ℓp-maximal regularity for a class of fractional difference equations on UMD spaces. The case 1 < α < 2.

Abstract

By using Blunck's operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue spaces of sequences for a discrete version of the Cauchy problem with fractional order $1 <\alpha< 2$. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the UMD-class.