What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact box dimension?

Let 1 < s < 2, and λk > 0 with λk → ∞ satisfy the Hadamard condition λk+1/λk λ > 1. For a class of Besicovich functions , the present paper investigates the intrinsic relationship between box dimension of graphs of their vth fractional integrals g(t) and uth fractional derivatives and the asymptotic behavior of {λk}. We show that: if 0 < v < 1, s > 1 + v, then for sufficiently large λ, holds if and only if ; if 0 < u < 2 − s, then for sufficiently large λ, holds if and only if .