Are all the functions whose fourier weight is concentrated on the small sized sets(or terms with low degree) computed by $\mathsf{AC}^0$ circuits ? 
Are all the functions whose fourier weight is concentrated on the small sized sets computed by AC0 circuits?
cc.complexity theory;circuit complexity;boolean functions;fourier analysis
No. Consider the following function on $\{0,1\}^n$:$$ f(x) = x_0 \land \cdots \land x_{n-\sqrt{n}-1} \land (x_{n-\sqrt{n}} \oplus \cdots \oplus x_{n-1}). $$Clearly this function is hard for AC0. On the other hand, the function is almost constant, so almost all of its Fourier spectrum is on the first level.If you want a balanced counterexample, consider$$ g(x) = x_0 \oplus \left[x_1 \land \cdots \land x_{n-\sqrt{n}-1} \land (x_{n-\sqrt{n}} \oplus \cdots \oplus x_{n-1})\right]. $$This function is almost always equal to $x_0$, so almost all of its Fourier spectrum is on the first two levels.