Asymptotic regularity of Daubechies' scaling functions

Let phi(N), N greater than or equal to 1, be Daubechies' scaling function with symbol (1+e(-i xi)/2)(N) Q(N)(xi), and let s(p)(phi(N)), 0 < p less than 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 than 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.