Let phi(N), N greater than or equal to 1, be Daubechies' scaling function w
ith symbol (1+e(-i xi)/2)(N) Q(N)(xi), and let s(p)(phi(N)), 0 < p less tha
n or equal to 1, be the corresponding L-p Sobolev exponent. In this paper,
we make a sharp estimation of s(p)(phi(N)), and we prove that there exists
a constant C independent of N such that
N - ln\Q(N)(2 pi/3)\/ln 2 - C/N less than or equal to s(p)(phi(N)) less tha
n or equal to N - ln\Q(N)(2 pi/3)\/ln 2.
This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana,
12(1996), 527-591) positively.