We consider Hill's equation y'' +(l -q)y=0 where q\in L1[0,p ]. We show that if ln-the length of the n-th instability interval- is of order O(n-k) then the real Fourier coefficients an,bn 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.

We consider Hill's equation y'' +(l -q)y=0 where q\in L1[0,p ]. We show that if ln-the length of the n-th instability interval- is of order O(n-k) then the real Fourier coefficients an,bn 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.