  Shifted variants of (dyadic) Hardy-Littlewood maximal function and Stein's
square function have played a significant role in the study of many important
operators such as Calderon commutators, (bilinear) Hilbert transforms,
multilinear multipliers, and multilinear rough singular integrals. Estimates
for such shifted operators have a certain logarithmic growth in terms of the
shift factor, but the optimality of the logarithmic growth has not yet been
fully resolved. In this article, we provide sharp vector-valued shifted maximal
inequality for generalized Peetre's maximal function, from which improved
estimates for the above shifted operators follow with optimal logarithmic
growths in a new way. We also obtain a vector-valued maximal inequality for the
shifted (dyadic) Hardy-Littlewood maximal operator.
