supported by"Maximal Regularity for Vector-valued Boundary Problems"from the National Natural Science Foundation of China(Grant No.10571099);Specialized Research Fund for the Doctoral Program of Higher Education and the Tsinghua Basic Research Foundation(Grant No.JCpy2005056).
We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0, 2π]d; X) using an induction argument based on a known result when d= 1.