When can we say that a distributional solution is indeed an analytic function?
2. Similarly, if for a $C^2$ domain $\Omega$ we have
$$
A f=0\;\text{ in } \Omega,
$$
when can we have the $f$ is $C^2$ up to the boundary $\partial\Omega$? Again I know that if $A$ is $\Delta$ then the result is true, as it happens also if $A=\Delta+1$, and again the regularity result holds also for distribution solutions. Is the same result true also for other operators? When can we say that a distributional solution is a $C^2$ function up to the boundary?