We consider Hill's equation y'' +($lambda$ -q)y=0 where $qin L^1[0,pi]$. We show that if $l_n$-the length of the n-th instability interval- is of order $O(n^{-k})$ then the real Fourier coefficients $a_n$, $b_n$ of q are of the same order for(k=1,2,3), which in turn implies that $q^{(k-2)}$, the (k-2)th derivative of q, is absolutely continuous almost everywhere for k=2,3.