For a fixed $1 < p < \infty$, we say that a weight $\omega : \mathbb{R}^n → [0, \infty)$ belongs to $A_p$ (class of Muckenhoupt weights) if $\omega$ is locally integrable and there is a constant $C$ such that, for all balls $B$ in $\mathbb{R}^n$, we have
$${\displaystyle \left({\frac {1}{|B|}}\int _{B}\omega (x)\,dx\right)\left({\frac {1}{|B|}}\int _{B}\omega (x)^{-{\frac {q}{p}}}\,dx\right)^{\frac {p}{q}}\leq C<\infty ,}$$ where $|B|$ is the Lebesgue measure of $B$, and $q$ is a real number such that: $1 / p + 1 / q = 1$.
We say $ω : \mathbb{R}^n → [0, \infty)$ belongs to $A_1$ if there exists some $C$ such that
$${\displaystyle {\frac {1}{|B|}}\int _{B}\omega (y)\,dy\leq C\omega (x),}$$ for all $x \in B$ and all balls $B$. I found as examples of functions in $A_p$ just $|x|^{-\alpha}$ for $1< \alpha < \infty$ and $(1-p)n < \alpha <n$. But just this example. I want to know other examples of functions in $A_p$, for example, exponential functions, like $e^{-\alpha|x|^2}$ belongs to $A_p$?
Since $A_p$ is invariant under translations and dilations, from power weights we also get polynomials.
There is a relation between BMO and Muckenhoupt weights: if $f$ is a locally integrable function such that $e^f \in A_2$, then $f$ must be in BMO. The function $x^2$ grows too much to be in BMO, so no $A_2$ membership for Gaussians I am afraid. On the other hand, if $f$ is in BMO, then $e^{\delta f}$ is in $A_2$ for small $\delta$. See for example [Duoandikoetxea, Fourier Analysis, page 151].
Another way to see that Gaussians densities are not in $A_p$ is that Muckenhoupt weights are doubling measures: there exists $C>0$ such that $w(2B) \le Cw(B)$ for all balls $B$. If we consider balls touching the origin but centered far away from it, we see that Gaussian measures cannot be doubling.