Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D ^α u(t) = Au(t)+ f(t), 0 ≤ t ≤ 2π, where A is a closed operator in a Banach space X, α > 0 and D ^α is the fractional derivative in the sense of Weyl. Using known results on L ^p -multipliers, we give necessary and/or sufficient conditions for the L ^p -well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.